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Computation of rotation minimizing frames

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Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present a novel simple and efficient method for accurate and stable computation of RMF of a curve in 3D. This method, called the double reflection method, uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method has the fourth order global approximation error, thus it is much more accurate than the two currently prevailing methods with the second order approximation error—the projection method by Klok and the rotation method by Bloomenthal, while all these methods have nearly the same per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the RMF, though they have the same accuracy. We also investigate further properties and extensions of the double reflection method, and discuss the variational principles in design moving frames with boundary conditions, based on RMF. Categories and Subject Descriptors: G.1.0 (Numerical Analysis): General—Numerical algorithms; G.1.7 (Numerical Analysis): Ordinary Differential Equa- tions—Error analysis
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Computation of Rotation Minimizing Frame
Wenping Wang
University of Hong Kong
Bert J¨uttler
Johannes Kepler University
Dayue Zheng and Yang Liu
University of Hong Kong
Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graph-
ics, including sweep or blending surface modeling, motion design and control in computer ani-
mation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present
a novel simple and efficient method for accurate and stable computation of RMF of a curve in
3D. This method, called the double reflection method, uses two reflections to compute each frame
from its preceding one to yield a sequence of frames to approximate an exact RMF. The double
reflection method has the fourth order global approximation error, thus much more accurate than
the two currently prevailing methods with the second order approximation error — the projection
method by Klok and the rotation method by Bloomenthal, while all these methods have about
the same per-frame computational cost. Furthermore, this method is much simpler and faster
than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the
RMF, though they have the same accuracy. We also investigate further properties and extensions
of the double reflection method, and discuss the variational principles in design moving frames
with boundary conditions, based on RMF.
Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve,
surface, solid, and object representations;J.6[Computer-Aided Engineering]: Computer-aided design (CAD); G.1.6 [Nu-
merical Analysis]: Differential Geometry—approximation
General Terms: rotation minimizing frame, motion design, sweep surface, generalized cylinder,
differential geometry
Additional Key Words and Phrases: curve, motion, rotation minimizing frame
1. INTRODUCTION
1.1 Background
Let x(u)=(x(u),y(u),z(u))Tbe a C1regular curve in E3, the 3D Euclidean space. Denote x(u)=dx(u)/du
and t(u)=x(u)/||x(u)||, which is the unit tangent vector of the curve x(u). We define a moving frame
Authors’ address: Wenping Wang, Dayue Zheng and Yang Liu, Department of Computer Science, The University of Hong
Kong, Pokfulam Road, Hong Kong, China; email [wenping, dyzheng, yliu]@cs.hku.hk; Bert J¨uttler, Johannes Kepler University,
Institute of Applied Geometry, Linz, Austria; email Bert.Juettler@jku.at. The work of the first author was partially supported
by a grant from Hong Kong Research Grant Council.
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2·Wenping Wang et al.
(a) The Frenet frame of a spine curve. Only normal vectors
are shown.
(b) A rotation minimizing frame (RMF) of the same curve
in (a). Only reference vectors are shown.
(c) A snake modeled using the RMF in (b).
Fig. 1. An example of using the RMF in shape modeling.
associated with x(u) to be a right-handed orthonormal system composed of an ordered triple of vectors
U(u)=(r(u),s(u),t(u)) satisfying r(u)×s(u)=t(u) (see Figure 2). The curve x(u) in this context will be
called a spine curve.Sincet(u)isknownands(u)=t(u)×r(u), a moving frame is uniquely determined by
the unit normal vector r(u). Thus ris called the reference vector ofamovingframe.
From the differential geometry point of view, a readily available moving frame of a curve in 3D is the
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Computation of Rotation Minimizing Frame ·3
r
s
t
x(u)
Fig. 2. An orthonormal frame U(u)=(r(u),s(u),t(u)) attached to a spine curve x(u).
Frenet frame, whose three orthogonal axis vectors are defined as
t(u)= x(u)
x(u),s(u)= x(u)×x(u)
x(u)×x(u),r(u)=s(u)×t(u).(1)
Although the Frenet frame can easily be computed, its rotation about the tangent of a general spine curve
often leads to undesirable twist in motion design or sweep surface modeling. Moreover, the Frenet frame
is not continuously defined for a C1spine curve, and even for a C2spine curve the Frenet frame becomes
undefined at an inflection point (i.e., curvature κ= 0), thus causing unacceptable discontinuity when used
for sweep surface modeling [Bloomenthal 1990].
A moving frame that does not rotate about the instantaneous tangent of the curve x(u) is called a
rotation minimizing frame of x(u), or RMF, for short. It can be shown that the RMF is defined contin-
uously for any C1regular spine curve. Because of its minimal-twist property and stable behavior in the
presence of inflection points, the RMF is preferred to the Frenet frame in many applications in computer
graphics, including free-form deformation with curve constraints [Bechmann and Gerber 2003; Peng et al.
1997; Lazarus et al. 1993; Lazarus and Jancene 1994; Lazarus and Verroust 1994; Llamas et al. 2005], sweep
surface modeling [Bloomenthal and Riesenfeld 1991; Pottmann and Wagner 1998; Siltanen and Woodward
1992; Wang and Joe 1997], modeling of generalized cylinders and tree branches [Shani and Ballard 1984;
Bloomenthal 1985; Bronsvoort and Flok 1985; Semwal and Hallauer 1994], visualization of streamlines and
tubes [Banks and Singer 1995; Hanson and Ma 1995; Hanson 1998], simulation of ropes and strings [Barzel
1997], and motion design and control [J¨uttler 1999]. The RMF is also closely related to the problem of
designing stable motion of a moving camera tracking a moving target [Goemans and Overmars 2004], where
the rotation about the vector connecting the camera and the target should be minimized during camera
motion, subject to possible boundary conditions.
Discussions of the RMF and its applications can be found in the recent book by Hanson [Hanson 2005],
where the RMF is treated using a parallel transport approach.
A typical application of RMF in shape modeling is shown in Figure 1. Here a canonical snake surface
model is first defined along a straight line axis possessing an RMF generated by translation along the line.
Then a new axis curve (i.e., a spine curve) is designed to produce a novel pose of the snake. For comparison,
both Frenet frame and RMF of this same axis curve are shown in Figures 1(a) and 1(b). The RMF determines
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4·Wenping Wang et al.
Fig. 3. Sweep surfaces showing moving frames of a deforming curve: the Frenet frames in the first row and the RMF in the
second row.
a mapping from the space of the canonical model of the snake to the space around the new axis curve in
Figure 1(b); this mapping produces the snake in Figure 1(c). Note that the Frenet frame in this case exhibits
excessive rotation compared with the RMF, so it is less appropriate for shape modeling.
Next consider moving frames of a deforming spine curve x(u;t), as frequently encountered in computer
animation (see Figure 3). While the Frenet frame does not always experience abrupt twist for a given static
spine curve, the Frenet frame of the deforming spine curve often suddenly exhibits a radical twist at an
instant during deformation, especially when the spine curve has a nearly curvature vanishing point (i.e., an
inflection point). In contrast, the RMF of the deforming spine curve x(u;t) always varies smoothly and
stably over time as well as along the spine curve. The different behaviors of these two moving frames are
illustrated in Figure 3, visualized as sweep surfaces, through a sequence of snapshots of a deforming spine
curve. Here by continuous deformation we mean that the rate of change in both position (i.e., x(u;t)/∂t)
and unit tangent (i.e., t(u;t)/∂t) are bounded for any (u, t) in their finite intervals of definition. Note that,
this assumption is reasonable in practical application but does not imply that the normal vector r(u)of
x(u;t) (see Eqn. (1)) changes continuously with respect to time t, thus explaining the potential instability
of the Frenet frame.
Computation of the RMF is more difficult than that of the Frenet frame. The RMF is first proposed
and formulated as the solution of an ordinary differential equation in [Bishop 1975] and later in [Shani
and Ballard 1984; Klok 1986]. Exact (i.e., closed form) RMF computation is either impossible or very
involved for a general spine curve. Hence, a number of approximation methods have been proposed for
RMF computation. These methods fall under three categories: 1) discrete approximation;2)spine curve
approximation;and3)numerical integration. The discrete approximation approach is versatile for various
applications in computer graphics and computer animation, even when only a sequence of points on a path
(i.e., spine curve) is available, while the approach based on spine curve approximation is useful for surface
modeling in CAGD applications. We will see that direct numerical integration of the defining ODE of RMF
is relatively inefficient and therefore not well suited for RMF computation. The new method we are going
to propose is based on discrete approximation.
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Computation of Rotation Minimizing Frame ·5
1.2 Problem formulation
The RMF computation problem as solved by the discrete approximation approach is formulated as follows.
Let U(u) denote an exact RMF of a C1regular spine curve x(u)in3D,u[0,L], with the initial condition
U(0) = U0, which is some fixed orthonormal frame at the initial point x(0). Suppose that a sequence of
points xi=x(ui) and the unit tangent vectors tiat xiare sampled on the curve x(u), with ui=ih,
i=0,1,...,n,whereh=L/n is called the step size. The goal of discrete approximation is to compute
a sequence of orthonormal frames Uiat xithat approximates the exact RMF frame U(u)atthesampled
points, i.e., each Uiis an approximation to U(ui), i=0,1,2...,n.
Error measurement is needed to evaluate and compare different approximation schemes. Suppose that the
exact RMF U(u) has the same initial frame as the approximating frame sequence at x(u0), i.e., U(0) = U0.
Then the approximation error between U1and U(h) is called the one-step error. The approximation errors
at intermediate sampled points are normally accumulated to give a large error at the end of the spine curve.
However, due to error fluctuation, the maximum error may not always occur at the endpoint x(L). Therefore,
we define the global error Egto be the maximum error of frame approximation over all the sampled points
x(ui), i.e.,
Eg=n
max
i=0 {|(Ui,U(ui))|},(2)
where |(Ui,U(ui))|measures the magnitude of the angle between the reference vectors riand r(ui)of
frames Uiand U(ui).
We shall present a new discrete approximation method, called double reflection method, for RMF com-
putation. The main idea is based on the observation that the rigid transformation between two consecutive
frames for RMF approximation can be realized by two reflections, each being a reflection in a plane. The
resulting method is simple, fast, and highly accurate – its global approximation error is of order O(h4),
where h=L/n is the step size. This compares favorably with the second order (i.e., O(h2)) approximation
error of two prevailing discrete approximation methods, i.e., the rotation method [Bloomenthal 1990] and
the projection method [Klok 1986]. The accuracy of the double reflection method matches that of using
the standard 4-th order Runge-Kutta method to integrate the defining differential equation of RMF, but is
much simpler and faster than the latter.
In the following we shall first review related works in Section 2 and present necessary preliminaries in
3. The double reflection method is presented and analyzed in Section 4. Then we present experimental
verifications in Section 5, discuss extensions in Section 6 and conclude the paper in Section 7.
Readers interested only in implementation may skip to Section 4.1 for a simple description of the double
reflection method; the pseudo code is given in Table I in Section 4.
2. RELATED WORK
2.1 Discrete approximation
In discrete approximation an RMF is approximated by a sequence of orthogonal frames located at sampled
points xion the spine curve x(u). The projection method, as originally proposed in [Klok 1986], computes
an approximate RMF for modeling a sweep surface. Suppose that the the sampled points xiand the unit
tangent vectors tiof x(u)atthesampledpointsxiare provided as input. For RMF computation, the
projection method projects, along the direction x1x0, an initial reference vector r0in the normal plane of
the spine curve at x0to the next reference vector r1on the normal plane at x1. Then this step is repeated to
generate on the subsequent normal planes a sequence of reference vectors ri, which, together with the tangent
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vectors ti, define a sequence of orthonormal frames that approximate an exact RMF. The projection method
is empirically demonstrated to have the second order of approximation error [Chung and Wang 1996]. Note
that the above projection between normal planes is not length preserving. Therefore the reference vectors
rineed to be normalized to give unit vectors.
Another popular discrete approximation method is the rotation method [Bloomenthal 1990; Siltanen and
Woodward 1992; Poston et al. 1995]. The rotation method also needs as input the sampled points xion the
spine curve and the unit tangent vectors tiof the spine curve at xi. Consider the first two sampled points
x0and x1. Given the initial frame U0at x0, suppose that we need to compute the next frame U1at x1
from the boundary data (x0,t0;x1,t1). To minimize the rotation about the tangent of the spine curve, this
method rotates U0into U1about an axis b0perpendicular to t0and t1,thatis,b0=t0×t1; the rotation
angle θis such that the frame vector t0of U0is brought into alignment with the frame vector t1of U1, i.e.,
θ= arccos(t0·t1). Here, for frame computation, we ignore the translational difference between the origins
of U0and U1. The rotation method has the second order global approximation error [Poston et al. 1995].
A major problem with the rotation method is its lack of robustness for nearly collinear data. When
the two consecutive tangent vectors t0and t1are collinear, the rotation axis becomes undefined, since
b0=t0×t1= 0; but, since no rotation is needed in this case, we just need to set U1:= U0. However,
numerical problems will be experienced when t0and t1approach each other, i.e., becoming closer and closer
to being collinear; this happens, for example, when the spine curve is densely sampled for high accuracy
RMF computation. In this case some threshold value has to be used to avoid the degeneracy of the rotation
vector b0by treating nearly collinear data as collinear data. But if a spine curve is so densely sampled
that all consecutive data segments are deemed as collinear due to thresholding, then there will be a large
accumulated error in the computed RMF, because the spine curve will be treated as a straight line and all
the frames Uiwill be set to be identical to the initial frame U0. We note that this numerical problem for
nearly collinear data does not exist with the double reflection method we are going to propose.
2.2 Methods based on spine curve approximation
If a spine curve is first approximated by some simple curves whose RMF can be computed exactly or more
accurately, then the RMF of this simple approximating curve can be taken as an approximation to the
RMF of the original spine curve. An intuitive argument for this idea is that if two spine curves are close
to each other, then their RMFs should also be. This type of intuition lacks rigorous justification and could
be unreliable for moving frames defined by differential properties; recall that the Frenet frames of two spine
curves close to each other can be radically different. However, for RMF it is proved by Poston et al [Poston
et al. 1995] that the RMF of a spine curve ˜
x(u) approaches the RMF of another spine x(u) if and only if
the unit tangent vector ˜
t(u)of˜
x(u) approaches the unit tangent vector t(u)ofx(u).
Discrete approximation methods, such as the projection method or the rotation method, can be regarded
as the simplest methods based on spine curve approximation, using a polygon to approximate the spine
curve. A G1spline curve composed of circular arcs is used to approximate an input spine curve in [Wang
and Joe 1997] to compute an approximate RMF for modeling sweep surfaces in NURBS form. The spine
curve is approximated by PH curves using Hermite interpolation in [J¨uttler and M¨aurer 1999] for generating
sweep surfaces in rational representation. Exact description of the RMF of a PH curve and its rational
approximation are provided in [J¨uttler 1999; Farouki 2002; Farouki and Han 2003; Choi et al. 2004]. A
closely related technique is to approximate the rotation minimizing motions (RMM) by affine motions (cf.
[Pottmann and Wagner 1998]) and rational motions from the point of view of spherical kinematics [J¨uttler
1998].
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Computation of Rotation Minimizing Frame ·7
2.3 Numerical integration
Since the RMF is defined by a vector-valued ODE of the type y=f(x,y) [Bishop 1975; Shani and Ballard
1984; Klok 1986; Pottmann and Wagner 1998], naturally one may consider computing the RMF using a
numerical method to directly solve this ODE. Suppose that the classical fourth order Runge-Kutta method
is used. Then the RMF thus computed has the 4-th order global approximation error, which is the same as
that of the double reflection method that we are to propose. However, this general approach to solving the
ODE does not take into account the special geometric property of the problem of RMF computation and
therefore has severe drawbacks.
Firstly, the Runge-Kutta method requires the spine curve x(u)tobeC2, since the right hand side f
of the ODE is a function of the second derivative of x(u) (cf. Eqn. (6) in Section 3). This requirement is
unnecessarily restrictive, since the RMF is continuously defined for any C1spine curve. Secondly, deriving
and evaluating the second derivative of x(u) can be tedious and costly, rendering the method inefficient. In
the RMF computation problem under consideration, only the sampled points xiand the tangent vectors ti
are available as input. But both first and second derivatives of the spine x(u) are required by the Runge-
Kutta method. This mismatch between the input data of the RMF computation problem and the data it
requires makes the Runge-Kutta method not well suited for RMF computation.
Another problem is that the Runge-Kutta method does not strictly enforce the orthogonality between
the solved reference vectors riand the tangent vectors ti, even though in the initial conditions r0=r(0) is
orthogonal to t0=t(0). Therefore each rihas to be projected onto the normal plane of the spine curve to
make it perpendicular to ti; this adds further to the cost of the method.
Another method is based on the observation that the RMF and the Frenet frame differ by a rotation
determined by the torsion in the normal plane of the spine curve. Let θ(u) be the angle of this rotation. Let
τ(u) be the torsion of the spine curve x(u). Then θ(u) is given by [Guggenheimer 1989]
θ(u)=u
u0
τ(v)x(v)dv. (3)
With this formula, θ(u) may be computed with some quadrature rule and used to compute the RMF by
compensating the rotation of the Frenet frame. However, at inflection points of a spine curve, the Frenet
frame itself becomes discontinuous and exhibits abrupt change, and the torsion τ(u) becomes ill-defined (i.e.,
unbounded), making it difficult to evaluate the integration (3) accurately; therefore in this case the method
becomes unstable.
3. PRELIMINARIES
3.1 Definition by differential equations
First we introduce the rotation minimizing frame under weak assumptions on a spine curve, using differential
equations. These results will later be connected to the classical results from differential geometry. Generally,
we assume the spine curve x(u)tobeaC1regular curve, i.e., x(u)= 0 in its domain of definition, but
higher differentiability is needed for analysis of approximation orders. Again we use t(u)=x(u)/||x(u)||
to denote the unit tangent vector.
Consider a one-parameter family of unit vectors f(u) perpendicular to the tangent vector t(u). Such
a vector function f(u) is said to exhibit the minimal rotation, and therefore called a rotation minimizing
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vector, if it is a solution to the following system of differential–algebraic equations (DAE)
f(u)φ(u)t(u)=0
f(u)·t(u)=0
(4)
for the functions f(u)=(f1(u),f
2(u),f
3(u))and some function φ(u). Here the first equation (in vector
form) constrains the evolution of f(u) to be parallel to the tangent, and the second equation serves to preserve
orthogonality.
A rotation minimizing vector f(u) is not necessarily differentiable for a C1spine curve x(u); (e.g., consider
the case of a C1curve composed of a circular arc and a straight line segment). In view of this, one may
adopt the following weak form of the DAE (4)
f(u)u
0
φ(v)t(v)dv=0
f(u)·t(u)=0
(5)
which does not involve any derivative of f(u).
If the spine curve is of the C2class, then the above DAE is equivalent to the ODE
f(u)=[t(u)×t(u)] ×f(u)(6)
since
φt=(f·t)t=(f·t)t=[t(u)×t(u)] ×f(u)(7)
A rotation minimizing frame (RMF) is determined by a rotation minimizing vector. Specifically, we have
Definition 1: [Rotation minimizing frame] Given a C1curve x(u)E3,u[0,L], a moving orthonormal
frame U(u)=(r(u),s(u),t(u)),wherer(u)×s(u)=t(u),iscalledarotation minimizing frame (RMF) of
x(u)if t(u)=x(u)/||x(u)|| and r(u)is a solution of Eqn. (5) (or Eqn.(4) if x(u)is C2) for some initial
condition U(0) = U0.Herer(u)is called the reference vector of the RMF U(u).
Since the frame vector t(u)ofU(u) is always constrained to be the unit tangent vector of x(u), U(u)
is uniquely determined by its reference vector r(u), which is a rotation minimizing vector. The third frame
vector is given by s(u)=t(u)×r(u).
The evolution defined by DAE (4) preserves the inner product of two vectors. Indeed, if vectors f(u)
and g(u) both satisfy Eqn.(4) with associated functions φ(u)andψ(u), then
d
dt(f·g)=f·g+f·g=(φt)·g+f·(ψt)=0 (8)
Hence, the inner product (f·g) is a constant. From this we have the following observations:
Corollary 3.1. If two vectors f1(u)and f2(u)satisfy Eqn. (4) and the three vectors f1(0),f2(0) and
t(0) form a right–handed orthonormal frame, then f1(u),f2(u)and t(u)define an RMF of the spine curve
x(u).
Corollary 3.2. Suppose that r(u)is a rotation minimizing vector of a spine curve x(u).Thenanother
normal vector ˜
r(u)of x(u)is a rotation minimizing vector of x(u)if and only if ˜
r(u)keeps a constant angle
with r(u).
Or, equivalently,
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Computation of Rotation Minimizing Frame ·9
Corollary 3.3. Suppose that U(u)=(r(u),s(u),t(u)) is also an RMF of a spine curve x(u).Then
another right-handed orthonormal moving frame ˜
U(u)=(
˜
r(u),˜
s(u),t(u)) of x(u)is an RMF of x(u)if and
only if ˜
U(u)keeps a constant angle with U(u).
Finally, we note that the RMF is determined only by the geometry of a spine curve and independent of
any particular parameterization x(u) of the curve.
3.2 Some differential geometry
In this subsection we shall use the arc-length parameterization x(s) of the spine curve. Using the Frenet
formulas one may express (6) as
f(s)=κ(s)b(s)×f(s),(9)
where κ(s)andb(s) are the curvature and the binormal vector of x(s). The vector
ωRMF(s)=κ(s)b(s) (10)
is the angular velocity of the RMF.
The angular velocity of the Frenet frame is the so–called Darboux vector [Kreyszig 1991]
ωFrenet(s)=κ(s)b(s)+τ(s)t(s) (11)
This shows that, compared to the RMF, the Frenet frame involves an additional rotation around the tangent,
whose speed equals the torsion τ. This observation explains the integral formula (3) for computing the RMF
by correcting the “unwanted” rotation of the Frenet frame. The Frenet frame coincides with the RMF for
planar curves, for which τ0.
The RMF is also closely related to developable surfaces and principal curvature lines of a surface.
Suppose that U(u)=(r(u),s(u),t(u)) is an RMF of a curve x(u). Then the surface D(u, v)=x(u)+vr(u)
is developable. Let g(u) be the edge of regression of the developable surface D(u, v). Then the spine curve
is an involute of the curve g(u). This observation suggests a natural (but restrictive) way of modeling a
developable ribbon surface along a spine curve using the RMF.
Suppose that x(u) is a principal curvature line of a surface S. Then the consistent unit normal vector
of Salong the curve x(u) is a rotation minimizing vector of x(u), thus determining an RMF of x(u). This
follows from the well known fact that the normals of Salong x(u) form developable surface if and only if
x(u) is a principal curvature line of S. It therefore also follows that the spine curve x(u) is a principal
curvature line of the developable D(u, v) defined in the last paragraph.
Another important property of the RMF is that it is preserved by conformal transformation of E3,which
is a fact that we will formally prove in a forthcoming paper. This means that, given a spine curve x(u)E3
and a conformal mapping Cof E3,theRMFofx(u)ismappedbyCto the RMF of the transformed spine
curve C(x(u)). In other words, the operation of computing RMF of a curve and a conformal transformation
commute. This property will be needed later in the analysis of the approximation order of our new method
for computing the RMF.
Note that the group of conformal mappings in 3D is exactly the group generated by translations, rota-
tions, uniform scalings and sphere inversions (reflections with respect to spheres). Since a straight line is
mapped to a circle by a sphere inversion, in the above the transform of a unit vector vin RMF is understood
to be the unit tangent vector of the circle which is the image of the straight line associated with v.
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4. DOUBLE REFLECTION METHOD
In this section we shall first give an outline of the double reflection method, and, through a study of the
RMF of a spherical curve, explain why the method works well. Then we shall give a procedural description
of the method that has an optimized number of arithmetic operations, and finally present an analysis of the
approximation order of the method. The double reflection method is straightforward and can very easily be
described; however, its justification takes interesting geometric arguments that do not appear to be trivial.
4.1 Outline of method
Given boundary data (x0,t0;x1,t1) and an initial right-handed orthonormal frame U0=(r0,s0,t0)atx0,
the next frame U1=(r1,s1,t1)atx1for RMF approximation is computed by the double reflection method
in the following two steps.
Step 1 :LetR1denote the reflection in the bisecting plane of the points x0and x1(see Figure 4). Use R1
to map U0to a left-handed orthonormal frame UL
0=(rL
0,sL
0,tL
0).
Step 2 :LetR2denote the reflection in the bisecting plane of the points x1+tL
0and x1+t1(see Figure 5).
Use R2to map UL
0to a right-handed orthonormal frame U1=(r1,s1,t1). Output U1.
An efficient implementation of the above steps is given by the pseudo code in Table I.
r0
t0
x0
rL
0
tL
0x1
R1
Fig. 4. The first reflection R1of the double reflection
method.
r0
t0
x0
rL
0
tL
0x1
r1
t1
R2
Fig. 5. The second reflection R2of the double reflection
method.
4.2 Geometric interpretation
The reasons why the double reflection method described above computes an accurate approximation of an
RMF are the following: 1) the double reflection method is designed to produce an exact RMF when the
spine curve is a spherical curve; and 2) any spine curve x(u) with boundary data (x0,t0;x1,t1)canwell
be approximated by a spherical curve ˆ
x(u) interpolating the same boundary data. Therefore the double
reflection method should compute an accurate approximation of the exact RMF of an arbitrary spine curve.
Furthermore, we note that any two consecutive frames in a sequence of approximate RMF are related to
each other by a rigid motion. Since, as well known, a rigid motion can be realized by the composition of two
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Computation of Rotation Minimizing Frame ·11
reflections in a plane, we seek these two simple reflections in our implementation to realize the desired rigid
motion. This explains the efficiency of the double reflection method.
In the remaining of this section we shall provide a geometric argument about the intuition and mechanism
behind the double reflection method and discuss its properties. First consider the RMF of a spherical curve.
The next lemma indicates that there is a simple explicit characterization of the RMF of a spherical curve.
(We will treat a planar curve as a special case of a spherical curve where the radius is infinite.)
Lemma 4.1. Let x(u),u[0,h],beacurvesegmentlyingonasphereSor a plane P(see Figure 6).
Let n(u)be the outward unit normal vector of the sphere Salong the curve x(u)or a unit (constant) normal
vector of the plane P.ThenanRMFofx(u)is given by ¯
U1=(
¯
r,¯
s,t1),where
¯
r(u)=n(u)and¯
s(u)=t(u)×n(u).(12)
Proof. Suppose that x(u) is on a sphere. Without loss of generality, suppose that the sphere Sis centered
at the origin and has radius r. It is clear that r(u)=n(u), s(u)=t(u)×n(u)andt(u) form a right-handed
orthonormal moving frame. Since n(u)=1
rx(u), r=n=1
rx, which is parallel to t(u). Therefore, r
satisfies Eqn. (4), i.e., it is a rotational minimizing vector. Hence, by Definition 1, U(u)=(r,s,t)isan
RMF of x(u).
The proof is similar when x(u) is a plane curve. .
Lemma 4.1 suggests that, given the initial frame U0at x0,theRMFU1of a spherical curve x(u)at
the point x1does not depend on the in-between shape of x(u), but depends only on the boundary data
(x0,t0;x1,t1). This will be referred to as the path independence property, as stated below.
Lemma 4.2. [Path independence property] 1Let x(u)and y(v)be two curve segments, u[0,h
1]and
v[0,h
2], on a sphere (or a plane) sharing the same boundary data (x0,t0;x1,t1).LetU(u)and V(v)
denote the RMFs of x(u)and y(v), having the same initial frame U0, i.e., U(0) = V(0) = U0.Then
U(h1)=V(h2).
Proof. We will only consider the case of x(u)andy(u) being on a sphere S; the case of their being
on a plane can be proved in a similar way. First suppose that the initial frame U0is the special frame
¯
U0=(
¯
r0,¯
s0,t0)where¯
r0is the unit outward normal vector of the sphere Sat x0and ¯
s0=t0ׯ
r0. Then,
by Lemma 4.1, the RMFs ¯
U1and ¯
V1of x(u)andy(u)atx1are the same, i.e., ¯
U1=¯
V1=(
¯
r1,¯
s1,t1), where
¯
r1is the unit outward normal vector n1of the sphere Sat x1and ¯
s1=t1ׯ
r1.
Now suppose that the initial frame U0=(r0,s0,t0) is arbitrary. Let α0be the angle between U0and
¯
U0. Then, by Corollary 3.3, U(h1)and ¯
U1,astwoRMFsofx(u) at the endpoint x1, keep the same angle
α0. Similarly, the angle between the V(h2)and ¯
V1,astwoRMFsofy(v) at the endpoint x1,isalsoα0.It
follows that U(h1)=V(h2), since ¯
U1=¯
V1..
Next we show that the double reflection method yields the exact RMF for a spherical curve.
Theorem 4.3. Let x(u)be a curve segment, u[0,h], on a sphere or a plane with boundary data
(x0,t0;x1,t1).LetU(u)be an RMF of x(u).LetU0=U(0) and U1=U(h). Then, given boundary data
(x0,t0;x1,t1)and the initial frame U0, the double reflection method produces the frame U1.
Proof. Again we will only consider the case of the curve x(u) being on a sphere S; the case of a plane
can be proved similarly. First consider the special case of U0=¯
U0=(
¯
r0,¯
s0,t0), as defined in the proof of
1This property is equivalent to the fact that the integral b
aτ(s)dsvanishes for closed spherical curves [Kreyszig 1991].
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
12 ·Wenping Wang et al.
r
s
t
Fig. 6. An RMF of a spherical curve.
x0
x(u)
x1
ˆ
x(u)
Fig. 7. Spherical pro jection of a curve segment.
Lemma 4.2. Then, by Lemma 4.1, U1=¯
U1=(
¯
r1,¯
s1,t1). Here, ¯
r0and ¯
r1are unit outward normal vectors
of the sphere Sat x0and x1, respectively. Recall that in the double reflection method (cf. Section 4.1) the
first reflection R1is in the bisecting plane (denoted as H1)ofx0and x1,andR1maps ¯
U0to a left-handed
frame ¯
UL
0=(
¯
rL
0,¯
sL
0,tL
0). Because the two normals ¯
r0and ¯
r1of Sat x0and x1are symmetric about the
plane H1,wehave¯
rL
0=¯
r1.
Let H2denote the bisecting plane of the two points x1+tL
0and x1+t1. Clearly, ¯
rL
0(or ¯
r1)iscontained
in H2. Since the second reflection R2of the double reflection method is in the plane H2, it preserves ¯
rL
0=¯
r1.
Furthermore, by its construction, R2maps tL
0to t1. Therefore, R2maps ¯
UL
0to ¯
U1=(
¯
r1,¯
s1,t1). Hence,
the theorem holds in the special case of U0=¯
U0.
Now consider an arbitrary initial frame U0=(r0,s0,t0). Let α0denote the angle between U0and ¯
U0.
Let Rdenote the composition of R1and R2, i.e., the total rotation effected by the double reflection method.
Clearly, Rmaps U0to a right-handed orthonormal frame ˆ
U1=(
ˆ
r1,ˆ
s1,ˆ
t1) such that ˆ
t1=t1. Therefore,
ˆ
U1and ¯
U1differ by a rotation in the normal plane of x(u)atx1. Furthermore, since the rotation Ris
angle-preserving, the angle between ˆ
U1and ¯
U1is also α0,sinceRmaps ¯
U0to ¯
U1,andU0to ˆ
U1.Onthe
other hand, by Corollary 3.3, the angle between U1=U(h)and ¯
U1is also α0. It follows that ˆ
U1=U1, i.e.,
the exact RMF U1of the curve x(u)atx1is generated by the double reflection method. .
Not only the RMF of a spherical or plane curve x(u) is computed exactly by the double reflection method,
but also this computation does not make use of the sphere or the plane containing x(u). That is possible
because of the path independence property of the RMF of a spherical curve (cf. Lemma 4.2). Note that
when the curve segment x(u)isC1regular and parameterizes a line segment, since x(u) is a plane curve,
its RMF is computed exactly by the double reflection method, with no need of threshold as in the rotation
method to avoid numerical instability (see Section 2.1).
Now consider applying the double reflection method to computing the RMF of a general spine curve
x(u)E3,u[0,h], which has boundary data (x0,t0;x1,t1) and is not necessarily spherical or planar. In
general, there is a unique sphere Ssuch that x0and x1are on Sand t0and t1are tangent to Sat x0and x1.
Let ˆ
x(u) denote the projection of the curve x(u) onto the sphere Sthrough the center of S(see Figure 7).
Then it is easy to see that the curve ˆ
x(u) shares the same boundary data (x0,t0;x1,t1)withx(u), so it
follows from the basic results of Hermite curve interpolation that the approximation error x(u)ˆ
x(u)
between x(u) and its spherical projection ˆ
x(u) is of the order O(h4). Since x(u) is well approximated by
ˆ
x(u) and the double reflection method computes an exact RMF of the spherical curve ˆ
x(u), it is reasonable
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Computation of Rotation Minimizing Frame ·13
to expect that the double reflection method computes an accurate approximation to the RMF of the original
spine curve x(u).
Note that the above argument does not constitute a formal analysis of the approximation accuracy of the
double reflection method; it merely provides a geometric and intuitive understanding of why the method is
expected to work well for RMF computation. It will be proved in Section 4.7 that the global approximation
error of the double reflection method has the order O(h4).
4.3 Procedural description
The description of the double reflection method in Section 4.1, though simple in geometric terms, is not
for efficient implementation. In this section we will give a procedural description of the method, aiming at
minimizing the number of arithmetic operations required.
Since only transformation of vectors matters in RMF computation, we may just use the linear parts,
denoted by matrices R1and R2, of the two reflections R1and R2.SinceR1is a reflection in a plane with
normal vector v1x1x0,itcanbeshownthatitslinearpartis
R1=I2(v1vT
1)/(vT
1v1),(13)
where Iis the 3 ×3 identity matrix. We will call v1the reflection vector of R1. (Note that R1is none other
than the Householder transform used for QR matrix decomposition.)
The reflection R2has the reflection vector v2(x1+t1)(x1+tL
0)=t1tL
0,wheretL
0=R1t0.So
its linear part is
R2=I2(v2vT
2)/(vT
2v2).(14)
Let r0be the reference vector of U0.Thenr1=R2R1r0is the reference vector r1of the next frame U1.
With the known tangent vector t1, the remaining vector s1of U1=(r1,s1,t1)isgivenbys1=t1×r1.
The procedure of the double reflection method is given in Table I. For a given sequence of sampled points
xiand associated unit tangent vectors ti,withaninitialframeU0defined at x0, one just needs to apply the
two reflections R1and R2to successively generate the approximate RMF Uiat xi. Ineachstep,fromthe
current frame Ui, we form the first reflection R1following Eqn.( 13) and use R1to map the reference vector
rito rL
i, and also the tangent vector tito tL
i.ThenweusetL
iand ti+1 to form the second reflection R2
following Eqn. (14) and use R2to map rL
ito the reference vector ri+1 of the next frame Ui+1.
4.4 Degeneracy, stability and symmetry
By degeneracy we mean that either of the reflections R1and R2becomes undefined. Clearly, R1is undefined
if and only if x1x0=0,andR2is undefined if and only if x1+tL
0=x1+t1, i.e., the two points x0+t0and
x1+t1are symmetric about the bisecting plane of x0and x1;thisisequivalentto(x1x0)·(t1+t0)=0
and (x1x0)×(t1t0) = 0. Hence, for proper application of the double reflection method, we need to
ensure that the following two conditions are satisfied: (1) x1x0=0;and(2)(x1x0)·(t1+t0)=0or
(x1x0)×(t1t0)= 0. Both conditions are simple to test and can easily be satisfied provided that the
spine curve is sufficiently subdivided or sampled.
It has been commented earlier (cf. Section 2.1) that the rotation method suffers from numerically
instability when the vectors t0and t1are collinear or nearly so. Now we examine the stability of the double
reflection method for the same kind of local data (x0,t0;x1,t1) , i.e., when v1=x1x0,t0and t1are
collinear or nearly so. Two reflections are used in the double reflection method. The first reflection R1uses
v1=x1x0as the reflection vector, thus R1is numerically stable because v1canbeassumedtobea
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
14 ·Wenping Wang et al.
Tab l e I . Algorithm — Double Reflection
Input: Points xiand associated unit tangent vectors ti,i=0,1,...,n.
An initial frame U0=(r0,s0,t0).
Output: Ui=(ri,si,ti), i=0,1,2,...,n, as approximate RMF.
Begin
for i=0 to n1do
Begin
1) v1:= xi+1 xi; /*compute reflection vector of R1.*/
2) c1:= v1·v1;
3) rL
i:= ri(2/c1)(v1·ri)v1; /*compute rL
i=R1ri.*/
4) tL
i:= ti(2/c1)(v1·ti)v1; /*compute tL
i=R1ti.*/
5) v2:= ti+1 tL
i; /*compute reflection vector of R2.*/
6) c2:= v2·v2;
7) ri+1 := rL
i(2/c2)(v2·rL
i)v2; /*compute ri+1 =R2rL
i.*/
8) si+1 := ti+1 ×ri+1; /*compute vector si+1 of Ui+1.*/
9) Ui+1 := (ri+1,si+1,ti+1 );
End
End
nonzero vector, and this stability has nothing to do with whether v1is collinear with t0(or t1)ornot. The
second reflection R2uses the reflection vector v2=t1tL
0,wheretL
0is the image of t0under R1. Hence,
when t0and t1become collinear or nearly so, v2 would approach to t1+t02t1,sincetL
0approaches to
t1in this case. Hence, v2 well be defined, which ensure the numerical stability of the second reflection R2.
Clearly, the above argument for the stability of the double refection method applies also when the spine
curve is specified only by a sequence of sample points {xi}n
i=0 (see Section 6.1. In this case the tangent
vectors tihave to be estimated from the points xi, and nearly collinear data would results if the points xi
are densely sampled. As a consequence of its stability in the presence of nearly collinear data, the double
reflection method is free of the threshold problem which plagued the rotation method (cf. Section 2.1).
Hence, the double reflection method produces the RMF exactly (or accurately) in a numerically stable
manner even for a sequence of points on a spine curve which is a straight line (or nearly a straight line),
using the same unified procedure, i.e., free of threshold testing.
Finally, we note that the double reflection method is symmetric in the following sense. Given a sequence
of sampled points xi,i=0,1,...,n,onaspinecurvex(u), suppose that the Uiare the frames computed by
the double reflection method applied to x(u)withU0as the initial frame. Then the same sequence of frames
in the reversed order, i.e., Uni,i=0,1,...,n, will be generated by applying the double reflection method
starting from xn,usingUnas the initial frame. This symmetry property can be proved by examining the
basic steps of the double reflection method, but we will skip the proof. The projection method and the
rotation method also possess this symmetry property, while the Runge–Kutta method does not.
4.5 Axis-angle representation
It is instructive to derive the axis-angle representation of the rigid motion Rthat relates any two consecutive
orthonormal frames produced by the double reflection method. Using the notation in 4.3, the two reflection
vectors used in the double reflection method are v1=x1x0and v2=t1tL
0. Denote their normalized
vectors by ¯
v1=v1/v1and ¯
v2=v2/v2. Based on elementary geometric argument, it is easy to see that
the rotation axis vector of Ris v=¯
v1ׯ
v2and the rotation angle is α=2arcsin(¯
v).
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Computation of Rotation Minimizing Frame ·15
Substituting in the expression tL
0=R1t0,whereR1is given by Eqn. (13), it is straightforward to derive
the axis vector
v=γ(x1x0)×(t1t0),
where γ1=x1x02t1x12+[(x1x0)Tt0][(x1x0)Tt1]. Clearly, vapproaches to zero if the vectors
x1x0,t0and t1become nearly collinear. Therefore, regardless of its efficiency, if this axis-angle reflection
is used to compute the RMF for nearly collinear data, it will experience numerical instability and thus need
threshold testing, as in the case of the rotation method (see Section 2.1). In contrast, the double reflection
method avoids this instability elegantly by computing the same rigid motion using two reflections in a plane.
Hence, we conclude that the stability issue in the presence of collinear data is not inherent to the problem
of RMF computation; rather, it is due to a particular algorithm for solving the problem.
4.6 Invariance under conformal mappings
We have seen that conformal mappings in 3D preserve the RMF of a space curve (cf. Section 3.2). It turns
out that the approximate RMF computed with the double reflection method is also preserved by conformal
mappings, in the following sense. Suppose that the sampled points xiof a spine curve x(u)areusedto
compute the approximate RMF Uiof x(u). Then the images of Uiunder a conformal mapping Care the
same as the approximate RMF of the curve C(x(u)) that are computed by the double reflection method
using the sampled points C(xi).
This property follows easily from the fact that the basic step of the double reflection method is performed
on the sphere Sitouching the two ends of the data (xi,ti;xi+1,ti+1 ) and this sphere is preserved by any
conformal mapping C(which is the composition of a sequence of sphere inversions), i.e., the image C(Si)is
the sphere touching the transformed data (C(xi),C(ti); C(xi+1),C(ti+1)).
Since both exact RMF and approximate RMF computed with the double reflection method are preserved
by conformal mappings, and the conformal mapping is angle preserving, we conclude that the approximation
error of the double reflection method is invariant under conformal mappings.
The double reflection method is an ideal method from the viewpoint of discrete differential geometry.
Because the exact RMF of a smooth curve is preserved by conformal mappings, we naturally expect that a
good method acting on a discretization of the curve for computing its approximate RMF is invariant under
the same group of transformations. The double reflection method indeed satisfies this property. We note
that the projection method, the rotation method and the Runge–Kutta method do not possess this property.
4.7 Order of approximation
First consider an analytic curve segment with the arc length parameterization x(s), s[0,h], of length h.
Suppose that the initial frame U(0) = U0(r0,s0,y0)ofanRMFU(s)ofx(s) is given. We approximate
the frame U(h)atx1=x(h)bytheframeU1computed with the the double reflection method.
Theorem 4.4. The one-step error U(h)U1in RMF computation introduced by the double reflection
method has the order of O(h5).Specically,
r(h)r1=1
720 Kh5+O(h6).(15)
Here K=2κ12τ0+κ02τ03+κ1κ0τ1κ2κ0τ0is a bounded constant for a smooth curve, where κi=
(d/ds)iκ(s)|s=0,τi=(d/ds)iτ(s)|s=0 are the curvature, torsion and their respective derivatives at s=0.
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
16 ·Wenping Wang et al.
The proof of Theorem 4.4 is given in Appendix I. The constant Kin Eqn.(15) has an interesting geometric
interpretation. A spherical curve x(s) is characterized by the differential equation [Kreyszig 1991]
τ
κd
dsκ
κ2τ=0.
It is easy to verify that the numerator of this equation is
K(s)=2κ1(s)2τ0(s)+κ0(s)2τ0(s)3+κ1(s)κ0(s)τ1(s)κ2(s)κ0(s)τ0(s).
Therefore, K(s) = 0 if and only if x(s) is a spherical curve. Hence, intuitively speaking, K=K(0) measures
how close x(s) is to a spherical curve at s=0.
As an obvious corollary of Theorem 4.4, we have the next theorem that the RMF computation by the
double reflection method applied to a general regularly parameterized spine curve has the fourth order global
approximation error.
Theorem 4.5. Given a regularly parametrized spine curve x(u),u[0,M],letxi=x(ui),i=0,1,...,n,
be point s sampled on x(u)with equally spaced paramet er values, i.e., ui=ihand h=M/n. Then the global
error of the approximate RMF of x(u)computed by the double reflection method applied to the sequence {xi}
has the order O(h4).
5. COMPARISON AND EXPERIMENTS
We first give the numbers of operations used in the three methods (i.e., double reflection, projection, and
rotation) for computing RMF in order to compare the efficiency of these methods. To save space, the
detailed counting is referred to our technical report [Wang et al. 2007]. The double reflection method can
be implemented such that the per frame computation of the double reflection method costs 28 additions, 32
multiplications and 2 divisions. For the projection method [Klok 1986], the per frame computation needs 5
additions, 21 multiplications, 2 divisions and 1 square root to compute a new frame; considering the cost of
the square root, this is less than, but comparable to, the cost of the double reflection method.
A procedure of the rotation method is given in [Poston et al. 1995]. Given the two consecutive unit
tangent vectors t0and t1, the rotation axis is computed as (a, b, c)=t0×t1and the cosine of rotation angle
is cos α=t0·t1. Then the rotation matrix is given by
R=
cos αcb
ccos αa
bacos α
+1cos α
a2+b2+c2
a2ab ac
ab b2bc
ac bc c2
.
From here it is easy to see that the per frame computation of the rotation method can be implemented with
26 additions, 36 multiplications and 1 division.
The number of operations for the three methods are summarized in Table II. The three methods have
similar computational costs, as our tests show that a sqrt or a division is about six times more time consuming
than a multiplication; this makes sense because square root and division are approximated by a truncated
series in arithmetic hardware. The actual timing comparison will be given in the next subsection.
Another procedure of the rotation method is given in [Bloomenthal 1990], which uses 19 mults and a
square root to compute the rotation matrix Rafter using 6 mults to get the rotation axis t0×t1. Hence, that
version of the rotation method requires in total 40 multiplications and one square root to compute a new
frame, assuming that the tiare unit tangent vectors. In the subsequent experimental comparisons involving
the rotation method we will refer to the faster implementation in [Poston et al. 1995].
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Computation of Rotation Minimizing Frame ·17
Method #ofadds #ofmults #ofdivs #ofsqrt
Projection 15 21 2 1
Rotation 26 36 1 0
Double reflection 28 32 2 0
Table II. The operations counts of the three methods.
Fig. 8. Timings of the double reflection method, the pro-
jection method and the rotation method.
Fig. 9. Timings of Runge-Kutta method and the double
reflection method.
Two examples will be used to compare the double reflection method with the following existing methods:
the projection method, the rotation method and the 4-th order Runge-Kutta method, in terms of efficiency
and accuracy. All test cases were run on a PC with Intel Xeon 2.66 GHz CPU and 2.00 GB RAM.
Example 1: In the first example we use the four methods to compute the RMF of the spine curve,
whichisatorusknot,givenby
x(u)=[(0.6+0.3cos(7u)) cos(2u),(0.6+0.3cos(7u)) sin(2u),0.3sin(7u)]T,u[0,L] (16)
We compute the RMF using different step sizes h=0.01 2k,k=0,1,...;thatis,foreachfixedstepsize
h,thesampledpointsarex(ih), i=0,1,...,L/h.
The timings of computing the sequence of frames by the four methods are shown in Figures 8 and 9.
We see that the projection method, the rotation method and the double reflection method have similar time
costs. The Runge-Kutta method costs much more time than the double reflection method, since it needs
more function evaluations in each step than the other three methods.
To observe approximation errors, we need an exact RMF of the spine curve or an approximate RMF of
very high accuracy against which the computed approximate RMF by the four methods can be compared.
Since the exact RMF of the torus knot given by Eqn.(16) is difficult to obtain, we use the integration function
provided in Maple to get an approximate RMF of x(u) whose approximation error is known to be less than
1016. This highly accurate RMF is used in place of an exact RMF to measure the global approximation
error Egdefined in (2).
The global approximation errors ekof the four methods are shown in Figure 10 and also in Tables III
and IV, where ekis the error of using 2ksegments, k=6,7,...,11. These data confirm that the projection
method and the rotation method have the second order of global approximation error O(h2), and the Runge-
Kutta method and the double reflection method have the fourth order of global approximation error O(h4).
Example 2: In the second example we use the double reflection method to approximate the RMF of a PH
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
18 ·Wenping Wang et al.
Fig. 10. Global errors of the four methods for the torus
knot in Example 1.
Fig. 11. Global errors of the four methods for the PH
curve in Example 2.
Double reflection Runge-Kutta
#ofsegments error ek,ratioek/ek1error ek,ratioek/ek1
265.10E3, N.A. 3.58E2, N.A.
273.24E4, 0.063577 2.32E3, 0.064846
282.03E5, 0.062776 1.46E4, 0.062737
291.27E6, 0.062571 9.10E6, 0.062408
210 7.95E8, 0.062578 5.68E7, 0.062422
211 4.97E9, 0.062575 3.55E8, 0.062438
Table III. Global approximation errors ekof the double reflection method and by using the 4-th order Runge-Kutta method
for the torus knot in Example 1. The error ratios ek/ek1show that the approximation orders of these two methods are both
O(h4).
Projection method Rotation method
#ofsegments error ek,ratioek/ek1error ek,ratioek/ek1
261.56E1, N.A. 2.60E1, N.A.
279.03E2, 0.579295 1.91E1, 0.736606
282.26E2, 0.249757 4.76E2, 0.248776
295.64E3, 0.249939 1.19E2, 0.249668
210 1.41E3, 0.249983 2.97E3, 0.249906
211 3.52E4, 0.249995 7.42E4, 0.249971
Table IV. Global approximation errors ekof the projection method and the rotation method for the torus knot in Example 1.
The error ratios ek/ek1show that the approximation orders of these two methods are both O(h2).
(Pythagorean-hodograph) curve, whose RMF can be computed exactly by a closed-form formula [Farouki
2002]. Given two points x0= (1000,0,0)Tand x1= (1000,2000,4000)Twith associated un-normalized
tangent vectors ˆ
t0=(1,5,1)T,ˆ
t1=(3,2,5)T, we obtain a cubic PH curve x(u) as the spine curve using
G1Hermite interpolation, following [J¨uttler and M¨aurer 1999]. Let the Frenet frame of x(u)atu=0bethe
initial frame U0. Compared with the exact RMF of x(u) at the endpoint x1=x(1), we obtain the errors
of the approximate RMF computed by the four methods. These errors are shown in Figure 11. The errors
of the double reflection method and the rotation method are also given in Table V and their color coded
surface representations in Figure 12. These data confirm again the fourth order approximation error O(h4)
of the double reflection method.
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Computation of Rotation Minimizing Frame ·19
Double reflection Rotation method
#ofsegments error ek,ratioek/ek1error ek,ratioek/ek1
269.29E9, N.A. 1.78E4, N.A.
275.94E10, 0.063919 4.47E5, 0.250721
283.75E11, 0.063181 1.12E5, 0.250321
292.36E12, 0.062926 2.80E6, 0.250151
210 1.48E13, 0.062789 7.00E7, 0.250073
211 9.25E15, 0.062521 1.75E7, 0.250036
Table V. Global approximation errors ekof the double reflection method and the rotation method for the PH curve. The error
ratios ek/ek1confirm again the O(h4) global error of the double reflection method and the O(h2) global error of the rotation
method.
(a) Double reflection method. (b) Rotation method. (c) Frames computed by double reflection.
(d) Error coding bar
Fig. 12. The color coded sweep surfaces showing the errors of the double reflection method and the rotation method for the
PH curve in Example 2, with 256 segments.
6. EXTENSIONS
6.1 Spine curve defined by a sequence of points
In some applications a spine curve is specified by a sequence of points xiin 3D, which we may assume to
lie on some unknown regularly parameterized spine curve, and we need to compute a sequence of frames Ui
which has minimal rotation about the spine curve. In order to apply the double reflection method in this
case, we need to furnish each data point xiwith a unit tangent vector ti.
In the following we assume that the given points xi,i=0,1,...,n, are sampled from a regular parametric
x(u) with equally spaced parameter values, i.e., xi=x(ui), where ui=u0+ih. In actual computation,
this underlying curve x(u) is not known, so the tangent vectors tiat xineed to be estimated from the given
points xi. The key requirement for computing the tiis that the approximation error of tito the true tangent
vector x(ui) is of the order O(h5), so the global error of the double reflection method for computing the
RMF based on the estimated tangent vectors will be of the order O(h4).
We use the following formulas to estimate the tangent vectors at an internal point xi, i.e., when 2 i
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
20 ·Wenping Wang et al.
n2, we have
ti=xi28xi1+8xi+1 xi+2.
For boundary points, i.e., when i=0,1,n1orn,wehave
t0=25x0+48x136x2+16x33x4,
t1=3x010x1+18x26x3+x4,
tn1=3xn+10xn118xn2+6xn3xn4,
tn=25xn48xn1+36xn216xn3+3xn4.
Using Taylor expansion, it is straightforward to verify that the error of approximation of the tito the
true tangent x(ui)isO(h5). After normalization, the error of the unit tangent vector ˜
ti=ti/tiis at most
O(h5). Hence, the global error of the double reflection method based on the local data (xi,˜
ti;xi+1,˜
ti+1)
is O(h4). This has also been confirmed by our numerical experiments, which are not included here due to
space limitation.
It is assumed above that there are at least 5 sample points xi, i.e., n4. If n<4, some other simpler
methodcanbeusedtoestimate the tangent vectors ti(which would necessarily have approximation errors
larger than O(h5)). We skip the discussion on this special case for the sake of brevity.
6.2 Using only tangent vectors
According to its defining equation (4), the RMF of a spine curve x(u) is entirely determined by the unit
tangent vector t(u). Thus it is natural to consider computing the RMF of x(u) using only the sampled
tangent vector ti=˙
x(ui). From a practical point of view, this treatment is also desirable when the points
x(ui) are overly densely sampled, which may make the first reflection vector v1=xi+1 xitoo small and
therefore make computation of the reflection R1less stable.
In order to apply the double reflection method in this case, all we need to do is provide a reflection vector
for the first reflection R1. Our analysis shows that the global approximation order O(h4) to the true RMF
of x(u) is preserved if the first reflection vector is chosen to be
v1= 13(ti+ti+1)(ti1+ti+2).(17)
Then the remaining steps of the double reflection method are the same. This assertion can be proved in a
similar way to that of proving Theorem 4.7. Note that the computation of v1in Eqn. (17) does not involve
subtraction between two close quantities, and therefore is numerically robust. Note, however, a different
treatment is needed to compute v0and vn1, such that an order O(h4) approximation to x1x0and
xnxn1are achieved. We skip the details here.
6.3 Variational principles for RMF with boundary conditions
In general, the RMF of a closed smooth spine curve does not form a closed moving frame. Therefore, when
a closed moving frame with least rotation is needed, it can be generated by adding a gradual rotation to the
RMF along the closed spine curve to make the resulting moving frame closed. Even for an open spine curve,
it is often required that its moving frame meet given end conditions while having a natural distribution of
rotation along the spine curve. So an appropriate additional rotation to the RMF needs to be computed in
this case. We study in this section how this additional rotation can properly be determined.
More specifically, consider a curve segment x(s), s[0,L], in arc-length parameterization. We would
like to find a one-parameter family of unit vectors g(s) orthogonal to the tangent vector t(s) and satisfying
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Computation of Rotation Minimizing Frame ·21
the boundary conditions
g(0) = g0and g(L)=g1(18)
The vector g(s) defines an orthonormal frame M=(t,g,t×g) along the spine curve.
We compare the frame M(s) with the RMF generated by a vector r(s) satisfying r(0) = g(0). Let
α(s)=(r(s),g(s)) be the angle between the two frames, where the sign is chosen such that it corresponds
to a rotation around the oriented line determined by the tangent vector t(s). In addition, assume that α(s)
is continuous and satisfies α(0) = 0. We will call M(s)themodified frame, since it is obtained by adding a
rotation of angle α(s) to the RMF. In this sense the RMF serves as a reference frame with respect to which
another moving frame is specified.
The boundary conditions (18) imply that
α(0) = 0 and α(L)=(r(L),g1)+2(19)
for a some fixed integer k. The angular velocity vector of the modified frame M(s)is
ωmodified(s)=κ(s)b(s)+α(s)t(s) (20)
The function s→ α(s) specifies the angular speed of the rotation of M(s) around the tangent of the curve
x(u). We now consider two possible ways of choosing α(s).
Minimal total angular speed. One may choose α(s) that minimizes the functional
L
0||ωmodified||ds=L
0κ(s)2+α(s)2dsMin (21)
and satisfies the boundary conditions (19). Let F(s, α, α)=κ2+α2. Then we have at hand a functional
of the angular function α(s). The moving frame M(s) corresponds to a curve on the unit quaternion sphere,
and minimizing the functional in (21) amounts to minimizing the length of this curve subject to that g(s)is
perpendicular to t(s); this is the computational approach taken in [Hanson 1998].
Here we will analyze this variational problem to give it a simple geometric interpretation as well as an
easy computational method. Solving Euler’s equation of the functional (21) using calculus of variations
yields
0=Fαd
dsFα=κ
(κ2+α2)3/2(κα ακ)=κ3
(κ2+α2)3/2α
κ
(22)
assuming κ= 0. It follows that
α(s)=(s) (23)
for some constant C, which can be determined from the boundary conditions and the total curvature.
Consequently, the angular speed of the additional rotation around the tangent is proportional to the curvature
of the curve. Hence, minimizing (21) makes the additional rotation more concentrated on curve segments of
higher curvatures.
The above analysis is only valid for curved segments with κ(s)≡ 0. For straight line segments, the
variational problem (21) does not have a unique solution. In fact, the integrand in this case simplifies to
|α|, and any monotonic function α(s) which satisfies the boundary conditions is a solution. Because of
this non-uniqueness of solution, optimization methods as used in [Hanson 1998] for minimizing (21) will
experience numerical problems with a spine curve that is close to a straight line. Based on our analysis, a
more efficient method is to compute the curvatures at sampled points of the spine curve, and then distribute
the additional rotation proportional to the curvatures along the curve, with respect to the RMF.
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
22 ·Wenping Wang et al.
Minimal total squared angular speed. One may also choose α(s) that minimizes
L
0||ωmodified||2ds=L
0
(κ(s)2+α(s)2)dsMin (24)
and satisfies the boundary conditions (19). Now, with F=κ2+α2, Euler’s equation gives α =0,or
α(s)=as for some constant a;thatis,the rotation of Mis linearly proportional to the arc length parameter s.
This choice of the additional rotation is not only easy to implement, but also free of the numerical problem
with the method based on minimizing (21); so it is recommended over the first one based on minimizing the
total angular speed. Note that this means of applying the additional rotation as proportional to arc-length
has been suggested in the literature (e.g. [Bloomenthal 1990; Wang and Joe 1997]), but here we provide
theoretical justification from the viewpoint of the variational principle through minimization of the total
squared angular speed.
Efficient implementation of the above methods of computing a moving frame with boundary conditions
is based on angle adjustment to the RMF, either according to curvature or arclength. When the RMF
is computed approximately, the resulting solution is only an approximate one. In this regard, the higher
accuracy of the double reflection method makes this solution more accurate than using the projection method
or the rotation method.
One may choose the integer kin (19) to minimize the rotation if the least deviation to the RMF is
desired, or choose kto design a moving frame with a specified amount of total twist along the spine curve.
Figure 13 shows comparison of the two methods above for computing frames meeting certain boundary
conditions. The method based on total angular speed minimization (i.e., rotation proportional to curvature)
and the method based in total squared angular speed minimization (i.e., rotation proportional to arclength)
are shown in the first row and the second row, respectively. In each row, the four figures are for the case
of using RMF computed by the double reflection method with no twist adjustment, the case of using the
minimal twist to close the frame, the case of a twist of 2π,andthecaseofatwistof4π. We see that the
twist is more concentrated in high curvature parts of the spine curve in the first row, while it is distributed
more uniformly along the curve in the second row.
In Figure 14, the support structure of a glass table, as a closed sweep surface, is modeled with a moving
frame meeting six conditions to make the structure have proper contact (i.e., along a line segment) with the
table at four locations and with the ground at the other four locations. These conditions are met by adjusting
an RMF by a twist linearly proportional to arclength between every two consecutive contact locations.
7. CONCLUDING REMARKS
We have presented a new discrete approximation method for computing the rotation minimizing frame of a
space curve. The method uses two reflections in a plane to compute the next frame from the current frame,
and is therefore called the double reflection method. This method is simple, fast, and more accurate than the
projection method and the rotation method, which are currently often used in practice. We have shown that
the approximation error of the double reflection method is O(h4), while the errors of the other two methods
are O(h2), where his the step size used to sample points on a spine curve of fixed length.
The double reflection method is also much superior to direct application of the standard 4-th order
Runge-Kutta method. Although the two methods have the same order of approximation error, the double
reflection method is simpler and faster, and requires only C1information of a spine curve, while the Runge-
Kutta method needs C2information. We have also discussed the applications of RMF in modeling moving
frames meeting boundary conditions.
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
Computation of Rotation Minimizing Frame ·23
(a) (b) (c) (d)
(e) (f) (g) (h)
Fig. 13. Comparison in computing a closed moving frame. Minimization of total angular speed is shown in row one. Minimization
of total squared minimization is shown in row two. In each row, from left to right, the four figures are for the case of RMF
computed by the double reflection method, the case of using the minimal twist to close the frame, the case of an additional
twist of 2π, and the case of an additional twist of 4π.
We conjecture that O(h4) is the maximum accuracy that can be achieved in RMF computation using
only the sampled position and tangent data (x0,t0;x1,t1) of a curve segment.
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8. APPENDIX I
Proof of Theorem 4.4. There are two parts in this proof. In the first part we derive an expression of
the order O(h5) term of the one-step error. In the second part we show that coefficient of this error term is
bounded for a regular curve, thus yielding the claimed order of magnitude.
We will obtain the error expression using the canonical Taylor expansion of the curve x(s)atx(0), which
can be derived from the Frenet formulas [Kreyszig 1991]. In a neighborhood of x(0), x(s) is approximated
by the series
x(s)=
s1
6κ2
0s31
8κ0κ1s4+···
1
2κ0s2+1
6κ1s3+1
24 (κ2κ3
0τ2
0κ0)s4+···
+1
6κ0τ0s3+1
24 (κ0τ1+2κ1τ0)s4+···
,(25)
where the Frenet frame at s= 0 is aligned with the axes of the Cartesian coordinates, and κi=(d/ds)iκ(s)|s=0,
τi=(d/ds)iτ(s)|s=0. With the help of computer algebra tools, we generate Taylor series for all quantities
needed for computing the variables listed in the procedure of the double reflection method (Table I). Due
to space limitation, only an outline of the derivation will be given.
Consider a segment of x(s)oflengthhstarting at the origin, i.e.,
(0,0,0)=x0=x(0),x1=x(h),(1,0,0)=t0=˙
x(0),t1=˙
x(h).(26)
Let r0=(0,C,S), where C2+S2= 1, be the reference vector of U0at x0. We compute the new reference
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
26 ·Wenping Wang et al.
vector r1using steps from (1) to (7) of the algorithm Double Reflection (see Table I):
v1=(h+O(h3),1
2κ0h2+O(h3),O(h3))
c1=h21
12 κ02h4+O(h5)
rL
0=(0h1
3(1+κ0τ0S)h2+O(h3),C1
2κ02Ch2+O(h3),S+O(h3))
tL
0=(1+ 1
2κ02h2+O(h3),κ0h1
3κ1h2+O(h3),1
3κ0τ0h2+O(h3))
v2=(2κ02h2+O(h3),2κ0h+5
6κ1h2+O(h3),5
6κ0τ0h2+O(h3))
c2=41
36 (τ02κ02+κ12)h4+O(h5)
r1=(0h1
2(1+κ0τ0S)h2+O(h3),C 1
2κ02Ch2+O(h3),S+O(h3))
On the other hand, using the angular velocity of the RMF (Eqn. (10)) we generate the Taylor expansion of
the reference vector r(h)oftheexactRMFU(h),
r(h)=r(s)s=0
+κ(s)b(s)×r(s)

=r(0)
s=0
h+d
ds(κ(s)b(s)×r(s))

=r(0)
s=0
h2
2+...
Using the Frenet formulas and the fact that the derivatives of r(s) are given by the previously generated
terms of the Taylor expansion, r(h) can be expressed solely by using derivatives of curvature and torsion at
s= 0, and by the initial value r(0) = (0,C,S). Finally, we compare the Taylor expansions of r(h)andr1
to obtain
r(h)r1=(O(h6),1
720 SKh
5+O(h6),1
720 CKh
5+O(h6)),
where
K=2κ12τ0+κ02τ03+κ1κ0τ1κ2κ0τ0(27)
Hence,
r(h)r1=1
720Kh5+O(h6)
Next, we need to show that the coefficient Kin the O(h5) term above is finite for a regular smooth
curve. This is a concern because the torsion τ0appearing in K(Eqn. (27)) and τ0can become unbounded
for a regular curve (see our technical report [Wang et al. 2007]). Note that only the curvature κ0, torsion τ0
and their derivatives are present in K.Since
κ(s)=¨
x(s)(0) = (˙
x(s)ר
x(s)) ·...
x(s)
¨
x(s)3
it is easy to see that, if a spine curve has non-vanishing curvature, then κ0=κ(0) is bounded from zero, and
τ0=τ(0) and its derivative are finite; consequently, Kwill be finite in this case.
We will use a conformal mapping to turn an arbitrary curve segment x(s), s[0,h], possibly with
vanishing curvature, into another curve segment with curvature bounded from zero. First take the osculating
plane of x(s)ats= 0. With a rigid motion we take this plane to be the x-yplane and have the point x(0)
positioned at the origin (0,0,0). Let Csdenote the inversion with respect to the sphere S1of radius 1 and
centered at (0,0,1). Then the plane x-yis mapped by Csto the sphere S2of radius 1/2 and centered at
(0,0,1/2). Clearly, Csis conformal, and the point x(0) = (0,0,0) is fixed by Cs.
Let κ0be the curvature of x(s)ats=0. Letxc(s) denote the transformed curve Cs(x(s)). With a bit
abuse of notation, we use xc(t), t[0,h
c], to denote arclength parameterization of the segment xc(s). At
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Computation of Rotation Minimizing Frame ·27
t=0,thecurvexc(t) has the normal curvature equal to 2, which is the reciprocal of the radius of S2,and
the geodesic curvature equal to κ0, which is the curvature of x(s)ats=0. (Thecurvexc(s) has the same
normal curvature and geodesic curvature at xc(0) as any spherical curve on S2that has the second order
contact with xc(s)atxc(0). ) It follows that the curvature of xc(u)atxc(0) is κc=(κ2
0+4)
1/2.
Clearly, κcis bounded away from zero. Hence, if we apply the double reflection method to the transformed
curve segment xc(t), t[0,h
c], according to the preceding analysis, the fifth order term of the approximation
error takes the form 1
720 Kch5
c;hereKcis finite, since κcis bounded away from zero. On the other hand,
because the approximation error produced by the double reflection method is invariant under a conformal
mapping (cf. Section 4.6), in the limit we have
K
720h5=Kc
720h5
c
When his sufficiently small, due to the regular nature of the mapping Csin the neighborhood of x(0), there
exists a constant d>0 such that hc<dh. It follows that
K=h5
c
h5Kc<d
5Kc
Hence, Kis also finite. This completes the proof that the local one-step error of the double reflection method
is of the order of O(h5).
ACM Transactions on Graphics, Vol. V, No. N, Month 20YY.
... We therefore use the rotation-minimizing frame (RMF) [45,46], which minimizes the frame spinning along the trajectory, and which is commonly used in computer graphics and 3D modeling [47]. To calculate the RMF, we use a simple and fast approximation method called double reflection method [48]. This method requires the trajectory sample points, the tangent vectors T in all sample points, and a coordinate frame in the first sample point (U 1 , V 1 , T 1 ) as an input. ...
... This method requires the trajectory sample points, the tangent vectors T in all sample points, and a coordinate frame in the first sample point (U 1 , V 1 , T 1 ) as an input. The coordinate frames in the remaining sample points along the trajectory are calculated recursively [48]. The coordinate frame (U 1 , V 1 , T 1 ) in the first sample point is chosen such that T 1 is the tangent, and the two remaining mutually perpendicular vectors U 1 and V 1 can be chosen arbitrarily in the plane that is perpendicular to T 1 . ...
... In a first step of the transformation, we numerically calculate the -coordinate sample points from the trajectory sample points. The RMF (U ( ) , V ( ) , T ( )) in the remaining trajectory points is then calculated numerically by the aforementioned double reflection method [48]. In the special case of a plane trajectory in the ( , )-plane with U 1 chosen to be parallel to the -axis, U ( ) is parallel to the -axis in all trajectory points. ...
... The sweep subject type is specified by the choice of the generator and the position. Sweeping surfaces are a considerable and essential types of surfaces in geometric modeling and are universally used in industrial design, which shows why these surfaces are one of the charming subjects of surface theory, as well as being applied in many areas of science such as computer-aided geometric design, computeraided design, and so on [8][9][10][11][12]. One of the paramount facts about the sweeping surface is that the sweeping surface can be a developable ruled surface [13,14]. ...
... Developable surfaces are the distinctive ruled surfaces that are rather interesting and have many applications in many subjects. Therefore, many geometers and engineers have investigated and obtained many properties of the ruled and developable surfaces (see, for example, [8][9][10][11][12][13][14][15][16]). However, to the authors' knowledge, there is no work devoted to discussing the notions of sweeping surfaces immersed in Lie groups. ...
... A moving orthonormal frame {ξ 1 , ξ 2 , ξ 3 }, through a space curve γ(s) in G, is a rotation-minimizing frame (RMF) with respect to ξ 1 if its angular velocity ω satisfies < ω, ξ 1 >= 0, or equivalently, the derivatives of ξ 2 and ξ 3 are both parallel to ξ 1 . A similar description holds when ξ 2 or ξ 3 is selected as the reference orientation [10,17,18]. ...
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This paper investigated the rotation minimizing frames that are related to the space curves and the sweeping surfaces that are traced by these frames in the three-dimensional Lie group. Then, the sufficient and necessary conditions for the sweeping surface to be a developable ruled surface were obtained. In particular, we mostly focused on the study of the resulting developable surface is a cylinder, cone, or tangent surface. Meanwhile, to support the results in the paper, some illustrative examples are presented.
... (The curvature associated with N is then κ 1 = − 1 r .) This is an important step in the implementation of the double reflection method for computing approximations of RM frames [17]. Another remarkable feature of RM frames along spherical curves is that they are path independent, i.e., if two spherical curves connect q 1 to q 2 in S 2 (p, r) and their normals at q 1 coincide, then their normals at q 2 must also coincide [17]. ...
... This is an important step in the implementation of the double reflection method for computing approximations of RM frames [17]. Another remarkable feature of RM frames along spherical curves is that they are path independent, i.e., if two spherical curves connect q 1 to q 2 in S 2 (p, r) and their normals at q 1 coincide, then their normals at q 2 must also coincide [17]. Notice that for the remaining RM vector field [16]. ...
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In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. Later, we characterize plane and spherical curves as curves whose position vector lies, up to a translation, on a moving plane spanned by their unit tangent and an RM vector field. Finally, as an application, we characterize Bertrand curves and slant helices as curves whose so-called natural mates are spherical and general helices, respectively.
... Using these center voxels {si}, the desired segment consists of, a smooth 3D-path (centerline) can be calculated utilizing, e.g., Bézier interpolation algorithm [10]. The calculated smooth 3D-path (centerline) consists of a user-defined number of points N3D-path serving as starting points for 3D frame calculation (Nframe=N3D-path≥Nvoxels) required for CPR view preparation [4,17]. ...
... In Fig. 4 (a), the sketch shows the virtual sphere around a segment point si with radius Rsphere. The calculated frame with vectors {u, v, t} spans up a local coordinate system oriented along the tubular structure [17]. The projected slice spanned up by {u, v} ( Fig. 4 (b)) indicates the new segment point s'i (center voxels*) calculated with the points rj from the including vessel surface vertices (green dots). ...
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An accurate planar visualization (curved planar reformation - CPR) of intracranial arteries is mandatory for an improved luminal and mural assessment especially in low resolution Magnetic Resonance Imaging (MRI) data sets acquired in standard clinical settings. CPR visualization methods based on the centerline of the desired structure are fast and easy to implement but the accuracy strongly depends on the spatial resolution of the 3D data set and the size of the desired vessel. In the manuscript, a novel algorithm for fast and robust centerline calculation in multi-contrast 3D MRI data is presented. It considers the extracted surface of the vessel structure for a more accurate centerline prediction resulting in an enhanced CPR visualization of small vessels.
... for some continuous function φ : [0, L] → R 3 with a reference vector n p (0) ∈ S 2 satisfying n p (0) ⊥ t p (0) (see, e.g., [9,46]). We denote by ...
... Let p (0) ∈ (S △ ) 3 be a three-dimensional cubic not-a-knot spline describing the initial guess for the spine curve. Accordingly, we compute a rotation minimizing frame (t p (0) , n p (0) , b p (0) ) along this spline using the double reflection method from [46]. Then we choose a one-dimensional cubic not-a-knot spline θ (0) ∈ S △ that describes the rotation function of the initial guess for the geometry adapted frame (t p (0) , n p (0) ,θ (0) , b p (0) ,θ (0) ) as in (2.5). ...
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Electromagnetic waves impinging on three-dimensional helical metallic metamaterials have been shown to exhibit chiral effects of large magnitude both theoretically and in experimental realizations. Chirality here describes different responses of scatterers, materials, or metamaterials to left and right circularly polarized electromagnetic waves. These differences can be quantified in terms of electromagnetic chirality measures. In this work we consider the optimal design of thin metallic free-form nanowires that possess measures of electromagnetic chirality as large as fundamentally possible. We focus on optical frequencies and use a gradient based optimization scheme to determine the optimal shape of highly chiral thin silver and gold nanowires. The electromagnetic chirality measures of our optimized nanowires exceed that of traditional metallic helices. Therefore, these should be well suited as building blocks of novel metamaterials with an increased chiral response. We discuss a series of numerical examples, and we evaluate the performance of different optimized designs.
... Given that the spatial path-parameterization conducted in all these works is based on the Frenet-Serret frame [11], the resultant equations of motion are not defined in inflection points, i.e., when the curvature vanishes, and thus, are only continuous for paths turning in one direction. Moreover, the undesired rotation of the Frenet-Serret frame with respect to its tangent component introduces a distortion in the representation of the environment [12]. ...
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This paper presents a two-stage prediction-based control scheme for embedding the environment's geometric properties into a collision-free Pythagorean Hodograph spline, and subsequently finding the optimal path within the parameterized free space. The ingredients of this approach are twofold: First, we present a novel spatial path parameterization applicable to any arbitrary curve without prior assumptions in its adapted frame. Second, we identify the appropriateness of Pythagorean Hodograph curves for a compact and continuous definition of the path-parametric functions required by the presented spatial model. This dual-stage formulation results in a motion planning approach, where the geometric properties of the environment arise as states of the prediction model. Thus, the presented method is attractive for motion planning in dense environments. The efficacy of the approach is evaluated according to an illustrative example.
... In a next step, local coordinate systems are created in each of these points. Using the tangent of the curve as local X direction, local Y and Z directions are calculated via the method of rotation-minimizing frames (Wang et al. 2008). Following, the FE nodes from the starting non-design cross-section of the spring are copied in every local coordinate system using translation and rotation operations. ...
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The presented paper describes a shape optimization workflow using Bayesian strategies. It is applied to a novel automotive axle system consisting of leaf springs made from glass fiber reinforced plastics (GFRP). Besides the primary objectives of cost and mass reduction, the assembly has to meet multiple technical constraints with respect to various loading conditions. The related large-scale finite element model is fully parameterized by splines, hence the general shape of the guide curve as well as the spring’s height, width and material properties can be altered by the corresponding workflow. For this purpose, a novel method is developed to automatically generate high-quality meshes depending on the geometry of the respective springs. The size and complexity of the model demands the implementation of efficient optimization techniques with a preferably small number of required response function evaluations. Therefore, an existing optimization framework is extended by state-of-the-art Bayesian methods, including different kernel combinations and multiple acquisition function approaches, which are then tested, evaluated and compared. To properly address the use of GFRP as spring material in the objective function, an appropriate cost model is derived. Emerging challenges, such as conflicting targets regarding direct material costs and potential lightweight measures, are considered and investigated. The intermediate steps of the developed optimization procedure are tested on various sample functions and simplified models. The entire workflow is finally applied to the complete model and evaluated. Concluding, ideas and possibilities in improving the optimization process, such as the use of models with varying complexity, are discussed.
... In geometry, there are many orthonormal frames such as Frenet, Bishop, RM frames, etc. for investigating the geometric structures of curves and surfaces, [1][2][3][4]. These frames describe the kinematic properties of a particle moving along a continuous, differentiable curve in Euclidean space. ...
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In this study, the ruled surface generated by the natural lift curve in 3 IR is obtained by using the isomorphism between unit dual sphere, 2 DS and the subset of the tangent bundle of unit 2-sphere, TM. Then, exploiting E. Study mapping and the isomorphism mentioned below, each natural lift curve on TM corresponds the ruled surface in. 3 IR Moreover, the singularities of this ruled surface are examined according to RM vectors and these ruled surfaces have been classified. Some examples are given to support the main results.
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Computer animation relies heavily on rigging setups that articulate character surfaces through a broad range of poses. Although many deformation strategies have been proposed over the years, constructing character rigs is still a cumbersome process that involves repetitive authoring of point weights and corrective sculpts with limited and indirect shaping controls. This paper presents a new approach for character articulation that produces detail-preserving deformations fully controlled by 3D curves that profile the deforming surface. Our method starts with a spline-based rigging system in which artists can draw and articulate sparse curvenets that describe surface profiles. By analyzing the layout of the rigged curvenets, we quantify the deformation along each curve side independent of the mesh connectivity, thus separating the articulation controllers from the underlying surface representation. To propagate the curvenet articulation over the character surface, we formulate a deformation optimization that reconstructs surface details while conforming to the rigged curvenets. In this process, we introduce a cut-cell algorithm that binds the curvenet to the surface mesh by cutting mesh elements into smaller polygons possibly with cracks, and then derive a cut-aware numerical discretization that provides harmonic interpolations with curve discontinuities. We demonstrate the expressiveness and flexibility of our method using a series of animation clips.
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In this paper, we define three different rotation-minimizing frames by rotating the moving frame around its coordinate axis vectors. Darboux vectors associated with these frames are obtained as special cases of the Darboux vector associated with the moving frame. Using these Darboux vectors and the moving frame we define six different developable surfaces. For each of these surfaces we give two invariants of curves on these surfaces to characterize their singularities. Moreover, we show that the base curves of these surfaces are contour generators with respect to an orthogonal projection or a central projection if and only if one of the invariants given for each surface is constantly equal to zero. Examples are provided to illustrate our theorems and results.
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We investigate the computation and properties of rotation minimizing frame (RMF), which is a moving orthonormal frame U(u) attached to a smooth curve x(u), called the spine curve, in 3D such that U(u) does not rotate about the instantaneous tan- gent of x(u). Due to its minimal-twist property, the RMF is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. In general, the RMF cannot be computed exactly and therefore one often needs to approximate the exact RMF by a sequence of orthonormal frames at sampled points on the spine curve. We present a novel simple and efficient method for accurate and stable computation of an RMF for any C1 regular curve in 3D. This method, called the double reflection method, uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method is highly accurate - it has the global fourth order approximation error, thus comparing favorably to the second order ap- proximation error of two currently prevailing methods - the projection method by Klok and the rotation method by Bloomenthal, while all these methods have com- parable per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard 4-th order Runge-Kutta method to integrate the defining ODE of the RMF, which yields the same accuracy as the double reflection method. We also present further properties and extensions of the double reflection method for various application scenarios. Finally, we discuss the variational principles in design moving frames with boundary conditions, based on the RMF.