Content uploaded by Miklós Hoffmann
Author content
All content in this area was uploaded by Miklós Hoffmann on Dec 14, 2018
Content may be subject to copyright.
PLEASE, NOTE, THAT THIS IS A PREPRINT. THE FINAL VERSION OF
THE ACCEPTED MANUSCRIPT AND ADDITIONAL MATERIALS AND
RESEARCH DATA CAN BE FOUND HERE:
https://doi.org/10.1016/j.gmod.2018.12.001
1
An improved skinning algorithm for circles and spheres
providing smooth transitions
Kinga Kruppaa,∗, Roland Kunklia, Mikl´os Hoffmannb
aFaculty of Informatics, University of Debrecen, Debrecen, Hungary
bInstitute of Mathematics and Informatics, Eszterh´azy K´aroly University, Eger, Hungary
Abstract
Recently there has been a growing interest in the topic of skinning of circles
and spheres, since modeling based on these objects has been found useful in
areas such as medical applications and character animation. Among others, an
efficient method was presented by Kunkli and Hoffmann [1], whilst Bastl et al.
also provided an effective skinning algorithm [2]. In this study, we outline the
major advantages and disadvantages of these methods, and we show that there
are major insufficiencies in [2] in terms of dynamic modeling. We provide a
new, improved skinning technique, which preserves the advantages of the two
aforementioned algorithms and gives a solution to their arisen problems: it
can be used for real-time modeling due to the smooth alteration of skins, and
provides good results even in extreme cases. We overcome the problem of self-
intersections, and we extend the method to branched systems of circles and
spheres.
Keywords: skinning, circle, sphere, interpolation
2010 MSC: 65D17
∗Corresponding author
Email addresses: kruppa.kinga@inf.unideb.hu (Kinga Kruppa),
kunkli.roland@inf.unideb.hu (Roland Kunkli), hoffmann.miklos@uni-eszterhazy.hu
(Mikl´os Hoffmann)
Preprint submitted to Graphical Models December 14, 2018
1. Introduction
One of the most important areas in Computer Aided Geometric Design is
curve and surface modeling from point-based datasets. Several, now widely
known methods — specifically interpolation or approximation of a sequence of
points — have appeared in this topic (see the classical works [3, 4, 5]). Another5
paradigm of modeling has also been applied for several years: surface generation
based on a set of input curves, called skinning or blending, as it can be seen e.g.,
in [6, 7, 8]. In this skinning technique, a set of curves — typically orthogonal
planar cross-sections of a future surface — is given along a skeleton or spine
curve, and these cross-sections are swept by the computed surface, called skin.10
This surface, usually a spline surface, interpolates or approximates the section
curves. The method has been successfully applied in several applications from
medical surface reconstruction to CAGD in the last decades [9, 10, 11].
However, it is possible to extend the ideas of point-based interpolation and
skinning of cross-sections to blending other planar or spatial objects as well.15
Besides the well-known and widely used control point based modeling methods,
a novel approach has appeared in the creative industry and in the scientific field
as well: curve and surface modeling based on circles and spheres. In this paper,
skinning is considered as a method of constructing interpolation curves for a set
of circles or constructing tubular-like surfaces to a sequence of spheres.20
This type of skinning is often mentioned along Medial Axis Transformation,
even though it is essentially different and is used for a reversed problem: the
final shape is given, and the aim is to find a skeleton that represents the shape.
In our case, the aim is to design a shape using a discrete set of input circles.
However, there are some fields and applications that make use of the connection25
of the two approaches. For example, circle skinning methods can give a good
approximation for the contour reconstruction shown in [12].
Looking at the problem from a practical point of view, skinning can be ap-
plied in fields like molecular biology [13, 14] and medical image processing, for
example for reconstructing cardiovascular systems [15, 16]. These methods gen-30
3
erally use deep algebraic and analytical tools, which may lead to time-consuming
computation, contrary to our aim of real-time design. The sphere-based surface
modeling principle has become particularly popular in the animation indus-
try for modeling 3D articulated shapes and tubular-like surfaces with several
branches. Some recent methods use subdivision techniques for computing the35
surface from a set of given spheres along a skeleton by linearly blending them
and creating a coarse mesh [17, 18]. A clear advantage of these methods is
the ready-to-use final subdivision surface (now standard in animation software)
but there are also drawbacks: the final surface does not necessarily touch (in-
terpolate) the given spheres, and it can be time-consuming to create the base40
mesh by an evolutionary approach to avoid shrinkages. Existing commercial
software, such as ZSpheres R
by Pixologic apply this skinning method, but with
serious drawbacks, as it is discussed in [19]. A similar technique is applied for
simplification of existing structures in [20].
Another approach of sphere-based surface modeling is to use polynomial45
curves and surfaces as skins. Several papers have been published in the last
couple of years providing sphere-based surface modeling algorithms. Some of
them also use evolutionary, iterative approach to find the optimal skin [15, 21,
22, 23], however, these methods are not only time-consuming, but they also
depend on the initial position of the skin and may fall into local minima. Other50
methods use direct real-time computation of the skin, where the central problem
is to find the most appropriate touching points on the given circles (in 2D) or
touching circles on the given spheres (in 3D) along which the skin will touch
the given dataset [1, 2, 19, 24, 25].
In this paper, we evaluate and improve these latter methods, fundamentally55
based on two different recent techniques proposed by Kunkli and Hoffmann [1]
and by Bastl et al. [2]. We provide a new, improved skinning algorithm based
on these methods: preserving their advantages while offering a solution to their
greatest insufficiencies, in terms of their behavior in extreme cases and the lack
of smooth transition in the designing phase.60
The paper is organized as follows. We briefly recall the two above mentioned
4
skinning methods and discuss their advantages and drawbacks in Section 2. In
Section 3 we introduce our new, improved solution, and the results of the three
skinning methods are compared to each other in various situations of input data
in Section 4. In Section 5 we discuss the impact of our method, while Section 665
shows the possibility of extension to 3D and branching. Finally, Section 7 closes
the paper with the conclusion.
Throughout this paper, the corresponding figures to these skinning methods
are distinguished with three different colors: the method of Kunkli and Hoff-
mann is marked with blue, the method of Bastl et al. is marked with red, and70
our new solution is marked with green.
2. Evaluation of recent skinning methods
2.1. The skinning method of Kunkli and Hoffmann
Given an ordered set of circles, the method proposed by Kunkli and Hoff-
mann [1] defines skinning as the construction of two G1continuous curves touch-75
ing each of the given circles at one point separately. The touching points are
determined by applying the solutions to the problem of Apollonius; then they
are separated into two groups for the “left” and for the “right” skin. Finally,
Hermite interpolation curves are constructed between every two neighboring
circles, for which the tangent lengths are computed with the help of the radical80
lines of the circle pairs.
This algorithm provides good results in general scenarios, and the skins
respond dynamically when the user modifies the positions or the radii of the
circles during modeling. Since the touching points are derived from the solution
of the problem of Apollonius, it also guarantees that the touching points never85
lie inside any of the circles. Despite these advantages, in some extreme cases
the method has insufficiencies: the skins can intersect each other or the given
circles.
5
2.2. The skinning method of Bastl et al.
For a given sequence of circles, in the paper published by Bastl et al. [2],90
skinning is the construction of two at least G1continuous curves touching each
of the circles either along a circular arc defined by two endpoints, or at one
point (as the arc degenarates into a midpoint if the positions of the circles
require). These touching points are defined with the help of outer bitangents.
For the construction of the Hermite interpolation curves, the lengths of the95
corresponding tangent vectors are based on the distance of the touching points.
The algorithm generally provides excellent output, however, there are some
major disadvantages in the aspect of dynamic modeling.
Given a sequence of circles C={c1, c2, . . . , cn}, n ∈N, let us consider three
consecutive circles ci−1,ci, and ci+1 with their corresponding disks di−1,di,100
and di+1. Let ci−1∩ci=∅and ci∩ci+1 6=∅. Let the points Qiand Pi
be defined on ci, originated from the left outer bitangents of ci−1and ci, and
ciand ci+1, respectively. The method of Bastl et al. defines the midpoint on
ciin different ways — thus the position of the midpoint and the skin itself
suddenly change even if we minimally modify the positions of the circles. When105
Qi/∈di+1, then the midpoint (noted as Wi1in Fig. 1 (a)) is defined by Piand
Qi. However, if we position ci+1 so that Qi∈di+1, then the midpoint (noted
as Wi2in Fig. 1 (b)) is redefined with the help of Piand Ni2, where Ni2is an
intersection point of ciand ci+1. The same problem happens when ciand ci−1
intersect. Furthermore, even the midpoint Wican get inside the adjacent circle110
(see Fig. 1 (c)). To resolve this, Bastl et al. introduce weights to specify the
location of the midpoint on the arc. Even though choosing an accurate weight
would solve this problem, there is no algorithmic, exact solution to determine a
local weight for each circle.
Another problem occurs while switching between using a midpoint and a115
circular arc defined by two touching points. The algorithm calculates the corre-
sponding tangent lengths differently in these two cases, and it does not change
them continuously. See Fig. 2 as an example: (a) shows the initial position of
circle ci+1, while (b) shows when it is minimally relocated.
6
(a) (b) (c)
Figure 1: In the method of Bastl et al., if Qi/∈di+1 then the midpoint is Wi1=Pi˙_Qi(a).
If we minimally change the position of ci+1, so that Qi∈di+1 , the midpoint is redefined as
Wi2=Pi˙_Ni2(b). This causes a sudden change in the shape of the skin. In (c), we can see
that the midpoint can get inside ci+1.
(a) (b)
Figure 2: The sudden and significant change of the tangent lengths using the method of
Bastl et al. At first (a), ci+1 has two touching points, thus 2 ·Di+1 is used for tangent
length. If we slightly modify the position of ci+1 (b), it will only have one touching point, a
midpoint. In this case, 2 ·\
Di+1 will be used, which is clearly much shorter than 2·Di+1. The
cropped images show the switching from two touching points (a) to using a midpoint (b).
3. The new skinning algorithm120
In this section, we present a new technique combining the advantages of the
two skinning methods we had examined while giving solutions to their disadvan-
7
tages as well. We provide an algorithm that allows smooth transition of skins
without sudden changes in real-time designing. Based on [1] and [2], we now
give extended definitions both for the admissible configuration of input circles125
and for the skins.
3.1. Definition of the problem
Definition 1. An ordered sequence of circles C={c1, c2, . . . , cn}, n ∈Nis
called an admissible configuration, if the corresponding disks D={d1, d2, . . . , dn}
fulfill the following conditions (see Fig. 3 (a)):130
•di6⊂
n
S
j=1,j6=i
dj, i ∈ {1,2, . . . , n}
•di∩dj=∅, i, j ∈ {1,2, . . . , n}, j /∈ {i−2, i −1, i, i + 1, i + 2}
•di−1∩di+1 6=∅=⇒di−1∩di+1 ⊂dii∈ {2,3, . . . , n −1}
Regarding three consecutive circles ci−1,ci, and ci+1, let us consider the
line segment connecting the centers of ci−1and ci. The intersection point of135
the line segment and circle ciis denoted with Si1. Similarly, Si2can be created
using the centers of ciand ci+1. To exclude those cases when the positions of
the circles would yield extremely sharp turnarounds, we extend the conditions
of Definition 1 with the following criteria (see Fig. 3 (b)):
•Si1/∈ci+1, i ∈ {2,3, . . . , n −1}140
•Si2/∈ci−1, i ∈ {2,3, . . . , n −1}
Definition 2. Given an admissible configuration of circles C={c1, c2, . . . , cn},
n∈N, let s(t)and ¯s(t)be two, at least G1continuous curves, called “left” skin
and “right” skin, satisfying the following criteria (see Fig. 4):
•Each circle ci(i∈ {1,2, . . . , n})has either two touching points Ui, Vi∈ci
145
where Ui, Vi∈s(t), or one touching point Wicalled “midpoint” where
Wi∈s(t). The tangent lines of ciand s(t)are identical at these points.
Analogously, in the case of ¯s(t), there exist either two points Ui, V ior a
midpoint Wi.
8
(a) (b)
Figure 3: Non-admissible configuration of input circles. Besides the now standard exclusion
of input circles displayed in (a), the scenario in (b) shows sharp turnarounds. One can easily
obtain admissible configuration by slightly repositioning e.g., the red circles.
(a) (b)
Figure 4: In (a), the touching points are correctly separated for the left and right skin, and
the directions of the tangent vectors are defined correctly. Meanwhile, (b) shows when the
skins do not satisfy such conditions.
•In case of s(t)(“left” skin), each circle ciis considered with a negative150
(clockwise) orientation, and so as the d
UiVicircular arc defined by Uiand
Vi. The relation UiViof the two points reflects the orientation as well.
The midpoint Wiis defined as Wi∈d
ViUi,||Vi−Wi|| =||Ui−Wi||, and
denoted as Wi=Vi˙Ui.
For ¯s(t)(“right” skin), the orientation is considered as positive (counter-155
clockwise), and the touching points and arcs can be defined similarly.
9
•In case of s(t), if the tangent vector(s) at the touching point(s) are ro-
tated by 90 degrees in clockwise direction, they point to the center of ci.
Analogously, the rotation is counterclockwise for the corresponding tangent
vectors of ¯s(t).160
•Ui, Vi, Wi/∈
n
S
j=1,j6=i
dj, where i∈ {1,2, . . . , n}. Same applies for Ui, V i,
and Wi.
The determination of the touching points and the tangent vectors at them
are of utmost importance in skinning. We now provide the computation for the
“left” skin; the “right” one can be computed analogously.165
3.2. Defining touching points
Given an admissible sequence of input circles, we consider three consecutive
circles ci−1,ci, and ci+1 with their corresponding disks di−1,di, and di+1,i∈
{2, . . . , n −1}. As stated in Definition 2, the skin touches cieither along a
circular arc defined by Uiand Vi, or at a single midpoint Wi. Let Qiand Pi
170
note the points on cioriginated from the left outer bitangents of circle ciand
ci−1, and ciand ci+1, respectively (see Fig. 5).
(a) (b)
Figure 5: Construction of the touching points using the outer bitangents. If UiVi, then
the skin touches cialong the circular arc [
UiVi. If not, then there is only one touching point:
the midpoint Wi.
If ci−1∩ci=∅, then Ui=Qi. Otherwise, we define the proper intersection
point Ni1. Let us note the center of ciby Oi(xi, yi), and the center of ci−1
10
by Oi−1(xi−1, yi−1). Now let Ni1be one of the intersection points for which175
it is true that by rotating the −−−−→
Oi−1Oivector in a counterclockwise direction
by less than 180 degrees, it has the same direction as −−−−−→
Oi−1Ni1(see Fig. 6).
Practically, Ni1(xNi1, yNi1) is the point of intersection for which the condition
(xi−xi−1)·(yNi1−yi−1)−(yi−yi−1)·(xNi1−xi−1)>0 holds.
180
Figure 6: Obtaining the suitable intersection point Ni1with the help of a counter-clockwise
rotation about the center of ci−1.
If ci∩ci+1 =∅, then Vi=Pi, else the proper intersection point Ni2, can be
determined analogously. Now we can define Uiand Viin the following way:
Ui=
Ni2,if Qi∈di+1
Qi,otherwise
Vi=
Ni1,if Pi∈di−1
Pi,otherwise
185
If UiVi, we have our final touching points Uiand Vi, and the oriented
circular arc d
UiVi. If UiVi, then a midpoint has to be used. Now let us
consider li1, the line going through the centers of ci−1and ci; correspondingly
li2is defined by the centers of ciand ci+1. We define Si1and Si2by intersecting
ciwith li1and li2. If Si1/∈[
PiQi, then V0
i=Vi, otherwise we define V0
ias190
follows:
V0
i=
Si1,if ci−1∩ci=∅
Ni1,otherwise
We can construct U0
ianalogously. If Si2/∈[
PiQi, then U0
i=Ui, else:
11
U0
i=
Si2,if ci∩ci+1 =∅
Ni2,otherwise
Finally we can define the new midpoint as W0
i=V0
i˙U 0
i. See Fig. 7 for all
possible variations.
To determine touching points for the first and last circles, it is considered as195
c1and cneach has one touching point originated from the left outer bitangents
of c1and c2, and cn−1and cn, respectively. Thus, U1=V1=P1and Un=
Vn=Qn. Algorithm 1 shows the pseudocode of finding the touching points on
circle ci.
3.3. Determining tangent lengths200
After defining all the contact points and arcs for the circles, we construct
cubic polynomial Hermite curves between each circle pair (ciand ci+1,i∈
{1, . . . , n −1}) whose endpoints are noted as Biand Ai+1. If UiVithen
Bi=Vi, else Bi=Wi. If Ui+1 Vi+1 then Ai+1 =Ui+1, else Ai+1 =Wi+1.
The corresponding tangent vectors are eiand fi, that are tangent to ciand ci+1
205
respectively. Their proper length is needed to be defined in order to provide a
solution and avoid the suddenly and dramatically changing tangent lengths. The
method of Bastl et al. [2] lacks the continuous transition while switching between
two touching points and one (as explained in Fig. 2), i.e., when the arc length
defined by Qiand Piis short in regard to the circumference of ci; while it behaves210
very well if the distance of the circles is quite large (see Fig. 8 (a)). Regarding
the method of Kunkli and Hoffmann [1], we can experience the opposite: even
though it provides continuously changing tangent lengths, it is not suitable
in such extreme cases (see Fig. 8 (b)). Considering these, we need a proper
combination of them to provide a solution. We prefer using the calculations215
of [1] when the arc length defined by Qiand Piis short, but we favor the
calculations of [2] when the distance of the circles is diverse.
First, let us recall the starting and ending tangent lengths computed by the
method of Kunkli and Hoffmann. Let tinote the radical line of circles ciand
12
(a) W0
i=Si1˙_Si2(b) W0
i=Si1˙_Qi(c) W0
i=Si1˙_Ni2
(d) W0
i=Pi˙_Si2(e) W0
i=Pi˙_Qi(f) W0
i=Pi˙_Ni2
(g) W0
i=Ni1˙_Si2(h) W0
i=Ni1˙_Qi(i) W0
i=Ni1˙_Ni2
Figure 7: Definition of the midpoint on cion several examples. Various mutual positions of
the circles require different ways to compute the location.
13
Algorithm 1 Defining touching points on circle ci
1: procedure getTouchingPoints(i)
2: if i= 1 then
3: U1=V1=P1
4: else if i=nthen
5: Un=Vn=Qn
6: else
7: Ui=Qi,Vi=Pi
8: if Pi∈di−1then
9: Vi=Ni1
10: if Qi∈di+1 then
11: Ui=Ni2
12: if UiVithen
13: Ai=Ui,Bi=Vi
14: else
15: U0
i=Ui,V0
i=Vi
16: if Si1∈[
PiQithen
17: V0
i=Si1
18: if ci−1∩ci6=∅then
19: V0
i=Ni1
20: if Si2∈[
PiQithen
21: U0
i=Si2
22: if ci∩ci+1 6=∅then
23: U0
i=Ni2
24: W0
i=V0
i˙U 0
i
25: Ai=Bi=W0
i
14
c1
c2
c3
c4
c5
c6
(a)
c1
c2
c3
c4
c5
c6
(b)
Figure 8: Inadequate tangent length calculations in previous methods. In (a), the
insufficiency of the method of Bastl et al.[2] can be seen: the right skin intersects c3because
the defined arc length is small. In (b), the generated skins between circles c4and c5get too
thin with the method of Kunkli and Hoffmann [1].
ci+1. Let dki1denote twice the distance of Biand ti; also let dki2denote twice220
the distance of Ai+1 and ti.
Bastl et al. compute the starting and ending tangent lengths in the following
way:
dbi1=
2· ||Bi−Vi||,if Bi=Wi
2· ||Bi−Vi+1||,otherwise
dbi2=
2· ||Ai+1 −Ui+1||,if Ai+1 =Wi+1
2· ||Ai+1 −Ui||,otherwise
Now let us consider the following notations:
•si1and si2: the arc length defined by Qiand Pion ci; and Qi+1 and Pi+1
on ci+1,
•dcici+1 : the distance of the centers of ciand ci+1,225
•davg =1
n−1
n−1
P
j=1
dcjcj+1 : the average distance of the centers of two neigh-
boring circles,
•ri: the radius of circle ci.
15
Let di1and di2denote the new starting and ending tangent lengths at Bi
and Ai+1, respectively:230
dij=λij·dbij+1−λij·dkij(1)
λij= min sij
2riπ·dcici+1
k·davg
,1
where k∈R+,λij∈[0,1], i∈ {1,2, . . . , n −1}and j ∈ {1,2}.
The value of k— as a kind of shape parameter — can be chosen by the
user. Our experience showed that choosing k= 0.5 results in good output in
the problematic positions as well.
The detailed algorithm of creating the skin between circles ciand ci+1 is235
shown in Algorithm 2.
To summarize, after finding the touching points on the circles (Algorithm 1),
one can contruct the skin by applying Algorithm 2 to all circle pairs from the
admissible set of input circles. The construction is showed in Algorithm 3.
4. Results and comparison240
As a comparison, let us examine different scenarios and generate the output
of each of the skinning methods.
Fig. 9 shows that the method of Kunkli and Hoffmann distorts the skin when
the circles are relatively far from each other: the skins intersect and they cut
into the circles as well. Our new algorithm solves these problems and generates245
appropriate skins for the given set of circles.
16
Algorithm 2 Construction of the skin for circle ciand ci+1
1: procedure computeSkin(i)
2: if Bi6=Aithen
3: Construct the circular arc [
AiBi
4: Let eibe the tangent vector at Bion ci
5: Let fibe the tangent vector at Ai+1 on ci+1
6: Let tibe the radical line of ciand ci+1
7: dki1= 2 ·distance(Bi, ti)
8: dki2= 2 ·distance(Ai+1 , ti)
9: if Bi=Aithen
10: dbi1= 2 · ||Bi−Vi||
11: else
12: dbi1= 2 · ||Bi−Vi+1 ||
13: if Ai+1 =Bi+1 then
14: dbi2= 2 · ||Ai+1 −Ui+1 ||
15: else
16: dbi2= 2 · ||Ai+1 −Ui||
17: keik=λi1·dbi1+ (1 −λi1)·dki1See Expression 1
18: kfik=λi2·dbi2+ (1 −λi2)·dki2
19: Construct the cubic Hermite interpolant given by {Bi, Ai+1,ei,fi}
Algorithm 3 Construction of the left skin for an admissible set of circles
Input: C={c1, c2, . . . , cn}, an admissible sequence of circles
Output: s(t), the left skin
1: for i=1;i≤n; i++ do
2: getTouchingPoints(i)see Algorithm 1
3: for i=1;i≤n−1; i++ do
4: computeSkin(i)see Algorithm 2
Regarding the method of Bastl et al., it does not allow dynamic modeling as
sudden changes happen both in terms of the touching points and the tangent
17
(a) (b)
Figure 9: Comparison of the method by Kunkli and Hoffmann [1] (a), and our new solution
(b). Even in such extreme cases, the new skins do not intersect the circles, nor the skins
themselves.
lengths. However, these problems do not arise using our skinning method. In
Fig. 10 (b), we can see that our algorithm changes the touching point on c3grad-250
ually when moving c2downwards; whilst Bastl et al. method is discontinuous in
the terms of the location of the touching point (see Fig. 10 (a)). Coming from
the construction, the new touching points do not get inside the circles; while
using the method of Bastl et al., they may fall inside other circles (see Fig. 11).
Fig. 12 shows that using our method, the change of the tangent lengths is in a255
continuous way.
For such problematic cases, we provide a more detailed comparison with
videos in [26].
5. Discussion
Regarding our new skinning technique, since the touching points are always260
out of the neighboring circles, and the tangent lines of the circles and the skins
are common at that touching points, we can prove that the skins of denser and
denser subsets of a family of circles converge to the envelope of the family (if
exists), at least locally.
Let us consider a continuous one-parameter family of circles having an enve-265
lope (we consider the skin and the envelope on one side of the family; the other
side works analogously). Also, let us consider one fixed member of the family,
18
(a)
c1c2
c3
c4
c1c2
c3
c4
(b)
Figure 10: Continuously moving c2downwards from its original position, one can expect
continuous alteration of the touching point. Using the method of Bastl et al. (a), if we mark
the locations of the touching point on c3while c2is being moved, we can see that the
movement of the touching point is discontinuous. This yields a sudden “jump” of the
touching point with a significant change in the shape of the skin as well. With our new
skinning method (b), the location of the touching point and the shape of the skin are
changing continuously.
ci. By picking two further members from the family of circles, on the opposite
sides of ci, let us denote them by ci−1and ci+1. Computing a skin along these
three circles, the resulting curves will evidently differ from the envelope in most270
cases. But let the two circles ci−1and ci+1 converge to the fixed circle ci— i.e.,
we consider circles from the family closer and closer to the original circle ciin
both sides — and let us suppose that these circles always form an admissible
set of circles in our terms. The computed touching point(s) on ciwill always
19
c1
c2
c3
c4
(a)
c1
c2
c3
c4
(b)
Figure 11: With the method of Bastl et al.(a), the midpoint is generated inside c4, and the
skin cuts into the circle. Our method (b) solves these kind of problems.
c2
c3
c4
c5
c6
c1
(a)
c2
c3
c4
c5
c6
c1
(b)
c2
c3
c4
c5
c6
c1
(c)
Figure 12: The method of Bastl et al. (red) changes the tangent length dramatically when
minimally moving a circle. Our method (green) continuously changes the tangent length,
thus it allows smooth dynamic design of the skin without intersecting the given circles. The
cropped pictures (b) and (c) show the generated skins for the slightly differently positioned
circle c5. (Note that some parts of the skins overlap, hence it is harder to separate them
visually.)
20
be out of the circles ci−1and ci+1, and in between the intersection points Ni1
275
and Ni2of ci−1and ci, and ciand ci+1, respectively. While the arc between
the two intersection points Ni1and Ni2is shrinking on ci, it always contains
the touching point(s). Applying the sandwich rule it is easy to see that the two
intersection points Ni1and Ni2, and the computed touching point(s) tend to
be identical if the family parameters of the two neighboring circles converge to280
the family parameter of the fixed circle ci. But the definition of the envelope
of the family of circles is the limit of points of intersections of nearby circles.
Therefore, the touching point of the envelope and the computed touching point
of the skin tend to be identical. Moreover, the length of the tangent vector
at the touching point on citends to zero due to the fact that in Expression 1,285
both dbijand dkijtend to zero. Since the tangent directions of the skin always
coincide with the tangent directions of the circle ciat the touching points, and
this also holds for the tangent direction of the envelope, these tangent lines also
tend to be identical. That is — locally — the skin converges to be the envelope
of the family.290
5.1. Avoiding intersections of the skins
The problem of intersecting skins (called self-intersections in [2]) and inter-
section with the input circles are interesting and important issues in skinning.
The method of Bastl et al. offers a solution to the latter problem. If the skinning
curves are considered as cubic B´ezier curves, then the skin can intersect the cir-295
cles when the control polygon is S-shaped. To resolve this, the exact position of
the midpoint is being changed along the circular arc, until the resulting control
polygon is C-shaped. This can be done by choosing a weight for the midpoint
to specify its exact position.
Building upon this idea, we now solve those cases when the skins intersect300
each other. Let us consider the two B´ezier curves (that are now C-shaped)
between two circles, and the starting and ending points: P0=Uiand P3=
Vifor the first (“left”), and R0=Uiand R3=Vifor the second (“right”)
curve. Let P0P1P2P3and R0R1R2R3denote the convex hulls of the control
21
polygons. Derived from the definition of the constructed skins (see Section 3.1)305
the following conditions are satisfied:
•P0P3∩R0R3=∅
•P0, P3/∈R0R1R2R3
•R0, R3/∈P0P1P2P3
By examining only the intersections of the control polygons, we avoid the costly310
operation of obtaining the intersections of the two curves. To avoid the intersec-
tion, thus we can specifically modify the lengths of the first and last sides of the
control polygons (P0P1,P2P3and R0R1,R2R3), so that they are shortened to
their closest intersection points. For details, see Algorithm 4. This way we can
reach that neither the control polygons (convex hulls) intersect each other, nor315
the curves. This implicitly solves those situations when the skins get too “thin”,
as the convex hulls can intersect even when the curves do not yet. Fig. 13 shows
the output of our method. Other than this algorithmic approach, the problem
of intersecting skins can be solved manually by changing the tangent lengths as
the user can adjust the shape parameter k.320
Even though [2] states that the B´ezier curve can always be transformed to
be C-shaped using weights to specify the location of the midpoint on the arc, in
such cases as in Fig. 2 (b), we can see that moving the midpoint along that short
arc does not yield a C-shape. Thus, in the future we plan to give a variation of
the Algorithm 4 so that it gives solution to S-shaped B´ezier curves as well.325
5.2. Generalization to convex planar shapes
Bastl et al. generalize their algorithm to closed convex parametric shapes,
and since our new technique finds the touching points in an analogous way —
using bitangents —, it is possible to extend it as well in principle. Obtaining Qi
and Piand the intersection points Ni1and Ni2is straightforward; only Si1and330
Si2are needed to be redefined — we need to define the center of the shape. To
resolve this, first we define the convex hull of the shape, then we determine the
22
(a) (b)
(c) (d)
Figure 13: In case of intersecting skins, we can apply Algorithm 4 step by step: (a) is the
original position of the skins between two circles; (b) and (c) shows when P1and R1are
moved. In this example P2does not need to be changed. In (d), the final result can be seen
after moving R2.
smallest circle cisthat contains the closed shape (using e.g., [27]), whose center
can be used as the center of the shape.
For the tangent lengths, dbi1and dbi2can be calculated directly. To deter-335
mine dki1and dki2we can use the radical lines of cisand ci+1s. Once the centers
of the shapes are defined, λi1and λi2can be computed as well.
6. Extensions
6.1. Skinning of spheres
Kunkli and Hoffmann defined a technique to extend their planar algorithm340
to higher dimensions. For an admissible sequence of spheres, the aim is to
construct a G1continuous surface that touches each sphere along a circle and
is tangent to the spheres. To create the skinning surface, touching circles have
to be defined: the method reduces the problem to the planar case and obtains
23
Algorithm 4 Solution to the intersecting skins
Input: control polygons P0P1P2P3and R0R1R2R3
Output: modified control polygons P0P1P2P3and R0R1R2R3
1: for j=0;j<4;j++do
2: Lj=P0P1∩RjR(j+1)%4
3: Mj=R0R1∩PjP(j+1)%4
4: P1=Ljwhere distance(P0, Lj)is the smallest
5: R1=Mjwhere distance(R0, Mj)is the smallest
6: for j=0;j<4;j++do
7: Lj=P2P3∩RjR(j+1)%4
8: Mj=R2R3∩PjP(j+1)%4
9: P2=Ljwhere distance(P3, Lj)is the smallest
10: R2=Mjwhere distance(R3, Mj)is the smallest
the touching circles based on the 2D solution. Patches between each pair of345
neighboring spheres are defined by Hermite interpolants: corresponding points
are defined on each pair of touching circles and get rotated along the circles
themselves. Tangent lengths are computed for the corresponding points on the
circles based on the radical plane of the two spheres (for details of computation
see [1]).350
The method of Bastl et al. uses the same approach to determine the touching
circles (in their case, it is possible to obtain one or two touching circles). Since
it is also based on the planar algorithm, and — as shown in Section 2.2 — the
localization of touching points has insufficiencies, it is clear that the problem
also occurs in higher dimension. However, they use a blending technique [28] to355
create patches between the spheres. The tangent length calculation that they
defined for their planar solution is not used in their spatial algorithm.
Our approach is to extend our skinning method to 3D, fundamentally based
on the technique defined in [1]. The requirements for an admissible configuration
of spheres are analogous to the ones declared in Section 3. We construct a G1
360
continuous surface that is tangent to the spheres and touches each sphere along
24
one or two circles. If a sphere has two touching circles, the spherical part defined
by them is also considered as a part of the skin. These circles are derived from
the touching points that we can acquire from the planar algorithm, and we use
Hermite curves to define the surface between two neighboring spheres.365
The tangent length calculation in Section 3.3 can be directly applied to
spheres; when needed, we can use surface area instead of arc lengths and cir-
cumference. Since the touching points have different tangent lengths for the
“left” and the “right” skins on each circle in the planar case, in 3D the cal-
culated tangent lengths have to be interpolated along the touching circles. As370
a final step, a spherical patch is calculated when a sphere has two touching
circles defined. Fig. 14 shows the output of our algorithm extended to three
dimensions.
Figure 14: Extension to 3D. The skins consist of Hermite interpolants (green) and spherical
parts (gray).
The final parametric surface is rational, since it can use the rational pa-
rameterization of the touching circle in its first parameter and will follow the375
polynomial parameterization of the Hermite arc in its second parameter. Here
we note, that the surface can be optimized to avoid unnecessary distorsion by
applying rotation-minimizing frames, but it certainly increases the computa-
tional time (for details see [24]).
6.2. Branching380
Branching is possible in both the planar and spatial cases. To create branches
in 2D, the approach of Bastl et al. can be directly applied, for details see [2].
Fig. 15 shows the output of our skinning algorithm for a complex set of circles
with several branches.
25
Figure 15: Branching in 2D — skinning a complex set of circles with our new method.
Based on [1], Bana et al. [19] provided an algorithm for branching in 3D: they385
define a boundary curve with the help of the touching circle on the sphere to
which the branch is connected. Applying this approach, however, one can face
an obstacle as a sphere often has two touching circles in our algorithm. Even
though it is possible to define the boundary curve in two parts, the resulting
joining surface and the original surface would not assure G1continuity.390
Bastl et al. also extended their algorithm to branch skinning surfaces, but
besides a few figures as examples, they did not provide exact description how
they can determine the new touching circles.
We now give an algorithm for branching in 3D. Let us consider neighboring
spheres si−1, si, si+1, and let sicbe a sphere of a new branch connecting to395
si(i∈ {2, . . . , n −1}). The idea is to create two touching circles on si, then
construct a third one to which the new skinning surface can join. In case the first
two circles intersect each other, we redefine them in a way that they touch each
other at one point — thus the skinning surface provides a smooth connection.
The detailed algorithm is as follows. As Fig. 16 (a) shows, touching circle ti1
400
is originated from applying the basic algorithm to sic,si, and si+1; while ti2is
obtained from si−1,si, and sic. Let Simbe the plane that contains the center
of siand is perpendicular to the intersection line of the planes of ti1and ti2. As
it can be seen in Fig. 16 (b), intersecting Simwith siyields circle cim. Then the
points Pi1and Pi2are defined by intersecting cimwith ti1and ti2, respectively.405
The touching circle to which siccan connect is ti3, which goes through Pi1and
26
(a) (b)
(c) (d)
(e)
Figure 16: The construction of the touching circles for branching. When the touching circles
ti1and ti2intersect, we provide an algorithm to redefine the touching circles to avoid the
intersection.
27
Pi2, and its center is the midpoint of the two points (see Fig. 16 (c)). If ti1and
ti2intersect, then we define the bisector plane of the planes of ti1and ti2which
intersects ti3. By intersecting this plane with cim, the resulting point that is
farther from sicis Mi, as in Fig. 16 (d). Then ti1is redefined so that it goes410
through Miand Pi1, and its center is the midpoint of the two points. ti2is
redefined analogously using Miand Pi2. As shown in Fig. 16 (e), ti1and ti2
touch each other only at Mi, and they touch ti3at Pi1and at Pi2, respectively.
Fig. 17 shows the output of our branching algorithm in 3D. In theory it is
possible to construct branching without selecting one specific sphere by com-415
puting a suitable joining surface between more than two spheres as well, but it
is beyond the scope of this paper. For a possible solution of this kind of problem
see [25].
Figure 17: Branched skinning surfaces using our algorithm. The skins consist of Hermite
interpolants (green) and spherical parts (gray).
7. Conclusion
We studied the main advantages and disadvantages of recently developed420
circle and sphere skinning methods, and we provided a new, improved algorithm
preserving the benefits of the method by Bastl et al. [2] and the one by Kunkli
and Hoffmann [1], whilst eliminating their drawbacks. We have seen that several
problems arose in the aspect of dynamic modeling, thus the aim of our new
skinning method was to give a solution to these inefficiencies. We have also425
extended the algorithm to the three-dimensional space and to branched shapes.
We introduced an algorithmic approach to the problem of intersecting skins as
28
well. The new, improved technique is fast, can be used for real-time modeling
due to the smooth alteration of skins, and provides good results even in extreme
cases.430
Acknowledgements
The first author was supported by the ´
UNKP–17–3 New National Excel-
lence Program Of The Ministry Of Human Capacities.
The second author was supported by the ´
UNKP–17–4 New National Ex-
cellence Program Of The Ministry Of Human Capacities.435
The third author’s research was supported by the grant EFOP-3.6.1-16-
2016-00001 (Complex improvement of research capacities and services at Esz-
terhazy Karoly University) and by the construction EFOP-3.6.3-VEKOP-16-
2017-00002. The project was supported by the European Union, co-financed by
the European Social Fund.440
References
[1] R. Kunkli, M. Hoffmann, Skinning of circles and spheres, Computer Aided
Geometric Design 27 (8) (2010) 611–621. doi:10.1016/j.cagd.2010.07.
003.
[2] B. Bastl, J. Kosinka, M. L´aviˇcka, Simple and branched skins of systems of445
circles and convex shapes, Graphical Models 78 (2015) 1–9. doi:10.1016/
j.gmod.2014.12.001.
[3] G. Farin, Curves and Surfaces for CAGD: A Practical Guide, 5th Edition,
Academic Press, 2002.
[4] G. Farin, J. Hoschek, M.-S. Kim, Handbook of Computer Aided Geometric450
Design, Elsevier, 2002.
[5] J. Hoschek, D. Lasser, Fundamentals of Computer Aided Geometric Design,
A.K. Peters, 1993.
29
[6] L. Piegl, W. Tiller, Surface skinning revisited, The Visual Computer 18 (4)
(2002) 273–283. doi:10.1007/s003710100156.455
[7] L. Piegl, W. Tiller, Surface approximation to scanned data, The Visual
Computer 16 (7) (2000) 386–395. doi:10.1007/PL00013393.
[8] X. Yang, J. Zheng, Approximate T-spline surface skinning, Computer-
Aided Design 44 (12) (2012) 1269–1276. doi:10.1016/j.cad.2012.07.
003.460
[9] C. D. Woodward, Skinning techniques for interactive B-spline surface in-
terpolation, Computer-Aided Design 20 (8) (1988) 441–451. doi:10.1016/
0010-4485(88)90002-4.
[10] D.-J. Yoo, Three-dimensional surface reconstruction of human bone using
a B-spline based interpolation approach, Computer-Aided Design 43 (8)465
(2011) 934–947. doi:10.1016/j.cad.2011.03.002.
[11] Y. Li, W. Chen, Y. Cai, A. Nasri, J. Zheng, Surface skinning using peri-
odic T-spline in semi-NURBS form, Journal of Computational and Applied
Mathematics 273 (2015) 116–131. doi:10.1016/j.cam.2014.05.026.
[12] F. F. Leymarie, P. Aparajeya, C. MacGillivray, Point-based medialness for470
movement computing, in: Proceedings of the 2014 International Workshop
on Movement and Computing, MOCO ’14, ACM, New York, NY, USA,
2014, pp. 31:31–31:36. doi:10.1145/2617995.2618001.
[13] M. L. Connolly, Analytical molecular surface calculation, Jour-
nal of Applied Crystallography 16 (1983) 548–558. doi:10.1107/475
S0021889883010985.
[14] H. Edelsbrunner, Deformable smooth surface design, Discrete & Computa-
tional Geometry 21 (1999) 87–115. doi:10.1007/PL00009412.
[15] J. Rossignac, B. Whited, G. Slabaugh, T. Fang, G. Unal, Pearling: 3D in-
teractive extraction of tubular structures from volumetric images, in: MIC-480
30
CAI Workshop on Interaction in Medical Image Analysis and Visualization,
2007.
[16] S. Piskin, G. Unal, A. Arnaz, T. Sarioglu, K. Pekkan, Tetralogy of Fallot
Surgical Repair: Shunt Configurations, Ductus Arteriosus and the Circle
of Willis, Cardiovascular Engineering and Technology 8 (2017) 107–119.485
doi:10.1007/s13239-017-0302-5.
[17] Z. Ji, L. Liu, Y. Wang, B-mesh: A modeling system for base meshes of
3d articulated shapes, Computer Graphics Forum 29 (7) 2169–2177. doi:
10.1111/j.1467-8659.2010.01805.x.
[18] X. Zhu, X. Jin, L. You, Analytical solutions for tree-like structure modelling490
using subdivision surfaces, Computer Animation and Virtual Worlds 26 (1)
(2015) 29–42. doi:10.1002/cav.1563.
[19] K. Bana, K. Kruppa, R. Kunkli, M. Hoffmann, KSpheres - an efficient
algorithm for joining skinning surfaces, Computer Aided Geometric Design
31 (2014) 499–509. doi:10.1016/j.cagd.2014.08.003.495
[20] J.-M. Thiery, ´
E. Guy, T. Boubekeur, Sphere-Meshes: Shape Approximation
Using Spherical Quadric Error Metrics, ACM Transactions on Graphics
(TOG) 32 (6) (2013) 178:1–178:12. doi:10.1145/2508363.2508384.
[21] G. Slabaugh, G. Unal, T. Fang, J. Rossignac, B. Whited, Variational Skin-
ning of an Ordered Set of Discrete 2D Balls, in: F. Chen, B. J¨uttler (Eds.),500
Advances in Geometric Modeling and Processing, GMP 2008, Springer,
2008, pp. 450–461.
[22] G. Slabaugh, J. Rossignac, B. Whited, T. Fang, G. Unal, 3D Ball Skinning
using PDEs for Generation of Smooth Tubular Surfaces, Computer-Aided
Design 42 (2010) 18–26. doi:10.1016/j.cad.2009.03.004.505
[23] B. Blaschitz, Skinning of Circles and Spheres by Geometric Optimization
in Minkowski Space, Journal for Geometry and Graphics 18 (2) (2014)
159–172.
31
[24] M. Hoffmann, J. Monterde, E. Troll, Blending of spheres by rotation-
minimizing surfaces, Journal for Geometry and Graphics 19 (2) (2015)510
189–200.
[25] M. Bizzarri, M. L´aviˇcka, J. Kosinka, Skinning and blending with rational
envelope surfaces, Computer-Aided Design 87 (2017) 41–51. doi:10.1016/
j.cad.2017.02.002.
[26] K. Kruppa, R. Kunkli, M. Hoffmann, Comparison of skinning methods515
website, https://skinning.github.io/skinning-methods/, [Online; ac-
cessed October 20, 2018].
[27] E. Welzl, Smallest enclosing disks (balls and ellipsoids), in: H. Maurer
(Ed.), New Results and New Trends in Computer Science, Springer Berlin
Heidelberg, Berlin, Heidelberg, 1991, pp. 359–370.520
[28] M. Bizzarri, M. L´aviˇcka, Parameterizing rational offset canal surfaces via
rational contour curves, Computer-Aided Design 45 (2013) 342–350. doi:
10.1016/j.cad.2012.10.017.
32
