Skinning of circles and spheres

Abstract

Skinning of an ordered set of discrete circles is discussed in this paper. By skinning we mean the geometric construction of two G1 continuous curves touching each of the circles at a point, separately. After precisely defining the admissible configuration of initial circles and the desired geometric properties of the skin, we construct the touching points and tangents of the skin by applying classical geometric methods, like cyclography and the ancient problem of Apollonius, finding touching circles of three given circles. Comparing the proposed method to a recent technique (Slabaugh et al., 2008, 2010), larger class of admissible data set and fast computation are the main advantages. Spatial extension of the problem for skinning of spheres by a surface is also discussed in detail.

Full text

Skinning of circles and spheres
R.Kunklia, M.Hoffmanna,b,1,∗
aDepartment of Computer Graphics and Image Processing, University of Debrecen, Egyetem sqr. 1,
H-4010 Debrecen, Hungary
bInstitute of Mathematics and Computer Science, K´aroly Eszterh´azy College, Le´anyka str. 4, H-3300 Eger,
Hungary
Abstract
Skinning of an ordered set of discrete circles is discussed in this paper. By skinning
we mean the geometric construction of two G1continuous curves touching each of
the circles at a point, separately. After precisely defining the admissible configura-
tion of initial circles and the desired geometric properties of the skin, we construct
the future touching points and tangents of the skin by applying classical geometric
methods, like cyclography and the ancient problem of Apollonius, finding touching
circles of three given circles. Comparing the proposed method to a recent technique
(Slabaugh et al.,2008; Slabaugh et al.,2009), larger class of admissible data set and fast
computation are the main advantages. Spatial extension of the problem for skinning of
spheres by a surface is also discussed in detail.
Keywords: interpolation, skinning, circles, spheres, cyclography
2000 MSC: 68U05
1. Introduction
Interpolation of geometric data sets is of central importance in Computer Aided
Geometric Design. If geometric data consist of points, then we have several, now stan-
dard methods to interpolate them (Farin,1997; Piegl and Tiller,1995; Hoschek and Lasser,1993).
If, however, the data set consists of other types of objects (e.g. circles), interpola-
tion is transferred to skinning, that is construction of a curve or surface which touches
each of the objects and somehow bounds the given data. Since this is a largely ill-
posed problem, constraints have to be defined for the data set as well as for the desired
solution.
In this paper we address the problem of skinning of a sequence of circles. This
problem is sometimes called 2D ball skinning, with a natural extension to 3D, where a
skinning surface of a set of given spheres is to find.
∗Corresponding author
Email addresses: rkunkli@gmail.com (R.Kunkli), hofi@ektf.hu (M.Hoffmann)
1Author supported by Bolyai Research Fellowship of the Hungarian Academy of Sciences
Preprint submitted to Elsevier May 3, 2010

Since the problem of skinning is not necessarily defined in a unique way in the
literature, here we formally describe what type of input is admissible for us and what
type of output we are searching for.
Definition 1. A sequence of circles C={c1,c2,c3, . . . , cn}(n∈N) is called admissible
configuration if the following conditions are fulfilled (didenotes the closed disk defined
by circle ci):
•di1
n
[
j=1,j,i
dj,i∈ {1,2, . . . , n}
•di∩dj=∅,i,j∈ {1,2, . . . , n},j<{i−2,i−1,i,i+1,i+1}
•if di−1∩di+1,∅, then di−1∩di+1⊂di
These assumptions also yield
ri<
i+1
[
j=i−1
dj,i=2, .., n−1
where riis the radical center of three consecutive circles ci−1,ci,ci+1. In Fig. 1 one can
see an admissible configuration and a configuration where some circles do not fulfill
the conditions.
Figure 1: An admissible configuration (left) and a non-admissible configuration (right) where circles c2−c4,
c6and c8do not fulfill the conditions: circles are not allowed to be entirely in the union of other circles while
radical center r3of three consecutive circles must be out of c3
Since the position of radical center will be of central importance in the algorithm,
we briefly remind the reader to the definition and computation of it. The radical axis (or
radical line) of two circles is the locus of points at which tangents drawn to both circles
have the same length. Since it is evidently orthogonal to the line passing through the
centers, it is enough to compute the distance of the axis to the centers:
d1=1
2




d+r2
1−r2
2
d




d2=1
2




d−r2
1−r2
2
d





2

where dis the distance of the two centers, r1,r2are the radii of the circles. By a
classical theorem of Monge the radical lines of three circles are either concurrent in a
point, known as radical center, or parallel iffthe three circle centers are congruent.
Now we define the desired output.
Definition 2. Given an admissible configuration of circles C={c1,c2,c3, . . . , cn}, we
are looking for two, at least G1continuous curves s(t) and ¯
s(t), called skins of the given
circles satisfying the following requirements (see also Fig. 2):
•There is a point of contact pi∈cifor all i=1, ..., nsuch that pi∈s(t) and tangent
lines of circle ciand s(t) are identical at pi. Analogously exist points ¯
pifor ¯
s(t).
•Tangent vector viof skin s(t) at pican be rotated to the direction of the center of ci
by 90◦in clockwise direction. Analogously this rotation is in counterclockwise
direction for tangent vectors of ¯
s(t).
•pi1
n
[
j=1,j,i
dj,and ¯
pi1
n
[
j=1,j,i
dj,i∈ {1,2, . . . , n}
For the sake of simplicity s(t) will be called ”left” skin and ¯
s(t) will be called ”right”
skin, based on the second requirement which ensures us that running along the skins, at
the points of contact the circles will always be in one (right or left) side of the curves.
Note, that due the last restriction, points of contact on the actual circle are required
to be out of other circles, which seemed to us a natural condition for skinning.
This or similar problem - beside its theoretical interest - frequently arises in ap-
plications like designing tubular structures, covering problems, molecule modeling
(Cheng and Shi,2005; Edelsbrunner,1999)). Medical image processing applies these
methods in e.g. blood vessel reconstruction (Whited et al.,2007; Slabaugh et al.,2009).
In computer animation, characters can also be constructed from a skeletal structure and
a corresponding geometric skin (Singh and Kokkevis,2000).
Figure 2: Two curves at left satisfying all the requirements to be skins. At right there are two curves which
also touch each circles but they do not fulfill the requirements: some of the touching points are inside of
other circles and separation of the two sides is not appropriate
The paper is organized as follows. After discussing the previous approaches, in
Section 3 we describe the new method and show that for an admissible configura-
tion of circles it always works. Detailed algorithm, several examples and comparison
3

to Slabaugh’s method can also be found in this section. Spatial extension to sphere
skinning by a surface is explained in Section 4. Conclusions and possible further im-
provements close the paper.
2. Previous work
The first and most natural approach of the problem would be the application of
the deep theoretical knowledge of the computation of envelope curves and surfaces,
dated back to Monge (Monge,1850), who first dealt with canal surfaces. Skin is def-
initely not an envelope, since this latter notion is defined for continuous data set, for
a one- or two-parameter family of curves or surfaces. The first requirement at De-
finition 1, however may be considered as the discrete version of envelope property.
An important contribution of this topic with computational aspects is the PhD thesis
of Josef Hoschek (Hoschek,1964). Since then a large number of papers have dealt
with envelope design, most of them with numerical computation (for the survey see
e.g. (Farin et al.,2002)). For circles and spheres, a recent exact solution is Peternell’s
method which is based on a cyclographic approach. In 2D cyclography defines a one-
to-one correspondence between the oriented circles of the plane and points in space
by cones. This way the sequence of given circles can be transformed to a sequence
of spatial points. An interpolating curve through these points can be defined by any
standard method and finally points of this spatial curve can be transferred back to
circles on the plane by the cones. The envelope of these circles is obtained as the
intersection of the plane and the envelope surface of the cones. Similar correspon-
dence works for spheres and points in 4D space. For a more detailed description, see
(Kruithof and Vegter,2006; Peternell et al.,2008) .
Figure 3: Given a set of discrete circles (black), classical interpolation may yield further circles (dashed
blue) in a way that the skin cannot be constructed for the original circles since the new set of circles do not
satisfy the requirements to form an admissible configuration (positions pointed by red arrows)
Although the papers mentioned above do not deal with skinning, one may try to
transfer the discrete data set to a one-parameter family of circles/spheres, having cen-
ters and radii as functions of a parameter. These functions can be achieved from the set
of discrete data by classical interpolating methods in the space, but this way the set of
4

new circles do not necessarily satisfy our requirements to be an admissible configura-
tion, as one can observe in Fig. 3.
A recent approach to the skinning problem for circles and spheres is Slabaugh’s
method (Slabaugh et al.,2008; Slabaugh et al.,2009). It is an iterative way to construct
the desired curves or surfaces. Let the discrete sequence of circles with centers ciand
radii ri,(i=1,...,n) be given. Initially pairs of Hermite arcs are defined between
the consecutive circles. Considering two neighbouring circles ciand ci+1, an initial
Hermite arcs are specified with touching points pi,pi+1and tangents vi,vi+1for the
skin. The final positions of these points and tangents are obtained by the end of several
iteration steps.
Figure 4: A one-sided (left) skin obtained by the method of Slabaugh (left figure, from
(Slabaugh et al.,2008)). For two-sided skin they constrain the points of contact to be separated by 180 de-
grees, but this way some of the points of contact may fall into other circles (middle figure). Right figure
shows the result of the proposed method: the green touching points of the left skin are almost identical to the
ones obtained by Slabaugh. The right (lower) skin has significantly different touching points.
The iteration itself is based on the minimization of a predefined energy function.
For computational reasons the positions of the touching points and the tangents are
transferred into one single variable, namely the angle αibetween the xaxis and the
radius pointing towards the touching point.
pi=oi+"ricosαi
risinαi#,
vi="−kisinαi
kicosαi#,
where kiis a predefined constant for each circle, half of the distance between the centers
oiand oi+1.
The method provides energy-minimized, C1continuous skin without any user in-
teraction, which, in this sense, the optimal solution, if it exists. But the method also
suffers from problems. The touching points are not guaranteed to be out of the circles,
especially not for two-sided skin, when the two touching points at each circle are con-
strained to be separated by 180◦degrees (c.f. Fig. 4). There are simple configurations
when it is theoretically impossible to find two diametrically opposite points on a circle
being out of other given circles, see e.g. the second circle in the leftmost figure of Fig.
9 or the lower right figure of Fig.5. Slabaugh’s method does not provide acceptable
skin by our definition, especially the last requirement in Definition 2 is not necessarily
fulfilled. From this point of view our method can handle a larger class of data sets. One
5

would try to omit this 180◦constrain from that method but then we are facing to solve
the separation of touching points, which is far from being trivial in a numerical itera-
tion. The same problem arises in 3D-ball skinning, where Slabaugh’s method allows
only great circles as possible touching circles. A further problem is that the conver-
gence of iteration to a global minimum is not proved and the number of iterations can
be over 100 which is time consuming. Moreover, the process has to be restarted after
any modification of data, thus this method is not suitable for real time modeling and
adjustment. For comparison to our method: the one-sided skin in Figure 4 (left) has
been computed by Slabaugh in 143 milliseconds ((Slabaugh et al.,2008), the two-sided
skin (right) has been computed by our method in 14 milliseconds (both at single core
3GHz CPU).
3. Our method
Now our task of skinning can be divided into the following steps:
•check if the given circles form an admissible configuration
•find appropriate points of contact for each circle ci,i=1, ..., n
•separate points into two classes, denoted by piand ¯
pifor left and right skin
•define tangent vectors viand ¯
vi
•compute the skins
The admissible criteriae of Definition 1 can be tested by elementary computation.
3.1. Localization of touching points
As we have learned from the previous section, the localization of possible touching
points on the given circles is essential for skinning. Our solution of finding the touching
points is based on the circles of Apollonius, which are touching circles to three given
circles. This ancient construction provides suitable touching points to each inner circle
in the data set, while the first and last circles are handled in a simple special way.
A classical result (Muirhead,1896) on the possible positions of three circles and
solutions of problem of Apollonius states that for three given circles ci−1,ci,ci+1,(i<
{1,n}) which satisfy our admissible conditions, exactly one of the following statements
holds:
•There exist exactly two circles touching externally by all the three given ones
•There exist exactly two circles touching internally by all the three given ones
•There exists exactly one circle touching externally and another one touching
internally by all the three given ones.
6

Figure 5: Considering three consecutive given circles (black), the touching points of the second circle are
computed by the two circles of Apollonius (red). Orientation of circles help to find the correct solutions. The
method always works for circles in admissible configuration
These touching circles together with their touching points can be found by a classi-
cal geometric method, the cyclography. Consider the three given circles to be oriented,
this yields eight different possibilities. The touching circle is always expected to have
the same orientation at each touching point as the given circles. This way the eight
different solutions can be distinguished by the original orientation of the given circles.
In two cases all the three given circles have the same orientation. The solutions of
these two cases provide the touching points piand ¯
piof the circle ci(c.f. Fig.5). These
points can be constructed and computed by known methods (M¨
uller and Krames,1928)
(Coaklay,1860). Moreover, it is also proved (M¨
uller and Krames,1928) that these touch-
ing points and the radical center of the three given circles are collinear (see Fig.6).
In the very special situation, when centers of the three circles are collinear and
having equal radii, simply the common tangent lines give the points of contact.
After defining the touching points at each inner circle, the first and last circles in the
sequence have to be handled as well. Touching points to these two circles can also be
defined by the common external tangent lines of the first two and the last two circles,
respectively.
3.2. Separation to left and right group
Finally two points have been localized at the circles ciwhich all satisfy the last
requirement of Definition 2. The next step is to separate them for ”left” and ”right”
classes, i.e. to identify which one should be denoted by piand which one by ¯
pi.
At first it is proved in (M¨
uller and Krames,1928) that these points of cican always
be separated by the circle with radical center rias center and intersecting orthogonally
the given three circles ci−1,ciand ci+1. Moreover it is mentioned that the radical center
7

Figure 6: Given three circles ci−1,ciand ci+1in admissible configuration, one can find the future points of
contact for skinning at ciby the two special solutions of the problem of Apollonius. The three possible
situations: one circle touching externally and another one touching internally by all the three given ones
(upper left); two circles touching externally (upper right); and two circles touching internally (middle).
Radical center and touching points are collinear.
and the two touching points are collinear. Thus the separation can easily be computed
by the following steps (see notations of Fig. 7): if the vector oi−1oican be rotated to
the direction of vector oi−1oi+1by a positive angle (in counterclockwise direction, with
less than 180◦) then the touching point being closer to the radical center riwill be in
the left group, i.e. will be denoted by pi. If the direction of rotation is opposite (as it
is for the next circle in Fig. 7) then the touching point closer to the radical center ri+1
is in the right group: ¯
pi+1. Special attention must pay to the first and last circle as well
as for circles with collinear centers. In these cases the vector oi−1oiis rotated to the
direction of oi−1piand the angle is similarly measured and evaluated as above.
3.3. Definition of tangent vectors
Finally we have two groups of well defined touching points on the circles. Between
each pair of points Hermite interpolation curve will be computed at each group, sepa-
rately. To these arcs, one has to define the length of the tangent vectors at these points
(the direction of the tangents is inherited from the actual circle). In Slabaugh’s method
the length of the tangent vectors was a simple function of the radius of the current
circle. This method works well if there is no large difference between the radii and
the distance of the consecutive circles. Contrary to that method we specify the length
of the tangent in a way that the radii and the distance of the circles, as significant in-
formation, are also incorporated. The radical line of two circles provides information
about the radii and the distance of the circles as well. Thus we use this line to obtain
8

Figure 7: Grouping of the constructed touching points into two groups, ”left” and ”right” (green and blue).
Detailed explanation can be found in the text.
unified information about the positions and size of the circles. For two consecutive
circles and touching points piand pi+1the distance of these points to the radical line
are computed, and the lengths of the tangent vectors at these points are settled to be
twice of these distances.
Figure 8: Definition of tangent vectors at the touching points by the help of radical line ri
3.4. Construction of the skin
Our final step is to construct the curve which is now a simple interpolation problem
for given points piand tangent vectors vi. At this point it is irrelevant that these data
are computed from a set of circles, so we have to emphasize that other interpolation
9

methods may work as well as our choice, the Hermite interpolation. We define a cubic
curve q(t), t∈[0,1], where
q(0) =pi,q(1) =pi+1,q0(0) =vi,q0(1) =vi+1,
and q(t)=H3
0(t)q(0) +H3
1(t)q(1) +H3
2(t)q0(0) +H3
3(t)q0(1),t∈[0,1],
where
H3
0(t)=2t3−3t2+1
H3
1(t)=−2t3+3t2
H3
2(t)=t3−2t2+t
H3
3(t)=t3−t2.
The Hermite interpolation arcs computed from these data serve as a G1continuous skin
of the given circles. Results can be seen in Fig. 9, comparison to Slabaugh’s method is
in Fig. 4 and Fig. 10.
Figure 9: Results of the proposed method. It works for rather complicated data sets and gives correct result
in the simplest case as well.
4. Extension to spheres
As aforesaid, we can define the problem in larger dimensions, too. Given an or-
dered set of (hyper)spheres, we are looking for a skinning (hyper)surface with similar
properties as it had in 2D. Definition 1 of admissible configuration of circles can di-
rectly be applied to spheres. The definition of skin is however, different from the one
given in Def. 2. We would like to obtain a G1continuous surface, which touches each
sphere along a circle, that is tangent to the spheres.
Definition 3. Given an admissible configuration of spheres C={s1,s2,s3, . . . , sn}, we
are looking for a G1continuous surface s(φ, t) called skin of the given spheres satisfying
the following requirements:
10

Figure 10: Comparison of Slabaugh’s method (left) and the proposed method (right). Note the difference in
the zoomed part below.
•There is a circle of contact (touching circle) cifor all i=1, ..., nsuch that the
skin s(φ, t) and sphere sihave common tangent planes at each point of ci. Circle
ciis an isoparametric curve of s(φ, t)
•ci1
n
[
j=1,j,i
dj,i∈ {1,2, . . . , n}.
We have to emphasize again that, due to the last restriction, circle of contact on the
actual sphere is required to be out of other spheres, which is a natural condition for
skinning from our point of view.
Steps of our solution are analogous to that ones applied in the planar case.
First of all, we have to localize the touching circles with centers ˜
oiand radii ˜
Ri
(i=1,...,n). For this step we can invoke the solution of the planar problem. Let us
consider a sphere si, where i=2, . . . , n−1, that is we exclude the first and the last
spheres for a moment (Fig.11). Now consider the plane Pi, determined by the centers
oi−1,oi,oi+1of the considered sphere and its neighbours. Intersecting the spheres by
this plane we obtain three circles. With the help of the above mentioned planar method
with Apollonius circles, we can find two points in the middle circle. There exists
exactly one plane Ti(i=2, . . . , n−1), which passes through these two points and
orthogonal to plane Pi. The intersection of sphere siand this orthogonal plane Tiis the
touching circle for the future skinning surface. We can localize a circle by this method
on every sphere, which has two neighbours.
The Appollonius problem itself can also be generalized in 3D, where touching
spheres of three given spheres have to be found. The envelope of these spheres is the
Dupin cyclide, which surface is widely used in CAGD (for an overview, see (Farin et al.,2002;
11

Figure 11: Touching circle localization on the sphere si, where i,1,n.Ti⊥Pi, where Tiis the plane of
the touching circle and Piis the plane passing through the centers. Dashed line shows Apollonius circles
as solution of the planar problem in Pi. Note that in general the touching circle is not a main circle of the
sphere si
Figure 12: Construction of touching circle on s1
Pratt,1990; Pratt,1995)). The touching circle we constructed now is identical to the one
in which the Dupin cyclide defined by the three given spheres si−1,si,si+1touches the
sphere si.
Touching circles for the first and the last spheres have to be defined in a different
way. Let us consider the first and the second spheres and the regular cone which touches
both spheres. The touching circle of this cone on the first sphere will be the circle for
the skinning surface as well (Fig. 12). Circle on the last sphere is defined analogously.
It directly follows from the planar construction, that this method always works in
every admissible case.
Now we obtained touching circle with center ˜
oiand radius ˜
Rion each sphere, thus
we can start to create the skin, following the ideas developed in the planar case: patches
are defined successively to each pair of spheres using Hermite interpolants through
corresponding points of the touching circles.
Consider the future patch si(φ, t) of the skin between touching circle cion sphere
siand touching circle ci+1on sphere si+1. Circle ciis the isoparametric curve si(φ, 0),
while ci+1is the isoparametric curve si(φ, 1) of this patch. At first we will define the
12

starting point on cias zi=si(0,0) and on ci+1as zi+1=si(0,1), then rotating them
by the same angle φalong the circles, corresponding pairs of points zφ
i=si(φ, 0),
zφ
i+1=si(φ, 1) will be defined. Lengths of tangent vectors are computed by the help of
the radical plane of the two spheres. Lines of all the tangents pass through the pole wi
of the plane of the touching circle with respect to the sphere si(i=1, . . . , n), see Fig.
13. To avoid unnecessary torsion, corresponding points are selected by the help of a
fixed spatial direction e(which can be e.g. the direction of the zaxis, not parallel to
any of the vectors wi−˜
oi).
Figure 13: Computation of tangent lengths is analogous to the spatial case, using radical plane Mi. Corre-
sponding points ziand zi+1are connected by isoparametric curve of the skin surface.
Let s: → {−1,1}and p: {si} → {−1,1}be functions defined by
s(x)=






−1 if x<0,
1 else
p(si)=








sD wi−oi
kwi−oik,oi+1−oi
koi+1−oikE if i,n,
sD wi−oi
kwi−oik,oi−oi−1
koi−oi−1kE else
where h,iis the standard inner product.
Figure 14: Result of the proposed algorithm. This data set has also been used by Slabaugh in
(Slabaugh etal.,2009), but, contrary to our algorithm, symmetric data set does not necessarily yield sym-
metric skin with Slabaugh’ method
13

Figure 15: A complex data set and its skin produced by the proposed algorithm. Note, that centers of spheres
are not coplanar.
Let ebe a fixed direction, e∦(wi−˜
oi), and let zibe defined by
zi=˜
oi+˜
Ri·e×p(si)·(wi−˜
oi)


e×p(si)·(wi−˜
oi)


(i=1,2, . . . , n).
Further corresponding points zφ
iof the touching circles are defined by rotating ziby
angle φaround the line passing through ˜
oiand having direction p(si)·(wi−˜
oi).
Let Mibe the radical plane of sphere siand si+1. We build our skinning surface up
from n−1 patches, where the ith patch is defined as:
si(φ, t)=H3
0(t)zφ
i+H3
1(t)zφ
i+1+
H3
2(t)·p(si)·2·d(Mi,zφ
i)·wi−zφ
i


wi−zφ
i


+
H3
3(t)·p(si+1)·2·d(Mi,zφ
i+1)·wi+1−zφ
i+1


wi+1−zφ
i+1


t∈[0,1], φ ∈[0,2π],i=1, . . . , n−1
where d is the Euclidean distance function and H3
iare the cubic Hermite-polynomials.
Results of the algorithm can be seen in Figure 14, 15 and 16.
5. Conclusion
An algorithm for skinning circles and spheres has been presented in the paper. Con-
trary to other approaches, our method works for large class of data sets, and it is proved
that for any admissible configuration the skin exists, however it does not minimize en-
ergy function and have only G1continuity. Applying classical tools from constructive
geometry, our method is robust and fast with real-time computing speed. Even if one
may wish to numerically minimize energy functions such as in Slabaugh’s work, the
presented skin can be applied as initialization of the iterative process, avoiding trivial
faults, local minima and consuming computation. Our technique provides a curve or
surface which can be modified in real time, and the shape is sensitive for the change of
radii and positions of the data set as well. Since the crucial point of the algorithm is
14

to find suitable touching points or touching circles, and it is independent of the current
interpolation method, extension of the algorithm using other types of surfaces can be a
future direction of research. Further improvements can be higher order continuity and
avoiding unnecessary intersection of the skin and the data set.
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Figure 16: The algorithm can handle distant spheres as well as sudden changes in size. Note that the upper
part of the stomach or the neck of the vase cannot be modeled by main circles as touching circles.
17