A NEW POTENTIAL FUNCTION FOR SELF INTERSECTING

GIELIS CURVES WITH RATIONAL SYMMETRIES

Yohan D. Fougerolle, Fr´ed´eric Truchetet

LE2i Laboratory, UMR CNRS 5158, University of Burgundy, 12 rue de la fonderie, 71200 Le Creusot, France

Johan Gielis

Genicap Lab BV, Wilheminaweg 1, 2042 NN, Zandwoort, The Netherlands

Keywords:

Gielis curves and surfaces, Implicit functions, Parametric functions, R-functions, Superquadrics, Symmetry.

Abstract:

We present a new potential ﬁeld equation for self-intersecting Gielis curves with rational rotational symme-

tries. In the literature, potential ﬁeld equations for these curves, and their extensions to surfaces, impose

the rotational symmetries to be integers in order to guarantee the unicity of the intersection between the

curve/surface and any ray starting from its center. Although the representation with natural symmetries has

been applied to mechanical parts modeling and reconstruction, the lack of a potential function for Rational

symmetry Gielis Curves (RGC) remains a major problem for natural object representation, such as ﬂowers

and phyllotaxis. We overcome this problem by combining the potential values associated with the multiple

intersections using R-functions. With this technique, several differentiable potential ﬁelds can be deﬁned for

RGCs. Especially, by performing N-ary R-conjunction or R-disjunction, two speciﬁc potential ﬁelds can be

generated: one corresponding to the inner curve, that is the curve inscribed within the whole curve, and the

outer -or envelope- that is the curve from which self intersections have been removed.

1 INTRODUCTION

Describing and modeling nature is fascinating and,

generally speaking, one of the most fundamental re-

search activities: whether to model physical behav-

iors or geometric structures, to describe or to recog-

nize natural shapes, every research community aims

at representing nature as accurately as possible. Clas-

sical models are largely based on isotropic spaces

with the Euclidean circle as (isotropic) unit circle.

In nature however, anisotropy is the rule and dif-

ferent ways of measuring or geometrizing exist. In

2003, halfway between the ﬁelds of botany and com-

puter graphics, Gielis et al. introduced the superfor-

mula (Gielis, 2003; Gielis et al., 2003), which can

be seen as a parametric formulation for generalized

circles or ellipses. Superellipses deﬁning anisotropic

unit circles led to notion of Minkowski distances and

Minkowski geometry (Thompson, 1996). Superel-

lipses have been extended to superquadrics in (Barr,

1981), which have found numerous applications due

to the limited number of shape parameters and their

ability to represent objects ranging from diamonds,

cubes, spheres, and any intermediate shape. More

interestingly, the superformula is now spreading to

other ﬁelds. For instance, it has been used in re-

cent work to study constant mean curvature surfaces

for anisotropic energies (Koiso and Palmer, 2008), in

clustering and data mining in (Morales and Bobadilla,

2008), and in ﬂuid dynamics in (Wang, 2008). Re-

cently, Natalini et al. have presented a numerical

algorithm to write down the explicit solution to the

Dirichlet problem for the Laplace equation in a star-

like domain (Natalini et al., 2008), and presented

closed form equations for various Gielis curves.

In 2005, based on the parametric formula pro-

posed in (Gielis, 2003), potential ﬁelds for 3D

Gielis surfaces have been proposed in (Fougerolle

et al., 2005). This representation has found appli-

cations for Gielis surface recovery for mechanical

parts (Bokhabrine et al., 2007). Unfortunately, one

of its major weakness is that the implicit ﬁeld equa-

tions require the rotational symmetries to be integers.

Such a restriction can be tolerated when manufac-

tured objects are represented. Unfortunately, as ini-

tially remarked in (Gielis, 2003), some natural objects

D. Fougerolle Y., Truchetet F. and Gielis J.

A NEW POTENTIAL FUNCTION FOR SELF INTERSECTING GIELIS CURVES WITH RATIONAL SYMMETRIES.

DOI: 10.5220/0001798200900095

In Proceedings of the Fourth International Conference on Computer Graphics Theory and Applications (VISIGRAPP 2009), page

ISBN: 978-989-8111-67-8

(a) (b)

Figure 1: a) Rose sepals. b) RGC with m = 5/2 and n

= n

= 0.43.

require the rotational symmetries to be rational num-

bers. Such symmetries are found in the phyllotaxy

of plants. Leaves are arranged in a helical or spiral

way around the stem. In rose, leaves are arranged

around the stem in a 5/2 arrangement, meaning that

the sixth leave will be precisely abovethe ﬁrst one and

the spacing between leaves is 144

◦

. In wild roses the

ﬁve petals are arranged in a plane, but the sepals in

the preceding (almost planar) whorl still show the he-

lical arrangement as they are still 144

◦

apart. This ar-

rangement can be seen using m = 5/2 in the superfor-

mula. It involves fusion of certain parts, while in the

center an open structure is created, giving rise to the

rose hip, as illustrated in Figure 1. Non-integer sym-

metries can be observed in biomolecules as well, e.g.

DNA and proteins in which non-integer symmetries

are observed frequently (Janner, 2001; Janner, 2005).

Complex objects that are deﬁned as Boolean opera-

tions between multiple globally deformed Gielis sur-

faces can be modeled and reconstructed (Fougerolle

et al., 2005; Bokhabrine et al., 2007). To transcribe

the Boolean predicates between 3D Gielis surfaces

into analytical equations, R-functions have been em-

ployed. The strategy adopted in this paper uses the

same tools and concepts, i.e. we use radial dis-

tance functions to build a 2D potential ﬁeld, and R-

functions for the transcription of Boolean combina-

tions into analytical equations. The difference is that

now we perform R-functions not to combine implicit

ﬁelds of several Gielis curves or surfaces, but the

multiple implicit values of the same rational Gielis

curve. Thus, we overcome the self-intersection issue

through an auto-R-function operation and build 2D

potential ﬁelds equation for RGCs, that can represent

for instance the envelope of the curve or its ”core”.

We present several advantages of this representation,

ranging from ﬂower modeling, from global shape to

petals, and its potential for further research directions,

such as parameter recovery and/or optimization.

The structure of the rest of paper is as follows: in

section 2 we recall the initial parametric deﬁnition of

Gielis curves, surfaces, and their associated potential

ﬁelds. In section 3 we brieﬂy present R-functions.

Using R-function and the initial Gielis formula, two

potential ﬁelds are presented for 2D RGCs in sec-

tion 4. Several strategies about extension to Rational

Gielis Surfaces are presented and discussed in section

5. We then present our future work and conclusions

in section 6.

2 GIELIS CURVES

In polar coordinates, the radius r(θ) of a Gielis curve

is deﬁned by:

r(φ) =



cos



mφ





sin



mφ





, (1)

with n

∈ R

, and a, b, and m ∈ R

∗

. Parameters a

and b control the scale, m represents the number of

rotational symmetries, n

, n

, and n

are the shape co-

efﬁcients. Regular polygons and superellipses can be

generated by setting the shape coefﬁcients to speciﬁc

values as shown in (Gielis, 2003).

Gielis only proposed the parametric formulation

for 2D curves. In the case of closed non self-

intersecting curve (m is positive integer), for a 2D

point P(x, y) one can deﬁne the following potential

ﬁeld:

(x,y) = 1−

−→

OPk

−→

OIk

= 1−

+ y

(θ)

. (2)

O is the center of the curve, and the point I =

r(θ(x,y)) corresponds to the intersection between the

curve and the half line [OP). If the symmetry param-

eter m is an integer, the intersection I is unique. If the

curve is closed, the sign of the potential ﬁeld F(x,y)

generated by equation 2 can be used to deﬁne a par-

tition of the 2D space. In this case, the set of points

where F(x,y) is positive corresponds to the inside of

the Gielis curve, the set of points where F(x,y) is neg-

ative corresponds to its outside, and the curve corre-

sponds to the zero-set of the potential ﬁeld.

In (Fougerolle et al., 2005), by setting a = b = 1 in

equation 1 and by considering a 3D Gielis surface as

the spherical product of two 2D Gielis curves, poten-

tial ﬁelds for 3D non self-intersecting Gielis surfaces

have been introduced as:

(x,y,z) = 1 −

(φ)

+ y

+ z

cos



(θ) − 1



+ 1

. (3)

Such representation of Gielis surfaces as iso-values

of a potential ﬁeld is crucial for Gielis surface recon-

struction from 3D data, because it is used to build var-

ious cost functions to be optimized.

A NEWPOTENTIAL FUNCTION FOR SELF INTERSECTING GIELIS CURVES WITH RATIONAL SYMMETRIES

3 R-FUNCTIONS

R-functions ﬁnd their origin in geometric algebra

based on logic and have been introduced by Vladimir

Logvinovich Rvachev in (Rvachev, 1967). Since

their introduction, R-functions have found direct ap-

plications in several ﬁelds, such as geometric mod-

eling and boundary value problems. For concise-

ness purpose, we brieﬂy present the most common R-

functions in this section. The reader is invited to refer

to the recent survey by Shapiro (Shapiro, 2007) for

more in depth presentation of R-functions and their

applications.

The simplest R-function is R

) and is de-

ﬁned by:

) =

1+ α



+ x

− 2αx



(4)

where α(x

) is an arbitrary symmetric function

such that −1 < α(x

) ≤ 1. Setting α to 1 leads

to the simplest and most popular R-functions:

R-conjunction min(x

) and R-disjunction

max(x

). Other useful R-functions with dif-

ferential and normalization properties, namely

and R

, are studied in detail in (Shapiro and

Tsukanov, 1999), and are respectively deﬁned by:

) =



+ x





+ x



(5)

where m is any even positive integer and

) = x

+ x



+ x



, (6)

for any even positive integer p. R

, R

, and R

func-

tions only handle two arguments. Rvachev introduced

the N-ary R-conjunction and R-disjunction to handle

more than two arguments, which are less restrictive

than R

, R

, and R

and more appropriate to RGCs.

The parameter m is an integer and corresponds to the

parameter used for R

functions.

i=n

i=1

(m)

≡

∑

i=1

(−1)

−|x

|)+

∏

i=1

+ |x

|).

(7a)

i=n

i=1

(m)

≡

∑

i=1

+|x

|)−

∏

i=1

(−1)

(|x

| − x

(7b)

If no speciﬁc constraint about differentiability is re-

quired, m can be set to zero, which leads to the

simpliﬁed version of N-ary R-conjunction and R-

disjunction deﬁned in equations 8a and 8b.

i=n

i=1

≡

∑

i=1

−|x

|)+

∏

i=1

+ |x

|). (8a)

(a) (b) (c) (d)

Figure 2: Examples of developing ﬂower buds. a) Ochna at-

ropurpurea: Development of ovary. b) Greyia sutherlandii

stage 1: sepals protect the newly formed petals. c) Greyia

sutherlandii stage 2: stamens develop with petals in the ﬁve

corners of the pentagon (sepals removed). d) Greyia suther-

landii stage 3: stamens and pistils develop.

i=n

i=1

≡

∑

i=1

+|x

|)−

∏

i=1

(|x

| − x

). (8b)

An R-function is a real-valued function character-

ized by some property that is completely determined

by the corresponding property of its arguments. More

speciﬁcally, the R-functions presented in this paper

have the property that their sign is completely deter-

mined by the signs of their arguments. In the follow-

ing section, we present how to use this property to

build a signed potential ﬁeld for RGCs.

4 POTENTIAL FIELD EQUATION

FOR RATIONAL GIELIS

CURVES

While RGC curves with multiple crossings and period

k (with k even) can be studied in a XY graph without

intersections, it is worthwhile to develop a potential

ﬁeld function which does take the intersections into

account and in which multiple function values do oc-

cur. As observed in rose, such intersections could in-

deed give rise to a center which can develop into the

rose hip. Alternatively, the multiple intersections can

lead to the separation of speciﬁc, isolated sectors in

which separate developmental processes may occur.

Such sectors are indeed observed using R-conjunction

of outer envelope and the complementary of the core

(Figure 5). At the bud stage, ﬂower development in-

volves the sequential initiation of various whorls in

well-deﬁned yet separated sectors. In Greyia suther-

landii (Figures 2(b), 2(c), and 2(d)) for example, af-

ter the formation of sepals in the ’corners’ petals are

formed and inbetween the formation of stamens oc-

curs.

We come back to the superformula initial formu-

lation and introduce rational symmetry: the symmetry

parameter m is no longer an integer and can be ratio-

nal, and a = b = 1. By deﬁnition, RGCs are self inter-

secting curves, i.e. the symmetry parameter m can be

GRAPP 2009 - International Conference on Computer Graphics Theory and Applications

Figure 3: Intersections between a RGC with m = 8/3 and a

half ray.

written as the ratio of two integers as m = p/q. The

parametric formulation of a RGC is then written as:

r(φ) =



cos



pφ





sin



pφ





, (9)

with p,q ∈ N

∗

. The parameter p is similar to m,

i.e. still represents the rotational symmetry number.

The parameter q corresponds to the maximum num-

ber of self intersections, and the angle θ now belongs

to [0,2qπ]. The condition for a RGC to be closed is to

verify that the radius r(θ) is identical for angles θ = 0

and θ = 2qπ, which can be written as r(0) = r(2qπ).

Figure 3 shows an example of a RGC with p = 8

and q = 3. For a given RGC, there exist multiple in-

tersections between a ray originated from its center

and the curve, i.e. there exist multiple polar coordi-

nates in [0,2qπ] that generate points lying both on the

curve and the ray, as illustrated in Figure 3, where

three intersections, noted I

and I

are detected.

The number of intersections depends on q and double

intersection points appear for angles θ = kp/q. For a

given point P(x,y), a ﬁrst angle θ

within [0,2π] can

be easily determined. The corresponding intersection

point I

can be computed as r(θ

). Other angles θ

and corresponding intersection points I

can easily be

determined using θ

= θ

+ 2kπ and I

= r(θ

). We

see that the several intersections can be simply com-

puted and correspond to the multiple values of the ra-

dius for θ

+ 2kπ values. For each intersection point

, equation 2 can be applied to associate a potential.

Now, the last problem to overcome is to build a

continuous potential ﬁeld from these k individual po-

tentials. One of the simplest idea is to consider the

maximum or minimum potential determined for in-

tersection points I

using equation 4 with α = 1. This

approach generates a potential ﬁeld that is not smooth

everywhere, as illustrated in Figures 4(b),4(d), 4(f),

4(h), that have been obtained using min and max

R-functions. Therefore, such a technique suffers a

severe drawback, especially for reconstruction pur-

poses, where differentiable functions are often pre-

ferred. To obtain a differentiable potential ﬁeld, we

can combine each potential using R-functions pre-

sented in equations 5 or 6. Using R

may be an ad-

vantage if normalization property is desired, but it

also has a major drawback. With R

-functions, the

generated potential ﬁeld depends on the order the R-

functions. Indeed, for instance, it is easy to verify that

∧x

)∧ x

6= x

∧(x

∧x

), except over its zero-set.

Unicity of the potential ﬁeld can be obtained using

two techniques: sort individual values or restrict the

q parameter to be set to 2 to keep binary R-functions,

which is not satisfactory in both cases. Fortunately,

for multiple self intersections, n potential values can

be combined using n-ary R-functions as deﬁned in

equation 7. The justiﬁcation for the introduction of

equations 8a and 8b now clearly appears: using bi-

nary R-functions, such as R

, R

or R

, restricts the

parameter q in equation 9 to be equal to 2, whereas

with N-ary R-functions q ∈ N

∗

. The commutativity

of N-ary R-functions is obvious since these functions

are a sum of sums and products that are commutative.

Eventually, by replacing the arguments of equa-

tions 8a and 8b by the potential ﬁeld deﬁned in equa-

tion 2, we obtain the inner and outer potential ﬁelds

for 2D-RGCs, respectively deﬁned by:

i=q

i=1

) and

i=q

i=1

), (10)

where F

) corresponds to the potentials evalu-

ated for the multiple intersection points (x

) using

equation 2 combined through N-ary R-conjunction

and R-disjunction, respectively. Figure 4 illustrates

the relative intensity of the potential ﬁeld for various

RGCs and several R-functions.

(a) using Eq.8b (b) using R

(e) using Eq.8b (f) using R

(g) using Eq.8a (h) using R

Figure 4: Color coding of relative intensity of the potential

ﬁeld generated by auto R-function of a RGC with a = b = 1,

= 0.5, n

= n

= 3.5, p = 5. The RGC is in dark red.

First row: m = 5/2. Second row m = 5/3. From left

to right: relative potential ﬁeld intensity using the N-ary

R-disjunction, Maximum, N-ary R-conjunction, and Mini-

mum.

Additionally, the inner and outer potential ﬁelds

A NEWPOTENTIAL FUNCTION FOR SELF INTERSECTING GIELIS CURVES WITH RATIONAL SYMMETRIES

can be combined together, for example to represent

developmental processes in ﬂowers. For instance, one

can want to perform the equivalentof the Boolean dif-

ference between the inner and outer curve, simply by

using binary R-functions, as illustrated in Figure 5.

Such a representation ﬁnds direct biological meaning:

when a plant goes into ﬂowering, the ﬂower approxi-

mately becomes a planar structure, going from a spiral

phyllotaxy(givenby Fibonaccinumbers) into a planar

phyllotaxy or still with helical or spiral tendencies.

But in a meristem of the ﬂower, new entities have to

develop sequentially (ﬁrst sepals, followed by petals,

stamen and ﬁnally pistil). This means that in a pla-

nar arrangement a clear separation of areas/sectors is

necessary, as delineation of potential ﬁelds. And that

is precisely what distinguishes Figures 4 and 5. This

illustrates the potential of our approach to build from

bottom up (pure R-functions/logic, with RGC) a way

of modeling ﬂowers. This example also illustrates the

strength of our approach, thanks to R-functions, be-

cause such sectors may not be represented using Na-

talini’s approach.

(a) p = 5,q = 2 (b) p = 5,q = 3 (c) p = 5,q = 4

(d) p = 5,q = 2 (e) p = 5,q = 3 (f) p = 5,q = 4

Figure 5: R-conjunction between a RGC outer envelope and

the complementary of its inner envelope to represent vari-

ous petal shapes. First row: using R

-function. Second row:

using R

5 EXTENSION TO RATIONAL

GIELIS SURFACES

We present two approaches to extend RGCs to Ra-

tional Gielis Surfaces (RGSs). The ﬁrst one is using

spherical product, and the second is building revo-

lution surfaces using a RGC and a proﬁle. The ob-

jective of 3D extension through spherical product is

to generate a closed surface built upon two Gielis

curves, which can be used in solid modeling for ex-

ample. This operation requires one important restric-

(a) (b)

Figure 6: Rational Gielis Surfaces. a) Spherical product. b)

Revolution surface.

tion: the second generating curve cannot be ratio-

nal. The main problems arising when considering

a RGC as second generating curve are surface gen-

eration/continuity and potential ﬁeld evaluation. In

spherical product, half of the second curve, that cor-

responds to angles φ ∈ [−

], is rotated around the

rotation axis of the ﬁrst generating curve. In this case,

the 2qπ modulus to evaluate every intersection leads

to curve discontinuities, which is a major problem

for efﬁcient and simple tesselation algorithms and po-

tential ﬁeld evaluation. Thus, one solution to build

closed surfaces while preserving surface continuity

and signed implicit ﬁeld deﬁnition, is to perform the

spherical product between one RGC with one GC.

The implicit ﬁeld generated can then be evaluated by:

i=q

i=1

) and

i=q

i=1

), (11)

where F

(x,y) is the potential equation for 3D

Gielis Surfaces presented in equation 3. An example

of a mesh corresponding to such surface is illustrated

in Figure 6(a).

The second approach consists in deﬁning a pro-

ﬁle, by any known techniques such as NURBS for

instance, and to use this proﬁle as an elevation pro-

ﬁle. In this case, the surface created does not deﬁne

a closed object, which makes impossible the expres-

sion of an implicit ﬁeld. Nevertheless, such represen-

tation may be very useful for compact and efﬁcient

representation of elementary 3D ﬂower patterns, es-

pecially in entertainment or video-game industry. An

example of a revolution surface using RGC, using a

cubic polynomial proﬁle, is presented in 6(b).

6 CONCLUSIONS

We have presented new potential functions for closed

Rational Gielis Curves and possible strategies for

their extension to 3D surfaces. Our approach makes

use of R-functions to overcome the self-intersections

GRAPP 2009 - International Conference on Computer Graphics Theory and Applications

issue introduced by the rational rotational symmetry.

With this technique, several differentiable potential

ﬁelds can be deﬁned for RGCs and, more speciﬁ-

cally, one corresponding to the inner curve, that is the

curve inscribed within the whole curve, and one cor-

responding to the envelope. Such representation of-

fers promising perspectives, especially in botany with

classiﬁcation and morphology metrics: as illustrated

in Figure 5, combining inner and outer potential ﬁelds

through R-functions leads to the deﬁnition of sectors

that are directly related to the ﬂower developmental

process.

Among the numerous other research directions,

we consider the study of other possible 3D extensions

and their applications to solid modeling, boundary

value problems, and/or entertainment (fast 3D ﬂower

modeling and rendering, procedural ﬂower ﬁeld tex-

ture generation, etc). Another still highly challeng-

ing research concerns the shape and symmetry pa-

rameters recovery. To our knowledge, in the lit-

erature, there still exist very few papers dedicated

to Gielis curves parameters recovery using integer

symmetries, and none considering rational symme-

tries. The introduction of potential equations for such

closed curves therefore opens new research perspec-

tives in this ﬁeld. Moreover, due to the complexity of

the space parameters, deterministic methods, such as

Levenberg-Marquardt method, can only be applied in

restricted cases, with prior symmetry detections and

strong assumptions. Our current and future works

include the development of more suitable RGC po-

tential functions for optimization processes combined

with the study of appropriate stochastic algorithms for

efﬁcient RGCs parameters recovery and their applica-

tion to classiﬁcation, pattern recognition, and object

segmentation both in 2D and 3D.

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A NEWPOTENTIAL FUNCTION FOR SELF INTERSECTING GIELIS CURVES WITH RATIONAL SYMMETRIES