Mobius-invariant¨ curve and surface energies and their applications Shin YOSHIZAWA1 and Alexander BELYAEV2 1Image Processing Research Team, RIKEN, Japan, [email protected] 2Institute of Sensors, Signals & Systems, Heriot-Watt University, UK, [email protected] Abstract: Curvature-based surface energies are frequently used in mathematics, physics, thin plate and shell engineer- ing, and membrane chemistry and biology studies. Invariance under rotations and shifts makes curvature- based energies very attractive for modeling various phenomena. In computer-aided geometric design, the Willmore surfaces and the so-called minimum variation surfaces (MVS) are widely used for shape model- ing purposes. The Willmore surfaces are invariant w.r.t conformal transformations (Mobius¨ or conformal invariance), and studied thoroughly in differential geometry and related disciplines. In contrast, the mini- mum variation surfaces are not conformal invariant. In this paper, we suggest a simple modification of the minimum variation energy and demonstrate that the resulting modified MVS enjoy Mobius¨ invariance (so we call them conformal-invariant MVS or, shortly, CI-MVS). We also study connections of CI-MVS with the cyclides of Dupin. In addition, we consider several other conformal-invariant curve and surface energies involving curvatures and curvature derivatives. In particular, we show how filtering with a conformal- invariant curve energy can be used for detecting salient subsets of the principal curvature extremum curves used by Hosaka and co-workers for shape quality inspection purposes. Keywords: Willmore energy, minimum variation surfaces, Dupin’s cyclides, Mobius/conformal¨ invariance. 1 Introduction Mobius¨ invariance [27] and studied thoroughly in differ- The so-called Bernoulli-Euler’s elastica curves were ential geometry [10] and related mathematical disciplines first considered by Bernolli at the end of the 17th century in [7, 9, 3, 21, 15]. The Willmore energy (1) measures a devi- relation to a mechanical equilibrium problem for elastically ation from sphericity and is currently a subject of intensive bending shapes. Later, Euler introduced a curvature-based researches in geometric modeling [2, 24]; see also refer- formulation of elastica [5] and considered a curvature- ences therein. based energy k2 ds, where k and ds are the curvature and Moreton and Sequin´ introduced the so-called minimum arc length elementR of a curve, respectively. Since then the variation curves (MVC) [18] and surfaces (MVS) [19] elastica and related curvature-based curve and surface en- which bend shapes as smoothly as possible, where the ergies have been widely studied in mathematics, physics, bending smoothness is modeled by the variation of curva- 2 engineering, computer vision, biochemistry, architecture, tures, minimizing the MVC energy (dk/ds) ds and MVS and other disciplines. See [13] for a historical survey of energy R the elastica. (e2 + e2 ) dA (2) A natural extension of the elastica to surfaces is given ZZ max min by the well-known Willmore surfaces [28, 29] which min- t t imize the elastic bending energy (Willmore energy) where emax = ∂kmax/∂ max and emin = ∂kmin/∂ min are the first order principal curvature derivatives w.r.t. their corresponding principal directions tmax and tmin. The (k2 + k2 ) dA (1) ZZ max min MVC and MVS energies measure the magnitude of the rate of change of curvatures, and provide more aesthetic shapes where kmax, kmin, and dA are the principal curvatures than the Willmore surfaces. The MVS energy (2) also mea- and area element of a surface, respectively. We assume sures deviation from special surfaces called Dupin’s cy- that k k . The Willmore surfaces possess beau- clides (see Figure 1) which are characterized by the condi- min ≤ max tiful mathematical properties. In particular, they are in- tions emax =0= emin and which are often used as CAD variant under conformal transformations: the so-called primitives. Figure 1: Examples of Dupin cyclides. Various modifications of the MVC energy were consid- Proof for conformal invariance of (4). The principal ered in [12] including curvatures and directions (kmax, kmin, tmax, and tmin) 2 ∂kmax 2 2 ∂kmin 2 are obviously invariant under any rigid motions (transla- [emax +( ) + emin +( ) ] dA · dA (3) ZZ ∂tmin ∂tmax ZZ tions and rotations) because they do not depend on any surface parameterization and choice of a coordinate sys- where the factor dA is used to maintain the scale- tem. Consequently, e and e are also translational- invariance. RR max min and rotational-invariants. Since it is a straightforward com- Unfortunately, MVS, their modifications as those from putation, we omit the derivations of [12] and other high-order curvature-based energies used for 2 shape modeling applications (e.g. H dA [30] where α 1 α 1 |∇ | k = k, e = e (5) H is the surface gradient of theRR mean curvature) are not α α2 ∇ conformal invariant. α α α 1 where α is a constant scaling factor, k = kmax,kmin and In this paper , we introduce a novel conformal-invariant α α α e = emax,emin are the principal curvatures and their cor- surface energy (4) which we call the conformal-invariant α responding derivatives of the scaled surface r = αr as- MVC (CI-MVC) energy. We also study connections of sociated with k = k ,k and e = e ,e , respec- CI-MVS with the cyclides and curvature-based surface max min max min tively. The scale invariance property is shown by creases which also enjoy conformal invariance. In particu- lar, efficient filtering the surface creases with a conformal- α α (e )2 +(e )2 dAα = e2 + e2 dA invariant curve energy is demonstrated. All numerical ex- ZZ q max min ZZ q max min periments in this paper are performed on surfaces approxi- 2 rα mated by dense triangle meshes. where dAα = α dA is the area element of . Now it remains to demonstrate the inversion invariance of (4). 2 Conformal Invariant MVS Energy The conformal invariant minimum variation surfaces 2.1 Inversion Invariance (CI-MVS) are the minimizers of the energy functional Consider a surface ˜r obtained from r by the inversion w.r.t. the sphere of radius c centered at the origin of coor- 2 2 emax + emin dA (4) dinates ZZ q ˜r = c2r/r2, r2 = r r. where a conformal-invariant energy remains unchanged · Then the length and area elements of ˜r and r are related by under the conformal transformations consisting of transla- tions, rotations, scaling, and spherical inversions. Confor- c4 ds˜ = c2ds/r2, dA˜ = dA. (6) mal transformations have been employed in many CG and r4 CAD applications e.g. geometric algebra [14], pose and Direct computations (see [26, Art. 82-83] or Appendix A motion estimations [25], and surface classification [8]. In of this paper) show that the principal curvatures of ˜r are addition, spherical inversions are also useful for geometric given by computing tasks such as visibility computation of point set surfaces [16] and chord length parameterization of Bezier´ r2 (r n) k˜ = k 2 · , (7) surfaces [1]. See also Section 3 where we discuss why in- max − c2 min − c2 variance under conformal transformations (instead of only 2 r n ˜ r ( ) rigid motions) is important in aesthetic shape design. kmin = kmax 2 · , (8) − c2 − c2 1It is an extension of our previous work [31]. The main difference from [31] is the numerical comparisons of the proposed and conventional energies under spherical inversions. where n is a unit normal of r and k˜ k˜ [26, Art. 82- Now let us denote by u˜ and v˜ the arclength parameter- min ≤ max 83]. The curvature lines of r are mapped onto the curvature izations of the k˜max and k˜min curvature lines of ˜r, respec- lines of ˜r such that ˜tmax = tmin and ˜tmin = tmax, where tively, in a small vicinity of point P˜ ˜r, the inversion ˜t and ˜t are the principal directions corresponding to image of P r. Then, according to the∈ first formula of max min ∈ the principal curvatures k˜max and k˜min, respectively. (6), we have One can easily observe that (6), (7), and (8) imply the 2 2 2 2 inversion invariance of the energy du˜ = c dv/r , dv˜ = c du/r . (10) 2 In view of (9) and (10), we have (k˜max k˜min) dA˜ ZZ − 2 2 ∂k˜max ∂k˜max ∂v ∂k˜max r r = = = 2 ( 2 emin), corresponding to (1). Since kmaxkmin dA becomes a ∂˜tmax ∂u˜ ∂u˜ ∂v c − c constant depending only on topologyRR of r from the Gauss- ∂k˜ ∂k˜ ∂u ∂k˜ r2 r2 Bonnet theorem (see, for example, [22, p. 155]), minimiz- min = min = min = ( e ). ˜t ∂v˜ ∂v˜ ∂u c2 c2 max ing ∂ min − Consequently, the curvature derivatives e˜max = (k2 + k2 ) dA 2 k k dA ∂k˜ /∂˜t and e˜ = ∂k˜ /∂˜t relate to e and ZZ max min − ZZ max min max max min min min max emin by the following equations: is equivalent to minimizing (1). c2 r2 c2 r2 e˜ = e , e˜ = e . (11) r2 max − c2 min r2 min − c2 max Curvature Derivatives. Differentiating k˜max and k˜min in (7) and (8) w.r.t. the parameters v and u, respectively, Substituting (6) and (11) to (4) shows the inversion in- gives variance property: ˜ r r 2 r n r n ∂kmax r v r ∂kmin v + v 2 2 ˜ 2 2 = 2 · k 2 · · , e˜max +e ˜min dA = emax + emin dA. ∂v − c2 r min − c2 ∂v − c2 ZZ q ZZ q It seems that the cross MVS energy (3) can not be modi- 2 ∂k˜ r ru r r ∂k ru n + r nu fied such that it becomes inversion-invariant, since we have min = 2 · k max 2 · · . ∂u − c2 r max − c2 ∂u − c2 the terms depending on the principal directions of r: In a small vicinity of non-umbilical point P of r, let us 2 ˜ 2 c ∂kmax r ∂kmax 2 t r consider the lines of curvature parameterized by their arc 2 = 2 + 2 ( max )(kmax kmin), r ∂˜t − c ∂tmin c · − lengths.