Mobius-invariant¨ curve and 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 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 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 ) 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 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. Then the surface is locally represented in para- min r r c2 ∂k˜ r2 ∂k 2 metric form = (u,v) for which min = min + (t r)(k k ). 2 2 t 2 min min max r ∂˜tmax − c ∂ max c − ru = tmax, rv = tmin, 3 Aesthetic Shapes and Curvature-based Creases and the Rodrigues’curvature formula (see, for instance, The CI-MVS energy (4) is closely related to the cy- [22, p. 94]) gives clides of Dupin and curvature-based surface creases. The cyclides were introduced by Dupin at the beginning of the nu = k t , nv = k t . − max max − min min 19th century, and have been studied in connection with var-

Now differentiating k˜max and k˜min in (7) and (8) along ious shape modeling tasks. See [4] for a historical survey the non-corresponding curvature lines of r gives and [6] for recent applications of the cyclides in geomet- ric modeling as a CAGD primitive. The family of cyclides ˜ ˜ 2 ∂kmax ∂kmax r tmin r r includes , cylinders, cones, and tori. = = 2 2 kmin 2 emin + ∂v ∂tmin − c r − c Mathematically, the cyclides are characterized by the t n k t r r2 condition emax =0= emin, and obviously they are the 2 min − min min = e , − c2 − c2 min best possible aesthetic shapes in terms of the MVS (2) and CI-MVS (4) energies. This can be interpreted by a concept t n where min = 0 (because the principal directions live of surface creases, the curves on a surface along which the in the tangent plane of r) and similar computations give us surface bends sharply, described via extrema of emax and ∂k˜min ∂u : emin: ˜ 2 ˜ 2 ∂e ∂e ∂kmax r ∂kmin r max and min = emin, = emax. (9) emax = 0, t < 0 emin = 0, t > 0. ∂v − c2 ∂u − c2 ∂ max ∂ min r (top) and ˜r with c = 1.0 (bottom) T 0.0 T 0.302 T 7.3 ≥ ≥ 7 ≥ F = 1.53M, sr = 4.795, EB(r) = 2904, EM (r) = 1.812 10 , EC (r) = 4320, 6 × sr = 3.12, EB(˜r) = 2906, EM (˜r) = 9.444 10 , and EC (˜r) = 4321. ˜ × Figure 2: The images demonstrate how well our thresholding scheme of (14) eliminates the unessential features of inversion dual crest lines.

Such curvature extremum curves, also called ridges- [23]). For example, a conformally invariant curve energy ravines and crest lines, have been employed in quality con- on a surface trol of free-form surfaces [11] and also have numerous sci- T = e + e ds (14) ence and engineering applications (see [32] and references Z | max| | min| therein). It is interesting to mention that the ridge points p is useful for filtering the unessential curvature extremum and cyclides processes zero values for emax and emin but the cyclides do not have any creases since their second cur- curves detected on a surface by eliminating the curves vature derivatives of the cyclides are also zero everywhere whose (14) are smaller than a given threshold value. Since (14) measures deviation from the cyclides, the remaining ∂emax/∂tmax =0= ∂emin/∂tmin. This implies that the best aesthetic shapes in terms of a family of MVS energies curves are more salient than the traditional curvature ex- consist of a set of cyclide patches whose boundaries be- tremum curves. In addition, the above filtering scheme im- come a set of ridges representing shape features. Similar proves noise-robustness significantly when we use (14) for concept has been employed for aesthetic shape modeling a detection of the curvature extremum curves on a mesh from feature lines [19, 20]. approximating a smooth surface, see Figure 2. By (5) and (11), the ridges and cyclides are invariant Equations (12) and (13) also lead to some interesting under the conformal transformations. Therefore the con- relations between scale and inversion transformations and formal invariance property of aesthetic surfaces minimiz- curvature derivatives. For example, the rate of change ing the variation of curvatures is also preferable. Conse- of area by the scale and inversion transformations are in- quently, our CI-MVS energy (4) has advantages over the versely proportional to a ratio of the curvature derivatives: traditional MVS energy (2). ∂A˜ emin emax ∂Aα emax emin Using (5) and (11) we arrive at = = , = α = α , ∂A −e˜max − e˜min ∂A emax emin α 2 2 2 α emaxdsα = emaxds = e˜minds˜ , (12) emax emax e˜min α 2 2 − 2 α = = . e dsα = e ds = e˜ ds˜ (13) e emin e˜max min min − max min and can easily construct a number of conformal-invariant 4 Inversion Dual Crest Lines differential forms [32] (in principle, they can be derived The so-called crest lines are formed by the perceptually from a complete conformal-invariant system derived in salient ridge points and consist of the surface points where the magnitude of the largest (in absolute value) principal energy denoted by ES( ) under spherical inversions in Fig- curvature attains a maximum along its corresponding line ure 7. Numerical experiments suggest that CI-MVS is bet- of curvature [17]: ter than the SI-MVS in terms of Mobius¨ invariance prop- erty as expected from our theoretical analysis. ∂emax kmax > kmin , emax = 0, t < 0, | | ∂ max 6 Conclusion

∂emin The paper introduces a novel surface energy (CI-MVS kmin < kmax , emin = 0, > 0. energy) by modifying the minimum variation energy in or- −| | ∂tmin der to satisfy invariance under conformal transformations. The subset of crest lines of r such that The CI-MVS energy consists of the principal curvatures ∂emax and their first order derivatives w.r.t. corresponding prin- kmax > kmin , k˜min < k˜max ,emax = 0, < 0, | | −| | ∂tmax cipal directions, measures deviation from the cyclides, and provides better invariance properties than the conventional ˜ ˜ ∂emin energy. We also studied connections of the CI-MVS with kmin < kmax , kmax > kmin ,emin = 0, t > 0 −| | | | ∂ min the cyclides and curvature extremum curves from a view are dual to the subset of crest lines of ˜r such that of aesthetic shape design. In addition, we show how filter- ing with a conformally invariant curve energy on a surface ∂e˜ k˜ < k˜ ,k > k , e˜ = 0, min > 0, improves a detection of the salient surface creases. Future min max max min min ˜t −| | | | ∂ min work will aim at developing numerical methods for min- imizing the CI-MVS efficiently for the given initial and ˜ ˜ ∂e˜max kmax > kmin ,kmin < kmax , e˜max = 0, < 0. boundary conditions. | | −| | ∂˜tmax Note that the concave condition of ˜r for the convex con- Acknowledgements. This work was supported in part by dition of r, and vice versa is considered. Grants-in-Aid for Scientific Research of Japan (24700182 and 20113007). The models are courtesy of AIM@SHAPE 5 Numerical Experiments shape repository (Camel, Caesar, Gargoyle, Gearbox, In Figures 2-6, we examined our energies (4) and (14) Pegasos, and Rolling Stage), FarField Technology Ltd and dual crest lines on several surfaces approximated by (Hand), INRIA (Car), MPI Informatik (Max-Planck bust), the triangle meshes, where F is a number of triangles (M: MIT Computer Graphics Group (Elephant and Horse), Mega), and sr and sr are the sizes of r and ˜r (i.e. diago- ˜ Stanford University (Armadillo, Bunny, and Dragon), and nal length of bounding box), respectively. The differential University of Washington (Fandisk). quantities on the triangle meshes are approximated by us- ing the method in [32]. References Figures 3-5 show differences between the traditional ˇ r [1] B. BASTL, B. JUTTLER¨ , M. LAVI´ CKAˇ , AND Z. S´IR, and inversion dual crest lines. The dual crest lines of ˜ Curves and surfaces with rational chord length parame- r r are calculated on ˜ and then mapped onto for visualiza- terization, Computer Aided Geometric Design, 29 (2012), tion purpose. We obtain remarkably similar patterns for the pp. 231–241. inversion dual crest lines (see the bottom images of Figure [2] A. I. BOBENKO AND P. SCHRODER¨ , Discrete Willmore 4 and (c,d) of Figure 5). Figures 2 and 6 demonstrate how flow, in Eurographics Symposium on Geometry Processing, well (14) eliminates the unessential creases located on the 2005, pp. 101–110. parts close to the cyclides. [3] C.BOHLE,G.P.PETERS, AND U. PINKALL, Constrained Let EB( ), EM ( ), and EC ( ) be the Willmore (1), willmore surfaces, Calculus of Variations and Partial Differ- MVS (2), and CI-MVS (4) energies. The B and C E ( ) E ( ) ential Equations, 32 (2008), pp. 263–277. are numerically well preserved under spherical inversions [4] V. CHANDRU,D.DUTTA, AND C.M.HOFFMANN, On the compared with EM ( ) according to our numerical experi- ments. Our CI-MVS is much useful for evaluating surface geometry of Dupin cyclides, The Visual Computer, 5 (1989), pp. 277–290. quality compared with the other energies in terms of invari- ance property and shape aesthetics, although it is computa- [5] L. EULER, Additamentum ‘de curvis elasticis’, in Metho- tionally expensive than others. dus Inveniendi Lineas Curvas Maximi Minimive Probpri- We have also compared CI-MVS (4) energy intro- etate Gaudentes, Lausanne, 1744. duced in this paper with the scale-invariant MVS (SI-MVS) [6] S. FOUFOU AND L. GARNIER, Dupin cyclide blends be- [19, 12] tween quadric surfaces for shape modeling, Computer Graphics Forum, 23 (2004), pp. 321–330. Proc. of Euro- (e2 + e2 ) dA dA (15) ZZ max min ZZ graphics. [7] J. GRAVESEN AND M. UPGSTRUP, Constructing invariant [25] R. WAREHAM, J. CAMERON, AND J. LASENBY, Appli- fairness measures for surfaces, Advances in Computational cations of conformal geometric algebra in computer vision Mathematics, 17 (2002), pp. 67–88. and graphics, in Proceedings of International Conference [8] X. GU AND S.-T. YAU, Surface classification using confor- on Computer Algebra and Geometric Algebra with Appli- mal structures, in Proceedings of IEEE International Con- cations, Springer-Verlag, 2005, pp. 329–349. ference on Computer Vision, IEEE Computer Society, 2003, [26] C. E. WEATHERBURN, Differential Geometry of Three Di- pp. 701–708. mensions, volume I, Cambridge University Press, 1927. [9] J. GUVEN, Conformally invariant bending energy for hy- [27] J. H. WHITE, A global invariant of conformal mappings in persurfaces, Journal of Physics A: Mathematical and Gen- space, Proceedings of the American Mathematical Society, eral, 38 (2005), pp. 7943–7956. 38 (1973), pp. 162–164. [10] U. HERTRICH-JEROMIN, Introduction to Mobius¨ Differen- [28] T. J. WILLMORE, Note on embedded surfaces, Anal. Stunt. tial Geometry, Cambridge University Press, 2003. Ale Univ. Sect. I. a Math., 11B (1965), pp. 493–496. [11] M. HOSAKA, Modeling of Curves and Surfaces in [29] T. J. WILLMORE, Surfaces in conformal geometry, Annals CAD/CAM, Springer, Berlin, 1992. of Global Analysis and Geometry, 18 (2000), pp. 255–264. [12] P. JOSHI AND C. SEQUIN´ , Energy minimizers for [30] G. XU AND Q. ZHANG, Minimal mean-curvature-variation curvature-based surface functionals, Computer-Aided De- surfaces and their applications in surface modeling, in Pro- sign and Applications, 4 (2007), pp. 607–618. ceedings of International Conference on Geometric Model- [13] R. LEVIEN, The elastica: a mathematical history, Tech. ing and Processing, Springer-Verlag, 2006, pp. 357–370. Rep. UCB/EECS-2008-103, EECS Department, University of California, Berkeley, 2008. [31] S. YOSHIZAWA AND A. BELYAEV, Conformally invariant energies and minimum variation surfaces, in Proceedings [14] S. MANN AND L. DORST, Geometric algebra: a computa- of Asian Conference on Design and Digital Engineering, tional framework for geometrical applications, part 2, IEEE Niseko, Hokkaido, Japan, December 2012. Computer Graphics and Applications, 22 (2002), pp. 58–67. [32] S. YOSHIZAWA, A. BELYAEV, H. YOKOTA, AND H.-P. [15] F. C. MARQUES AND A. NEVES, Min-max theory and the SEIDEL, Fast, robust, and faithful methods for detecting Willmore conjecture, http://arxiv.org/abs/1202.6036, 2012. crest lines on meshes, Computer Aided Geometric Design, [16] R.MEHRA, P.TRIPATHI,A.SHEFFER, AND N.J.MITRA, 25 (2008), pp. 545–560. Visibility of noisy point cloud data, Computers and Graph- ics, 34 (2010), pp. 219–230. SMI’10. [17] O. MONGA,S.BENAYOUN, AND O. D. FAUGERAS, From partial derivatives of 3-d density images to ridge lines, in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, IEEE Computer Society, 1992, pp. 354 – 359. [18] H. P. MORETON AND C. H. SEQUIN´ , Surface design with minimum energy networks, in Proceedings of ACM Sympo- sium on Solid Modeling Foundations and CAD/CAM Ap- plications, ACM, 1991, pp. 291–301. [19] H. P. MORETON AND C.H.SEQUIN´ , Functional optimiza- tion for fair surface design, in Proceedings of ACM SIG- GRAPH, 1992, pp. 167–176. [20] A. NEALEN, T. IGARASHI, O. SORKINE, AND M. ALEXA, Fibermesh: designing freeform surfaces with 3d curves, ACM Transactions on Graphics, 26 (2007), pp. 41:1–10. Proc. of ACM SIGGRAPH. [21] T. RIVIERE´ , Analysis aspects of willmore surfaces, Inven- tiones Mathematicae, 174 (2008), pp. 1–45. [22] D.J.STRUIK, Lectures on Classical Differential Geometry: Second Edition, Dover Publications, Inc. New York, 1988. [23] C. P. WANG, Surfaces in Mobius¨ geometry, Nagoya Math- ematical Journal, 125 (1992), pp. 53–72. [24] M. WARDETZKY,M.BERGOU,D.HARMON,D.ZORIN, AND E. GRINSPUN, Discrete quadratic curvature energies, Computer Aided Geometric Design, 24 (2007), pp. 499– 518. Appendix A. Sphere Inversions and Surface Curva- Differentiating (16) and (17), and their inner products tures. Given a surface r = r(u,v), consider its spherical with n˜ lead the coefficients of the second fundamental form inversion with radius c w.r.t. the origin of coordinates of ˜r:

2 2 2 c c 2 c ˜r = ˜r(u,v)= r(u,v), L˜ = ˜ruu n˜ = (8 (ru r) r 2 (ruu r + E)r + r2(u,v) r6 − r4 c2 c2 2 2 2 r r ∂(r·r) r r 4 (ru r)ru + ruu) ( (r n)r n) where r = r (u,v) = . Since ∂u = 2 u = − r4 r2 r2 − ∂ r·r ∂r ( ) r r ∂r 2 2 ∂u r = 2rur and ∂v = 2 v = 2 ∂v r = 2rvr, we c 2 ru·r rv ·r = (L + (r n)E), have ru = r and rv = r such that the basic tangents −r2 r2 of ˜r are given by differentiating ˜r w.r.t. u and v: and similar computations give us ∂˜r c2 c2 = ˜ru = 2 (ru r)r + ru, (16) 2 ∂u − r4 r2 c 2 M˜ = ˜ruv n˜ = (M + (r n)F ), ∂˜r c2 c2 −r2 r2 = ˜rv = 2 (rv r)r + rv (17) 2 ∂v − r4 r2 c 2 ˜ rvv n r n N = ˜ ˜ = 2 (N + 2 ( )G), r ∂r r ∂r r −r r where u = ∂u and v = ∂v are the basic tangents of . Then, their inner products give the coefficients of the first where L = ruu n, M = ruv n, and N = rvv n are the fundamental form of ˜r: coefficients of the second fundamental form of r. 4 4 4 Substitute the above results into the following formulas 2 c c 2 c E˜ = ˜ru = E, F˜ = ˜ru ˜rv = F, G˜ = ˜rv = G, of the mean and Gaussian curvatures r4 r4 r4 2 r2 r r r2 1 E˜N˜ 2F˜M˜ + G˜L˜ r (r n) where E = u, F = u v, and G = v are the coef- H˜ = − = H 2 , ficients of the first fundamental form of r. Thus, the area 2 E˜G˜ F˜2 − c2 − c2 − element of ˜r is given by L˜N˜ M˜ 2 r4 r2 (r n)2 K˜ = − = K + 4 (r n)H + 4 ˜ ˜ ˜2 c4 c4 c4 c4 EG F dA˜ = E˜G˜ F˜2 dudv = dA, − 4 2 ZZ ZZ p − ZZ r 1 EN−2FM+GL LN−M where H = 2 EG−F 2 and K = EG−F 2 are the 2 where dA = √EG F dudv is the area element of r. mean and Gaussian curvatures of r, respectively. Let k˜max − Let n be the unit normal vector of r. The unit normal and k˜min be the principal curvatures of ˜r and then sub- vector of ˜r is given by stitute the above equations in H˜ = (k˜max + k˜min)/2, K˜ = k˜ k˜ , k˜ = H˜ + H˜ 2 K˜ , and k˜ = ˜ru ˜rv 2 max min max min n r r r r r r r n 2 p 2− 2 ˜ = × = (( v ) u ( u ) v) + , H˜ H˜ K˜ with kmax kmin, c > 0, and r 0 ˜ ˜ ˜2 −r2 − × EG F gives− p − ≥ ≥ p − By using the formulas of vector triple product: r2 (r n) k˜ = k 2 , a (b c)=(a c)b (a b)c, (a b) c =(c a)b (c b)a, max − c2 min − c2 × × − × × − r2 (r n) we obtain k˜ = k 2 min − c2 max − c2 2 2 n˜ = (r n) r + n = (r n)r n. where k and k are the principal curvatures of r. −r2 × × r2 − max min (a) (b) (c) Figure 3: (a): Input mesh r. (b): Transformed mesh ˜r with c =0.1. (c): Magnified image of (b). F =1.25M, sr =2.031, EB (r) = 8449, 12 16 EM (r)=8.1262 × 10 , EC (r) = 12585, s˜r = 0.5, EB (˜r) = 8468, EM (˜r)=3.3567 × 10 , and EC (˜r) = 12605. The difference between EC (r) and EC (˜r) is very small compared with the MVS energy difference (EM (r) and EM (˜r)).

(a) (b) (c) (d) Figure 4: The top (bottom) images (a) and (b,c,d) present the crest lines (inversion dual crest lines) of r and ˜r, respectively, where the input meshes are shown in images (a) and (b,c) of Figure 3. The traditional crest lines (top images) of r and ˜r do not coincide when their convex and concave lines are exchanged, while the inversion dual crest lines (bottom images) share common geometric patterns.

(a) (b) (c) (d) Figure 5: Filtered crest lines (inversion dual crest lines) are demonstrated in (a) and (b) ((c) and (d)) whose threshold value (14) is greater than 1.5. Images (a,b) and (c,d) correspond to top and bottom images of Figure 4, respectively. 14 c = 1.5, T 0.0, sr = 1.658, EB(˜r) = 8453, EM (˜r) = 1.0146 10 , and EC (˜r) = 12581. ≥ ˜ ×

5 13 5 c = 0.5, T 0.5, F = 1.35M, sr = 1.609, EB(r) = 1.387 10 , EM (r) = 7.81 10 , EC (r) = 1.073 10 , ≥ 5 × 14 × 5 × sr = 5.697, EB(˜r) = 1.382 10 , EM (˜r) = 7.857 10 , and EC (˜r) = 1.07 10 . ˜ × × ×

4 8 4 c = 10.0, T 2.0, F = 1.96M, sr = 200.2, EB(r) = 2.112 10 , EM (r) = 4.072 10 , EC (r) = 2.387 10 , ≥ 4 × 12 × 4 × sr = 82.63, EB(˜r) = 2.118 10 , EM (˜r) = 1.046 10 , and EC (˜r) = 2.408 10 . ˜ × × ×

4 4 4 c = 0.25, T 2.0, F = 2.06M, sr = 1.483, EB(r) = 3.774 10 , EM (r) = 3.774 10 , EC (r) = 6.821 10 , ≥ 4 × 10 × 4 × s˜r = 1.3, EB(˜r) = 3.775 10 , EM (˜r) = 4.78 10 , and EC (˜r) = 5.007 10 . (a) (b)× × (c)× (d)

Figure 6: Images (b) and (c) show the inversion dual crest lines of (a) and (d) which correspond to r and ˜r, respectively. The mesh model shown in the top image is generated by translating the model of Figure 3. Remarkably similar feature patterns are extracted on (b) and (c), and we can confirm inversion duality of them as well as small differences between EC (r) and EC (˜r). (a):c = 1.0, F = 0.27M. (b):c = 0.1, F = 0.87M.

(c):c = 0.5, F = 0.62M. (d):c = 2.0, F = 0.013M. (e):c = 100.0, F = 0.8M.

(f):c = 10.0, F = 0.35M. (g):c = 2.0, F = 0.25M.

(h):c = 20.0, F = 0.31M. (i):c = 20.0, F = 0.4M. (j):c = 100.0, F = 0.5M.

(a) (b) (c) (d) (e) 11 16 9 6 8 ES(r) 5.26 10 1.5 10 3.13 10 2.14 10 1.0 10 × 14 × 16 × 10 × 7 × 9 ES(˜r) 1.02 10 8.56 10 3.18 10 2.27 10 1.9 10 × 4 × 4 × 3 × 2 × 3 EC (r) 1.02 10 2.12 10 1.49 10 7.68 10 5.04 10 × 4 × 4 × 3 × 2 × 3 EC (˜r) 1.02 10 2.12 10 1.55 10 8.85 10 4.99 10 × × × × × (f) (g) (h) (i) (j) 9 7 10 8 11 ES(r) 1.5 10 3.46 10 2.85 10 2.74 10 8.64 10 × 11 × 8 × 13 × 11 × 14 ES(˜r) 1.02 10 7.86 10 1.86 10 8.32 10 2.17 10 × 3 × 3 × 3 × 3 × 4 EC (r) 7.88 10 1.75 10 1.91 10 3.11 10 2.44 10 × 3 × 3 × 3 × 3 × 4 EC (˜r) 7.93 10 1.81 10 2.06 10 3.36 10 2.23 10 × × × × × Figure 7: The images (a-j) and their corresponding values demonstrate how well our CI-MVS (4) energy is preserved under spherical inversions compared with the SI-MVS (15) energy. Here the left and right images correspond to the input and transformed meshes via spherical inversions.