Chapter 2 SUPERQUADRICS AND THEIR GEOMETRIC PROPERTIES In this chapter we dene superquadrics after we outline a brief history of their development. Besides giving basic superquadric equations, we derive also some other useful geometric properties of superquadrics. 2.1 SUPERELLIPSE A superellipse is a closed curve dened by the following simple equa- tion x m y m + =1, (2.1) a b where a and b are the size (positive real number) of the major and minor axes and m is a rational number p p is an even positive integer, m = > 0, where (2.2) q q is an odd positive integer. If m = 2 and a = b, we get the equation of a circle. For larger m, however, we gradually get more rectangular shapes, until for m →∞ the curve takes up a rectangular shape (Fig. 2.1). On the other hand, when m → 0 the curve takes up the shape of a cross. Superellipses are special cases of curves which are known in analytical geometry as Lame curves, where m can be any rational number (Loria, 1910). Lame curves are named after the French mathematician Gabriel Lame, who was the rst who described these curves in the early 19th century1. 1Gabriel Lame. Examen des dierentes methodes employees pour resoudre les problemes de geometrie, Paris, 1818. 13 14 SEGMENTATION AND RECOVERY OF SUPERQUADRICS y m =2 1 m =4 0.75 0.5 m = 3 0.25 5 x -1 -0.5 0.5 1 -0.25 -0.5 -0.75 -1 Figure 2.1. A superellipse can change continuously from a star-shape through a circle to a square shape in the limit (m →∞). Piet Hein, a Danish scientist, writer and inventor, popularized these curves for design purposes in the 1960s (Gardner, 1965). Faced with var- ious design problems Piet Hein proposed a shape that mediates between circular and rectangular shapes and named it a superellipse. Piet Hein designed the streets and an underground shopping area on Sergels Torg in Stockholm in the shape of concentric superellipses with m =2.5. Other designers used superellipse shapes for design of table tops and other furniture. Piet Hein also made a generalization of superellipse to 3D which he named superellipsoids or superspheres. He named su- perspheres with m =2.5 and the height-width ratio of 4:3 supereggs (Fig. 2.2). Though it looks as if a superegg standing on either of its ends should topple over, it does not because the center of gravity is lower than the center of curvature! According to Piet Hein this spooky stability of the superegg can be taken as symbolic of the superelliptical balance between the orthogonal and the round. Superellipses were used for lofting in the preliminary design of aircraft fuselage (Flanagan and Hefner, 1967; Faux and Pratt, 1985). In 1981, Barr generalized the superellipsoids to a family of 3D shapes that he named superquadrics (Barr, 1981). He introduced the notation common in the computer vision literature and also used in this book. Barr saw the importance of superquadric models in particular for computer graph- ics and for three-dimensional design since superquadric models, which compactly represent a continuum of useful forms with rounded edges, can easily be rendered and shaded and further deformed by parametric deformations. Superquadrics and Their Geometric Properties 15 Figure 2.2. A “superegg” (superquadric with m =2.5, height-width ratio = 4:3) is stable in the upright position because the center of gravity is lower than the center of curvature (Gardner, 1965). 2.1.1 LAME CURVES For Lame curves x m y m + =1, (2.3) a b m can be any rational number. From the topological point of view, there are nine dierent types of Lame curves depending on the form of the exponent m in equation (2.3) which is dened by positive integers k, h ∈ N (Loria, 1910). Lame curves with positive m are 2h 1. m = 2k+1 > 1 (Fig. 2.3 a) 2h 2. m = 2k+1 < 1 (Fig. 2.3 b) 2h+1 3. m = 2k > 1 (Fig. 2.3 c) 2h+1 4. m = 2k < 1 (Fig. 2.3 d) 2h+1 5. m = 2k+1 > 1 (Fig. 2.3 e) 2h+1 6. m = 2k+1 < 1 (Fig. 2.3 f) 16 SEGMENTATION AND RECOVERY OF SUPERQUADRICS 1 1 0.75 0.75 0.5 0.5 0.25 0.25 -1 -0.5 0.5 1 -1 -0.5 0.5 1 -0.25 -0.25 -0.5 -0.5 -0.75 -0.75 -1 (a) -1 (b) (c) (d) (e) (f) Figure 2.3. Lame curves with positive m. Only the rst two types (a) and (b) are superellipses. Superquadrics and Their Geometric Properties 17 3 2 1 -3 -2 -1 1 2 3 -1 -2 -3 (g) (h) (i) Figure 2.3 (continued). Lame curves with negative m Lame curves with negative m are 2h 7. m = 2k+1 (Fig. 2.3 g) 2h+1 8. m = 2k (Fig. 2.3 h) 2h+1 9. m = 2k+1 (Fig. 2.3 i) which are shown in Fig. 2.3. Only the rst type of Lame curves (Fig. 2.3 a) are superellipses in the strict sense, but usually also the second type (Fig. 2.3 b) is included since the only dierence is in the value of the exponent m (< 1or> 1). Superellipses can therefore be written as 2 2 x ε y ε + =1, (2.4) a b where ε can be any positive real number if the two terms are rst raised to the second power. 18 SEGMENTATION AND RECOVERY OF SUPERQUADRICS 2.2 SUPERELLIPSOIDS AND SUPERQUADRICS A 3D surface can be obtained by the spherical product of two 2D curves (Barr, 1981). A unit sphere, for example, is produced when a half circle in a plane orthogonal to the (x, y) plane (Fig. 2.4) cos m()= , /2 /2 (2.5) sin is crossed with the full circle in (x, y) plane cos ω h(ω)= , ω<, (2.6) sin ω x cos cos ω /2 /2 r(, ω)=m() h(ω)= y = cos sin ω , . ω< z sin (2.7) z r m y h ω x Figure 2.4. A 3D vector r, which denes a closed 3D surface, can be obtained by a spherical product of two 2D curves. Analogous to a circle, a superellipse 2 2 x ε y ε + = 1 (2.8) a b Superquadrics and Their Geometric Properties 19 can be written as a cosε s()= , . (2.9) b sinε Note that exponentiation with ε is a signed power function such that cosε = sign(cos )| cos |ε! Superellipsoids can therefore be obtained by a spherical product of a pair of such superellipses r(, ω)=s1() s2(ω) = (2.10) ε1 ε2 cos a1 cos ω = ε1 ε2 = a3 sin a2 sin ω ε1 ε2 a1 cos cos ω ε1 ε2 /2 /2 = a2 cos sin ω , . ε1 ω< a3 sin Parameters a1,a2 and a3 are scaling factors along the three coordinate axes. ε1 and ε2 are derived from the exponents of the two original su- perellipses. ε2 determines the shape of the superellipsoid cross section parallel to the (x, y) plane, while ε1 determines the shape of the superel- lipsoid cross section in a plane perpendicular to the (x, y) plane and containing z axis (Fig. 2.5). An alternative, implicit superellipsoid equation can be derived from the explicit equation using the equality cos2 + sin2 = 1. We rewrite equation (2.10) as follows: x 2 = cos2ε1 cos2ε2 ω, (2.11) a 1 y 2 = cos2ε1 sin2ε2 ω, (2.12) a 2 z 2 = sin2ε1 . (2.13) a3 Raising both sides of equations (2.11) and (2.12) to the power of 1/ε2 and then adding respective sides of these two equations gives 2 2 2ε x ε2 y ε2 1 + = cos ε2 . (2.14) a1 a2 Next, we raise both sides of equation (2.13) to the power of 1/ε1 and both sides of equation (2.14) to the power of ε2/ε1. By adding the respective 20 SEGMENTATION AND RECOVERY OF SUPERQUADRICS ε2 =0.1 ε2 =1 ε2 =2 ε1 =0.1 ε1 =1 ε1 =2 Figure 2.5. Superellipsoids with dierent values of exponents ε1 and ε2. Size pa- rameters a1,a2,a3 are kept constant. Superquadric-centered coordinate axis z points upwards! Superquadrics and Their Geometric Properties 21 sides of these two equations we get the implicit superquadric equation ! ε2 2 2 2 ε1 x ε2 y ε2 z ε1 + + =1. (2.15) a1 a2 a3 All points with coordinates (x, y, z) that correspond to the above equa- tion lie, by denition, on the surface of the superellipsoid. The function ! ε2 2 2 2 ε1 x ε2 y ε2 z ε1 F (x, y, z)= + + (2.16) a1 a2 a3 is also called the inside-outside function because it provides a simple test whether a given point lies inside or outside the superquadric. If F<1, the given point (x, y, z) is inside the superquadric, if F = 1 the corresponding point lies on the surface of the superquadric, and if F>1 the point lies outside the superquadric. A special case of superellipsoids when ε1 = ε2 x 2m y 2m z 2m + + = 1 (2.17) a b c was already studied by S. Spitzer2. Spitzer computed the area of superel- lipse and the volume of this special superellipsoid when m is a natural number (Loria, 1910). 2.2.1 SUPERQUADRICS The term superquadrics was dened by Barr in his seminal paper (Barr, 1981).