Sphere-geometric aspects of bisector surfaces

Transcription

1 Sphere-geometric aspects of bisector surfaces Martin eternell Vienna University of Technology, AGGM 2006, arcelona, September

2 Definition Smooth oriented objects and in R 3 (points, curves or surfaces) The bisector surface of and is the locus of centers of spheres tangent both to and. p b q Tangents of are the bisectors of tangents of and at corresponding points p and q AGGM 2006, arcelona, September

3 Elementary Examples in R 2 object 1 object 2 bisector point point line of symmetry or. line or. line line of symmetry point line parabola AGGM 2006, arcelona, September

4 Elementary Examples in R 2 The bisector of a point and a circle is a conic. Ellipse: is inside Hyperbola: is outside S(t) S(t) c(t) c(t) AGGM 2006, arcelona, September

5 oint-curve isector in R 2 The bisector of a point = (0, 0) and a curve = c(t) in R 2 is the envelope of all lines of symmetry S(t) : c(t) (x 1 c(t)) = 0. 2 y intersecting s normals with the lines of symmetry the bisector admits the parametrization b(t) = c(t) 1 2 where n = ( ċ 2, ċ 1 ) is a normal vector of. c(t) c(t) c(t) n(t) n(t), AGGM 2006, arcelona, September

6 Elementary examples of bisectors in R 3 point point plane of symmetry point plane paraboloid of revolution or. plane or. plane plane of symmetry AGGM 2006, arcelona, September

7 Elementary examples of bisectors in R 3 point line parabolic cylinder line plane quadratic cone line line hyperbolic paraboloid p(u) n( ) b(u, v) b q(v) AGGM 2006, arcelona, September

8 Offset-invariance of bisectors Let d and d be the offset surfaces of and at oriented distance d. The bisector of d and d is the bisector of and. AGGM 2006, arcelona, September

9 Geometrical optics Consider two smooth surfaces and and their bisector surface. An illumination L orthogonal to is reflected at to an illumination L = L perpendicu- lar to. The bisector is a mirror surface in that sense. AGGM 2006, arcelona, September

10 Curve-curve bisector in R 3 Let and be two curves in R 3 with parametrizations p(u) and q(v). The bisector surface is constructed by b(u, v) = N (u) N (v) S(u, v) N b q(v) where N and N and S is the plane of symmetry of and. p(u) N N The bisector construction is linear. Rational curves, possess a rational bisector. AGGM 2006, arcelona, September

11 Circle-circle bisector in R 3 Let and be two circles in R 3 with parametrizations p(u) and q(v). The bisector surface carries two families of conics in planes through the axes A and A of and. is a double lutel conic surface (Degen, 64, 65, 86, 98). p(u) A A q(v) AGGM 2006, arcelona, September

12 Some related work Choi,J.J., Kim,M-S. and Elber, G.: Computing lanar isector Curves ased on Developable SSI. Dutta, D. and Hoffmann, C. On the skeleton of simple CSG objects. Elber, G. and Kim, M-S.: The isector Surface of Rational Space Curves. Elber, G. and Kim, M-S.: Rational bisectors of CSG rimitives. Farouki, R.T. and Johnstone, J.K.: The bisector of a point and a plane parametric curve. Farouki, R.T. and Johnstone, J.K.: Computing point/curve and curve/curve bisectors. Farouki, R.T. and Ramamurthy, R.: Specified recision Computation of Curve/Curve isectors. Hoffmann, C.: A dimensionality paradigm for surface interrogations, Literature on medial axis, skeleton and voronoi diagrams. AGGM 2006, arcelona, September

13 3D-Sphere Geometry An oriented (or.) sphere S in R 3 is given by S S : (x m) 2 r 2 = 0, E e m and the orientation is determined by or. normals. oints are considered as spheres of radius 0. An oriented plane E in R 3 is given by f F f e E E : e 0 + e 1 x 1 + e 2 x 2 + e 3 x 3 = e 0 + e x = 0. We always assume that e 2 = 1. F S m E and S are said to be in oriented contact iff e 0 + e 1 m 1 + e 2 m 2 + e 3 m 3 + r = e 0 + e m + r = 0, e 2 = 1. AGGM 2006, arcelona, September

14 Lie-sphere geometry in R 3 A Lie-sphere is either an or. sphere or an or. plane or a point in R 3. A point-model of Lie-sphere geometry is given by the quadric model L 5, L : 2X 0 X 5 + X1 2 + X2 2 + X3 2 X4 2 = 0. Lie-spheres X in R 3 are mapped onto points X L 5 by the correspondence sphere (m, r) point (p) M = (1, m, r, 1 2 (m2 r 2 ))R, = (1, p, 0, 1 2 p2 )R, plane (e 0, e) E = (0, e, 1, e 0 )R, with e = 1. oints x R 3 are mapped onto points X in X 4 = 0. lanes E are mapped onto points E X 0 = 0. AGGM 2006, arcelona, September

15 Relations to Laguerre and Möbius geometry Z = (0, 0, 0, 0, 0, 1)R is not an image of a sphere, plane or point of R 3 but is considered as point (one-point compactification). The quadratic cone X 0 = 0, X1 2 + X2 2 + X3 2 X4 2 = 0 consists of images of oriented planes and is referred to as laschke cone (cylinder). The quadric L X 4 = 0 is projectively equivalent to S 3 and is a point model of the Möbius geometry in R 3. A line L corresponds to a pencil of touching spheres or parallel planes. X 0 = 0 Z S 3 X 4 = 0 L AGGM 2006, arcelona, September

16 Lie-transformations and oriented contact A bijective mapping in the set of Lie-spheres which preserves oriented contact of spheres is called a Lie-transformation. The Lie-transformations appear in the quadric model as projective transformations L L. Lie-transformations are not necessarily point-preserving. oint-preserving Lie-transformations are Möbius transformations. lane-preserving Lie-transformations are Laguerre transformations. AGGM 2006, arcelona, September

17 Oriented contact of Lie-spheres olar form of L X, Y = X 0 Y 5 + X 5 Y 0 + X 1 Y 1 + X 2 Y 2 + X 3 Y 3 X 4 Y 4, Two Lie-spheres X, Y are in oriented contact exactly if X, Y = 0, and X, X = 0, Y, Y = 0. Any oriented plane is in contact with, E, Z = 0. Oriented spheres or points are never in contact with, M, Z = 0. AGGM 2006, arcelona, September

18 General bisector construction Surfaces, in R 3 with images (u, v) and (s, t) in L 5. X is tangent to and exactly if, X = 0, u, X = 0, v, X = 0,, X = 0, s, X = 0, t, X = 0, and X, X = 0 (X L) holds, (X 0 0). Eliminating u, v and s, t gives us three equations F (X) = 0, G(X) = 0, X, X = 0. isector pre-image in L is two-dimensional. The bisector is the projection of onto X 4 = X 5 = 0. AGGM 2006, arcelona, September

19 isectors of Lie-spheres Lie-spheres, in R 3 which are not in oriented contact. The bisector pre-image is the intersection L 3 :, X =, X = 0. is a quadric (of revolution) of index 0 in general. is planar if, are spheres of same radius or oriented planes. AGGM 2006, arcelona, September

20 Dupin cyclides Lie-spheres,, R and image points,, R, lane E = R, lane F =, X =, X = R, X = 0, E F : X E and Y F = X, Y = 0 Conic C = E L and conic D = F L. Family of spheres C corr. to C envelope a Dupin cyclide Φ. Family of spheres D corr. to = D envelope Φ too. Φ admits two generations as canal surface. AGGM 2006, arcelona, September

21 Dupin cyclides special cases Or. planes, and R = C are tangent planes of a cone (cylinder) of revolution Φ. D is the family of spheres touching Φ with centers at Φ s axis. AGGM 2006, arcelona, September

22 Lie-canal surfaces A one-parameter family of Lie-spheres C(u) is called a Lie-canal surface and corresponds to a curve C(u) L. The envelope Φ of C(u) is either a canal surface, a developable surface if C X 0 = 0 or a curve if C X 4 = 0. AGGM 2006, arcelona, September

23 isectors of Lie-canal surfaces Lie canal surfaces and with image curves (u), (v) L. Contact conditions, X = 0, u, X = 0,, X = 0, v, X = 0, define a two parameter family of lines G(u, v) in 5. isector pre-image (u, v) = G(u, v) L. AGGM 2006, arcelona, September

24 isector of two Lie-canal surfaces The bisector surface of two Lie canal surfaces, R 3 can be constructed in an elementary way (square roots). The construction is linear if and are both curves or developable surfaces. AGGM 2006, arcelona, September

25 isector of Lie-sphere and general surface General surface and Lie-sphere with images (u, v) and. Contact conditions, X = 0, u, X = 0, v, X = 0,, X = 0. define a two parameter family of lines G(u, v) in 5. isector pre-image (u, v) = G(u, v) L. AGGM 2006, arcelona, September

26 isector of Lie-sphere and general surface The bisector of a general surface R 3 and a Lie-sphere can be constructed in an elementary way. If is a point or an or. plane the construction is linear. If is a rational offset surface, is rational too. ( is shrunk to a point) AGGM 2006, arcelona, September

27 Summary We have presented a general concept for the geometric interpretation of bisector surfaces of two objects in R 3. oints, or. planes and spheres can be treated similarly. Elementary construction of the bisector of two Lie canal surfaces. Elementary construction of the bisector of a Lie sphere and a general surface. Thank you for your attention! AGGM 2006, arcelona, September