THE DIFFERENTIAL GEOMETRY OF PARAMETRIC PRIMITIVES

Transcription

1 THE DIFFERENTIAL GEOMETRY OF PARAMETRIC PRIMITIVES Ken Turkowski Media Technologies: Graphics Software Advanced Technology Group Apple Computer, Inc. (Draft Friday, May 18, 1990) Abstract: We derive the expressions for first and second derivatives, normal, metric matrix and curvature matrix for spheres, cones, cylinders, and tori. 26 January 1990 Apple Technical Report No. KT-23

2 The Differential Geometry of Parametric Primitives Ken Turkowski 26 January 1990 Differential Properties of Parametric Surfaces A parametric surface is a function: where x F( u) x [ x y z] is a point in affine 3-space, and u [ u v] is a point in affine 2-space. The Jacobian matrix is a matrix of partial derivatives that relate changes in u and v to changes in x, y, and z: J ( x,y,z) ( u,v) x u x v y u y v z x u u z x v v The Hessian is a tensor of second partial derivatives: H 2 x 2 ( x, y,z) ( u, v) ( u,v) u 2 2 x v u 2 y u 2 2 y v u 2 z u 2 2 z v u 2 x u v 2 x v 2 2 y u v 2 y v 2 2 z u v 2 z v 2 2 x u 2 2 x v u 2 x u v 2 x v 2 The first fundamental form is defined as: x G JJ t u x u x v x u x u x v x v x v Apple Computer, Inc. Media Technology: Computer Graphics Page 1

3 and establishes a metric of differential length: ( dx) 2 ( du)g ( du) t so that the arc length of a curve segment, u u( t), t 0 < t < t 1 is given by: t 1 ds t 1 t 1 t 1 s dt ẋ t 0 dt dt ẋ t 0 dt ug u t 2 dt t 0 t 0 1 ( ) The differential surface area enclosed by the differential parallelogram ( δu,δv) is approximately: δ S ( G) 1 2 δ uδv so that the area of a region of the surface corresponding to a region R in the u-v plane is: S R 1 ( G) 2 dudv The second fundamental matrix measures normal curvature, and is given by: n 2 x u 2 D n H n 2 x v u n 2 x u v n 2 x v 2 The normal curvature is defined to be positive a curve u on the surface turns toward the positive direction of the surface normal by: κ n ud u t ug u t The deviation (in the normal direction) from the tangent plane of the surface, given a differential displacement of u is: x n ud u t Apple Computer, Inc. Media Technology: Computer Graphics Page 2

4 Reparametrization If the parametrization of the surface is transformed by the equations: then the chain rule yields: or where is the new Jacobian matrix of the surface with respect to the new parameters u and v, and is the Jacobian matrix of the reparametrization. The new Hessian is given by where u u ( u,v) and v v ( u,v) ( x,y,z) ( u, v ) ( u,v) ( u, v ) J PJ x, y,z J ( ) ( u, v ) P ( u,v) ( u, v ) H PHP T +QJ ( u, v) u 2 Q ( u, v) v u ( u,v) u v ( u,v) v 2. The new fundamental matrix is given by: G PGP T and the new curvature matrix is given by: D PDP T u u u v ( x,y,z) ( u,v) v u v v Apple Computer, Inc. Media Technology: Computer Graphics Page 3

5 Change of Coordinates For simplicity, we have defined several primitives with unit size, located at the origin. Related to the reparametrization is the change of coordinates x x ( x), with associated Jacobian: C x x x x x y x z y x y y y z z x z y z z When the change of coordinates is represented by the affine transformation: x x y x z x x y y z y A x z y z z z x o y o z o the Jacobian is simply the submatrix: x x y x z x C x y y z y x z y z z z Regardless, the Jacobian and Hessian transform as follows: J JC, H HC The normal is transformed as: n nc 1 t ( nc 1t C 1 n t ) 1 2 The denominator arises from the desire to have a unit normal. The first and second fundamental matrices are then calculated as: G J J t JCC t J t D H ( n HC ) nc 1 t Not very pretty. But certain types of transformations can be applied easily. For a uniform scale with arbitrary translations, r 0 0 C 0 r 0 r I 0 0 r ( ) ( nc 1t C 1 n t ) 1 2 HCC 1 n t ( nc 1t C 1 n t ) 1 2 H n ( nc 1 t C 1 n t ) 1 2 D ( nc 1t C 1 n t ) 1 2 Apple Computer, Inc. Media Technology: Computer Graphics Page 4

6 so that J rj, H rh, n n, G r 2 G, D r D For rotations (and arbitrary translations), the Jacobian matrix CR is orthogonal, so the inverse is equal to the transpose, yielding: J JR, H HR, n nr, G G, D D Combining the two, we have the results for a transformation that includes translations, rotations and uniform scale: J rjr, H rhr, n nr, G r 2 G, D rd or in terms of the composite matrix C r R: J JC, H HC, n nc, G ( C) 2 3 G, D ( C ) 1 3 D ( C) 1 3 Apple Computer, Inc. Media Technology: Computer Graphics Page 5

7 Sphere Given the spherical coordinates: [ x y z] [ r sinφ cosθ r sin φ sinθ r cosφ ] we have the Jacobian matrix: ( x,y,z) ( θ,φ ) the Hessian tensor: y x 0 xz yz 2 ( x, y,z) ( θ,φ ) ( θ,φ ) yz xz [ x y 0] 0 yz xz 0 [ x y z] the first fundamental form: G x2 + y r 2 the normal: n x y z r r r and the second fundamental form: D x2 + y 2 0 r 0 r Apple Computer, Inc. Media Technology: Computer Graphics Page 6

8 Unit Sphere Angle Parametrization Given the unit spherical coordinates with 0 θ < 2π, 0 ϕ < π, we parametrize the sphere: [ x y z] [ sin φ cosθ sin φ sinθ cosφ] This yields the Jacobian matrix: J θφ y x 0 xz yz the Hessian tensor: H θφ yz xz [ x y 0] 0 yz xz 0 [ x y z] the first fundamental form: the normal: G θφ x2 + y n [ x y z] and the second fundamental form: ( ) 0 D θφ x2 + y Angle Parametrization With the reparametrization θ 2π u, ϕ π v, we have the Jacobian: P 2π 0 0 π Applying the chain rule, we have: J uv 2π y 2π x 0 πxz πyz π Apple Computer, Inc. Media Technology: Computer Graphics Page 7

9 H uv 2π yz xz 4π 2 [ x y 0] 2π 0 yz xz 0 π 2 [ x y z] ( ) 0 G uv 4π2 0 π 2 ( ) 0 D uv 4π 2 0 π 2 Changing coordinates to yield a sphere of arbitrary radius, we find that the expressions for the Jacobian, the Hessian, and the metric matrix remain the same, because x, y, and z scale linearly with r. The curvature matrix changes to: ( ) D uv 4π 2 0 r 0 π 2 r Apple Computer, Inc. Media Technology: Computer Graphics Page 8

10 Cone Angle Parametrization Given the unit conical parametrization: we have the Jacobian matrix: the Hessian tensor: the first fundamental form: the normal: [ x y z] [ zcosθ zsinθ z] y x 0 J θ z x y z z 1 [ x y 0] y x 0 z z H θ z y x z z 0 [ 0 0 0] 0 G θ z + z 2 0 z x n θ z z 2 and the second fundamental form: y z 2 D θ z z z Unit Parametrization For the parametrization: we have: [ x y z] [ rvcos2 π u rv sin2 πu vh] 2π y 2π x 0 J uv hx hy h rz rz H uv 4π 2 [ x y 0] 2π h [ rz y x 0 ] 2π h rz y x 0 [ ] [ ] Apple Computer, Inc. Media Technology: Computer Graphics Page 9

11 4π 2 G uv 0 ( ) 0 h 2 ( + z 2 ) z 2 n uv 1 h 1+ h 2 rz 2 x h 2 y rz 1 D uv 4π 2 rz h Apple Computer, Inc. Media Technology: Computer Graphics Page 10

12 Cylinder Angle Parametrization Given the cylindrical parametrization: we have the Jacobian matrix: the Hessian tensor: the first fundamental form: the normal: [ x y z] [ cosθ sinθ z] J θφ y x H θφ [ x y 0] [ 0 0 0] [ 0 0 0] [ 0 0 0] G θφ n [ x y 0] and the second fundamental form: D θφ Unit Parametrization With the parametrization: we have the Jacobian matrix: the Hessian tensor: the first fundamental form: the normal: [ x y z] [ r cos2 π u r sin2πu hv] 2π y 2π x 0 J uv 0 0 h [ ] H uv 4π 2 x 4π 2 y 0 [ ] [ 0 0 0] [ 0 0 0] G uv 4π2 r h 2 Apple Computer, Inc. Media Technology: Computer Graphics Page 11

13 n x y 0 r r and the second fundamental form: D uv 4π 2 r Apple Computer, Inc. Media Technology: Computer Graphics Page 12

14 Torus Angle Parametrization Given the torus parametrization: we have the Jacobian matrix: [ ] [ x y z] ( R+ r cosφ) cosθ ( R+ r cosφ )sin θ r sin φ J θφ y x 0 xz yz R the Hessian tensor: H θφ yz [ x y 0] xz 0 x 1 yz R xz 0 R y 1 z the first fundamental form: G θφ x2 + y r 2 the normal: 1 n x R r y 1 R r z r and the second fundamental form: D θφ x2 + y 2 R 1 0 r 0 r R 2 x 2 y 2 + z 2 r 2 0 2r 0 r using the torus s implicit equation: ( R) 2 + z 2 r 2 Apple Computer, Inc. Media Technology: Computer Graphics Page 13