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 The Differential Geometry of Parametric Primitives Ken Turkowski 26 January 1990 Differential Properties of Parametric Surfaces A parametric surface is a function: where is a point in affine 3-space, and 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: The Hessian is a tensor of second partial derivatives: The first fundamental form is defined as: Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 1 G = JJ t = ∂ x ∂ u • ∂ x ∂u ∂x ∂u • ∂ x ∂v ∂ x ∂v • ∂ x ∂u ∂x ∂ v • ∂ x ∂v           H = ∂ 2 x, y,z ( ) ∂ u, v ( ) ∂ u,v ( ) = ∂ 2 x ∂u 2 ∂ 2 y ∂u 2 ∂ 2 z ∂u 2       ∂ 2 x ∂u∂v ∂ 2 y ∂u∂v ∂ 2 z ∂u∂v       ∂ 2 x ∂v∂u ∂ 2 y ∂ v∂u ∂ 2 z ∂ v∂u       ∂ 2 x ∂ v 2 ∂ 2 y ∂ v 2 ∂ 2 z ∂v 2                   = ∂ 2 x ∂u 2 ∂ 2 x ∂u∂v ∂ 2 x ∂v∂u ∂ 2 x ∂v 2             J = ∂ x,y,z ( ) ∂ u,v ( ) = ∂ x ∂ u ∂ y ∂ u ∂ z ∂ u ∂ x ∂ v ∂ y ∂ v ∂ z ∂ v           = ∂x ∂u ∂x ∂v           u = u v [ ] x = x y z [ ] x = F u ( ) and establishes a metric of differential length: so that the arc length of a curve segment, is given by: The differential surface area enclosed by the differential parallelogram is approximately: so that the area of a region of the surface corresponding to a region R in the u-v plane is: The second fundamental matrix measures normal curvature, and is given by: The normal curvature is defined to be positive a curve u on the surface turns toward the positive direction of the surface normal by: The deviation (in the normal direction) from the tangent plane of the surface, given a differential displacement of is: Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 2 ˙˙ x • n = ˙ uD ˙ u t ˙ u κ n = ˙ uD ˙ u t ˙ uG ˙ u t D = n• H = n• ∂ 2 x ∂u 2 n • ∂ 2 x ∂u∂v n • ∂ 2 x ∂ v∂u n • ∂ 2 x ∂v 2             S = G ( ) R ∫∫ 1 2 dudv δ S≈ G ( ) 1 2 δ uδv δu,δv ( ) s = ds dt t 0 t 1 ∫ dt = ˙ x t 0 t 1 ∫ dt = ˙ x t 0 t 1 ∫ dt = ˙ uG ˙ u t ( ) t 0 t 1 ∫ 1 2 dt u = u t ( ) , t 0 < t < t 1 dx ( ) 2 = du ( ) G du ( ) t 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 and , and is the Jacobian matrix of the reparametrization. The new Hessian is given by where . The new fundamental matrix is given by: and the new curvature matrix is given by: Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 3 ′D = PDP T ′G = PGP T Q = ∂ u,v ( ) ∂ ′u 2 ∂ u,v ( ) ∂ ′u ∂ ′v ∂ u,v ( ) ∂ ′v ∂ ′u ∂ u,v ( ) ∂ ′v 2           ′H = PHP T +QJ P = ∂ u,v ( ) ∂ ′u , ′v ( ) = ∂ u ∂ ′u ∂v ∂ ′u ∂u ∂ ′v ∂v ∂ ′v           ′v ′u ′J = ∂ x, y,z ( ) ∂ ′u , ′v ( ) ′J = PJ ∂ x,y,z ( ) ∂ ′u , ′v ( ) = ∂ u,v ( ) ∂ ′u , ′v ( ) ∂ x,y,z ( ) ∂ u,v ( ) ′u = ′u u,v ( ) and ′ v = ′v u,v ( ) 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 , with associated Jacobian: When the change of coordinates is represented by the affine transformation: the Jacobian is simply the submatrix: Regardless, the Jacobian and Hessian transform as follows: The normal is transformed as: The denominator arises from the desire to have a unit normal. The first and second fundamental matrices are then calculated as: Not very pretty. But certain types of transformations can be applied easily. For a uniform scale with arbitrary translations, Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 4 C = r 0 0 0 r 0 0 0 r           = r I ′D = ′H • ′n = HC ( ) • nC −1t ( ) nC −1t C −1 n t ( ) 1 2 = HCC −1 n t nC −1t C −1 n t ( ) 1 2 = H •n nC −1t C −1 n t ( ) 1 2 = D nC −1t C −1 n t ( ) 1 2 ′G = ′J ′J t = JCC t J t ′n = nC −1t nC −1t C −1 n t ( ) 1 2 ′J = JC, ′H = HC C = x x y x z x x y y y z y x z y z z z           A = x x y x z x x y y y z y x z y z z z x o y o z o             C = ∂ ′x ∂ x = ∂ ′x ∂ x ∂ ′y ∂ x ∂ ′z ∂ x ∂ ′x ∂ y ∂ ′y ∂ y ∂ ′z ∂ y ∂ ′x ∂ z ∂ ′y ∂ z ∂ ′z ∂ z                 ′x = ′x x ( ) so that For rotations (and arbitrary translations), the Jacobian matrix C=R is orthogonal, so the inverse is equal to the transpose, yielding: Combining the two, we have the results for a transformation that includes translations, rotations and uniform scale: or in terms of the composite matrix : Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 5 ′J = JC, ′H = HC, ′n = nC C ( ) 1 3 , ′G = C ( ) 2 3 G, ′D = C ( ) 1 3 D C = r R ′J = rJR, ′H = rHR, ′n = nR, ′G = r 2 G, ′D = rD ′J = JR, ′H = HR, ′n = nR, ′G = G, ′D =D ′J = rJ, ′H = rH, ′n = n, ′G = r 2 G, ′D = r D Sphere Given the spherical coordinates: we have the Jacobian matrix: the Hessian tensor: the first fundamental form: the normal: and the second fundamental form: Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 6 D = − x 2 + y 2 r 0 0 −r         n = x r y r z r       G = x 2 + y 2 0 0 r 2       ∂ 2 x, y,z ( ) ∂ θ,φ ( ) ∂ θ ,φ ( ) = − x −y 0 [ ] − yz x 2 + y 2 xz x 2 + y 2 0       − yz x 2 + y 2 xz x 2 + y 2 0       − x −y −z [ ]               ∂ x,y,z ( ) ∂ θ ,φ ( ) = − y x 0 xz x 2 + y 2 yz x 2 + y 2 − x 2 + y 2         x y z [ ] = r sinφ cosθ r sin φ sinθ r cosφ [ ] Unit Sphere Angle Parametrization Given the unit spherical coordinates with , we parametrize the sphere: This yields the Jacobian matrix: the Hessian tensor: the first fundamental form: the normal: and the second fundamental form: Angle Parametrization With the reparametrization , we have the Jacobian: Applying the chain rule, we have: Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 7 J uv = −2π y 2π x 0 πxz x 2 + y 2 πyz x 2 + y 2 −π x 2 + y 2         P = 2π 0 0 π       θ = 2π u, ϕ = πv D θφ = − x 2 + y 2 ( ) 0 0 −1       n = x y z [ ] G θφ = x 2 + y 2 0 0 1       H θφ = − x −y 0 [ ] − yz x 2 + y 2 xz x 2 + y 2 0       − yz x 2 + y 2 xz x 2 + y 2 0       − x −y −z [ ]               J θφ = − y x 0 xz x 2 + y 2 yz x 2 + y 2 − x 2 + y 2         x y z [ ] = sin φ cosθ sin φ sinθ cosφ [ ] 0 ≤ θ < 2π, 0 ≤ ϕ < π 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: Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 8 D uv = − 4π 2 x 2 + y 2 ( ) r 0 0 −π 2 r         D uv = −4π 2 x 2 + y 2 ( ) 0 0 −π 2       G uv = 4π 2 x 2 + y 2 ( ) 0 0 π 2       H uv = 4π 2 −x −y 0 [ ] 2π − yz x 2 + y 2 xz x 2 + y 2 0       2π − yz x 2 + y 2 xz x 2 + y 2 0       π 2 − x −y −z [ ]               Cone Angle Parametrization Given the unit conical parametrization: we have the Jacobian matrix: the Hessian tensor: the first fundamental form: the normal: and the second fundamental form: Unit Parametrization For the parametrization: we have: Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Apple Computer, Inc. Media Technology: Computer Graphics Page 9 H uv = 4π 2 −x −y 0 [ ] 2π h rz −y x 0 [ ] 2π h rz −y x 0 [ ] 0 0 0 [ ]           J uv = −2π y 2π x 0 hx rz hy rz h         x y z [ ] = rvcos2 π u rv sin2 πu vh [ ] D θ z = − z 2 0 0 0         n θz = x z 2 y z 2 − 1 2       G θ z = x 2 + y 2 0 0 x 2 + y 2 + z 2 z 2         = z 2 0 0 2       H θ z = −x −y 0 [ ] − y z x z 0       − y z x z 0       0 0 0 [ ]           J θ z = − y x 0 x z y z 1         x y z [ ] = zcosθ zsinθ z [ ] [...]...Turkowski The Differential Geometry of Parametric Primitives 4π 2( x2 + y2)  Guv =  0   nuv =  h2x  1+ h2  rz 1  4π 2 rz − Duv =  1 + h2   0 Apple Computer, Inc 26 January 1990  2 2 2 2  h(x + y + z)   z2 0 h2y rz  −1   0  0 Media Technology: Computer Graphics Page 10 Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Cylinder Angle Parametrization Given the. .. matrix: −2π y 2π x 0 Juv =  0 h  0  the Hessian tensor: [ −4π 2 x −4π 2y 0] [0 0 0]  Huv =   [0 0 0] [0 0 0]   the first fundamental form: 4π 2 r 2 0  Guv =   h2   0 the normal: Apple Computer, Inc Media Technology: Computer Graphics Page 11 Turkowski x n= r The Differential Geometry of Parametric Primitives y r 26 January 1990  0  and the second fundamental form:  −4π 2r Duv... 0 Media Technology: Computer Graphics Page 12 Turkowski The Differential Geometry of Parametric Primitives 26 January 1990 Torus Angle Parametrization Given the torus parametrization: [ x y z] = [ (R + r cosφ ) cosθ (R + r cosφ ) sin θ r sin φ ] we have the Jacobian matrix: −y   Jθφ = − xz  2 2  x +y   x2 + y2 − R  x yz − 0 x2 + y2 the Hessian tensor:   [ − x −y 0]  Hθφ =   xz   yz... = [ cosθ z] sinθ we have the Jacobian matrix: − y x 0  Jθφ =    0 0 1 the Hessian tensor: [ − x −y 0] [ 0 0 0]  Hθφ =   [ 0 0 0]   [0 0 0] the first fundamental form: 1 0 Gθφ =   0 1 the normal: n = [x y 0] and the second fundamental form: −1 0 Dθφ =    0 0 Unit Parametrization With the parametrization: [ x y z] = [ r cos2 π u r sin2 πu hv] we have the Jacobian matrix: −2π... − y1 − 2  −z  2 2  x +y  x +y        the first fundamental form: x2 + y2 0  Gθφ =   r 2  0 the normal: R   1− 2 2 x +y x n= r    1− y R x +y 2 2 r   z  r   and the second fundamental form: Dθφ  x2 + y2   R  − 1 − 2  0 2 = r  x +y     0 −r     R2 − x 2 − y2 + z2 − r 2 = 2r  0   0  −r  using the torus’s implicit equation: ( ) 2 x + y − R +z = r . on the surface turns toward the positive direction of the surface normal by: The deviation (in the normal direction) from the tangent plane of the surface, given a differential displacement of. D Sphere Given the spherical coordinates: we have the Jacobian matrix: the Hessian tensor: the first fundamental form: the normal: and the second fundamental form: Turkowski The Differential Geometry of Parametric. du ( ) t 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