The differential Geometry of parametric primitives

The differential Geometry of parametric primitives - THE...

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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

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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 ( 29 u , v ( 29 ∂ u , v ( 29 = 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 ( 29 u , v ( 29 = 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 ( 29
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 ( 29 R ∫∫ 1 2 dudv δ S G ( 29 1 2 δ u δ v δ u

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The differential Geometry of parametric primitives - THE...

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