Korea University Jung Lee, Computer Graphics Laboratory 3D Game Engine Design David H. Eberly 8.3 Special Surfaces 2001/11/13

Jung Lee, Computer Graphics Laboratory Korea University 2 Bezier Rectangle Patches  Popular with game programmers Mathematical simplicity Ease of use  Equation : ( 0  i 0  n 0, 0  i 1  n 1 ) : 3D control points The first-order partial derivatives of the patch

Jung Lee, Computer Graphics Laboratory Korea University 3 Bezier Rectangle Patches Evaluation  Trade-off of speed vs. accuracy  Bernstein polynomial computation n 0 +1 evaluations of B n0,i0 (s) n 1 +1 evaluations of B n1,i1 (t) n 0 n 1 multiplications for pairs of the evaluated polynomials  de Casteljau algorithm Repeatedly computes convex combinations Generally stable Use more floating-point operations

Jung Lee, Computer Graphics Laboratory Korea University 4 Comparison For Bilinear Interpolation  Case where n 0 =1 and n 1 =1  Bernstein form 2 subtractions, 9 additions, and 16 multiplications  de Casteljau form 2 subtractions, 9 additions, and 18 multiplications

Jung Lee, Computer Graphics Laboratory Korea University 5 Degree Elevation  Degree-elevated to (n 0 +1, n 1 )  Degree-elevated to (n 0, n 1 +1)  Degree-elevated to (n 0 +1, n 1 +1) Degree-elevated control points

Jung Lee, Computer Graphics Laboratory Korea University 6 Degree Reduction  Let the degree-reduced surface m 0  n 0, m 1  n 1 Constraint  The endpoints of the two curves are same  Two choices for the extension to rectangle patches  The four corner points match between the two patches Least-squares fit to construct the remaining control points Process  Degree-reduce the four boundary curves first  Compute the remaining interior control points  Least-squares fit

Jung Lee, Computer Graphics Laboratory Korea University 7 Bezier Triangle Patches  Slightly more complicated than Bezier rectangle patches  Useful for creating models of arbitrary complexity  Equation : (i 0, i 1, i 2  0, i 0 + i 1 + i 2 = n)  Evaluation Computing the coefficients  To minimize arithmetic operations by saving intermediate products and sums

Jung Lee, Computer Graphics Laboratory Korea University 8 Degree Elevation  Multiply the original patch by Degree-elevated patch  Degree-elevated control points

Jung Lee, Computer Graphics Laboratory Korea University 9 Degree Reduction  Let the degree-reduced surface m  n Constraint  The endpoints of the two curves are same  Two choices for the extension to triangle patches  The three corner points match between the two patches Least-squares fit to construct the remaining control points Process  The boundary curves are reduced first  Compute the remaining interior control points  Least-squares fit

Jung Lee, Computer Graphics Laboratory Korea University 10 Bezier Cylinder Surfaces (1 / 2)  Bezier rectangle/triangle patches More curvature variation than is needed  Cylinder/developable surfaces Curved in 1D and flat Two choice of generation  Taking a curve and linearly translating  Sweeping operation  Taking a line segment and moving it  One endpoint is constrained to the originally specified curve Evaluation, degree elevation/reduction can be applied to  Initial curve for cylinder surfaces  Boundary curves for generalized cylinder surfaces

Jung Lee, Computer Graphics Laboratory Korea University 11 Bezier Cylinder Surfaces (2 / 2)  Cylinder Surface : Bezier curve : Linear translation : The first-order partial derivatives :  Generalized cylinder surface Specifying two Bezier curves  With the same number of control points Blending between the two curves Equation :  Control points : for 0  i  n The first-order partial derivatives

Jung Lee, Computer Graphics Laboratory Korea University 12 Nonparametric B-Spline Rectangle Patches  How to interpolate a rectangle lattice of scalars for 0  i 0  n 0, 0  i 1  n 1 Componentwise interpolation  B-spline polynomial of degree d for 0  k 0  d for 0  k 1  d : intermediate tensor  For applications The intermediate tensor can be cached

Jung Lee, Computer Graphics Laboratory Korea University 13 Quadric Surfaces  Discussion of the general quadratic equation A : 3  3 nonzero symmetric matrix : 3  1 vector c : scalar : 3  1 vector that represents the variable quantities  A = R T DR D : diagonal matrix  Diagonal entities are the eigenvalues of A R : rotational matrix  Rows are corresponding eigenvectors

Jung Lee, Computer Graphics Laboratory Korea University 14 Eigendecomposition  Rewritten quadratic equation,, Characterize the surface type based on the number of nonzero eigenvalues  Three : elliptic cone, ellipsoid, hyperboloid  Two : line, union of two planes, elliptic cylinder, hyperbolic cylinder, hyperbolic paraboloid  One : parabolic cylinder Determine that the solution is degenerate from  Point, line, plane

Jung Lee, Computer Graphics Laboratory Korea University 15 Tube Surfaces  Swept surface Sweeping a region of space by a planar object along a specified central curve  Most common planar object : circle  Radius varies with time : r(t)  Resulting surface is called a tube surface  Equation : : columns of an orientation matrix R(t)  pp.286 t : curve parameter   [0, 2  )  Surface of revolution  The central curve is a straight line