COMPUTING
Lecture 12 - Spline surfaces (notes)

Lecture 12 - Spline surfaces (notes) - Lecture 12...

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Lecture 12: Introduction to Surface Construction Non-Parametric Surfaces We now turn to the question of how to represent and draw surfaces. As was the case with constructing spline curves, one possibility is to adopt the simple solution of non-parametric Cartesian equations. A quadratic surface would have an equation of the form: ( x y z 1 ) a b c d b e f g c f h j d g j 1 x y z 1 = 0 which multiplies out to: ax 2 + ey 2 + hz 2 + 2 bxy + 2 cxz + 2 fyz + 2 dx + 2 gy + 2 jz + 1 = 0 and the nine scalar unknowns { a, b, . . . j } can be found by specifying nine points P i = ( x i , y i , z i ) , i = 1 , . . . , 9 , through which the surface must pass. This creates a system of nine linear equations to solve. Notice that because of the symmetry of the matrix there are only nine unknowns, not sixteen. The constant term can always be taken as 1 without loss of generality. This method however suffers the same limitations as the analogous method for curves. It is difficult to control the surface shape since there is only one quadratic surface that will fit the points. Parametric Surfaces It simple to generalise from parametric curves to parametric surfaces. In the case of spline curves, the locus of a point was a function of one parameter. However a point on a surface patch is a function of two parameters, which we write as P ( μ, ν ) . One way to do this is to define a parametric surface using a matrix formulation: P ( μ, ν ) = ( μ ν 1 ) a b c b d e c e f μ ν 1 Multiplying out we get: P ( μ, ν ) = a μ 2 + d ν 2 + 2 b μν + 2 c μ + 2 e ν + f (1) The values of the constant vectors { a , b , . . . , f } determine the shape of the surface. The surface has edges given by the four curves for which one of the parameters ν and μ is constant at either 0 or 1. These are the quadratics: P (0 , ν ) = d ν 2 + 2 e ν + f P (1 , ν ) = a + 2( b + e ) ν + 2 c + d ν 2 + f P ( μ, 0) = a μ 2 + 2 c μ + f P ( μ, 1) = a μ 2 + 2( b + c ) μ + d + 2 e + f The unknown values in the matrix ( a , b , c . . . ) are all vectors whose values can be computed by substituting in six points to be interpolated for given values of μ and ν . Just as we did for the spline curves, we need to specify the values of μ and ν where the knots are located. For example, one possibility is shown on the right. These values will create a surface that is similar to the one in Figure 1. μ ν P 0 0 0 P 1 0 1 P 2 1 0 P 3 1 1 P 4 1/2 0 P 5 1/2 1 Interactive Computer Graphics Lecture 12 1
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P 1 P 0 P 2 P 3 P 4 P 5 μ = 0 curve ν = 1 curve ν = 0 curve μ = 1 curve Figure 1: A quadratic surface determined by choosing 6 points Substituting the various choices of μ , ν for these points into equation 1 we obtain a set of linear equations: P 0 = f P 1 = d + 2 e + f P 2 = a + 2 c + f P 3 = a + 2 b + 2 c + d + 2 e + f P 4 = a / 4 + c + f P 5 = a / 4 + b + c + d + 2 e + f We can solve these equations to find the values of the constants { a , b , . . . , f } that define the patch. This is more flexible than the non-parametric formulation, but it still does not provide us with a very useful spline since there are no intuitive ways of using it to create a particular shape.
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  • Spring '14
  • Parametric equation, Coons Patch

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