Lecture23-Surfaces

Lecture23-Surfaces - CS 455 Computer Graphics Surfaces...

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    CS 455 – Computer Graphics Surfaces
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Parametric Bicubic Surfaces An extension to curves by adding another dimension s = 0.2 s = 0.4 s = 0.6 s = 0.8 s = 1.0 s = 0.0 t = 0.0 t = 0.2 t = 0.4 t = 0.6 t = 0.8 t = 1.0
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Curve Equation Remember, the equation of the curve is: Or, equivalently: Where G is the geometry vector, and is a constant G M T t Q = ) ( G M S s Q = ) (
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The Surface Equations If we allow G to be a function, we get i.e., the geometry can now change, based on t = = ) ( ) ( ) ( ) ( ) ( ) , ( 4 3 2 1 t G t G t G t G M S t G M S t s Q
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The Surface Equations • For a fixed t 1 , Q ( s, t 1 ) is a curve, since G ( t 1 ) is a constant. • Taking a new t 2 that is near in value to t 1 will give another curve which is slightly different from the first. Repeating the process for some number of parameters t , with 0 t < 1 will give a group of curves that define a surface.
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The Surface Equations • Each of the G i ( t ) functions are cubics, and can be represented as: where i i G M T t G = ) ( = 4 3 2 1 i i i i i g g g g G
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Surface Equations Taking the transpose of this gives us If we substitute this back into our original equation, and expand to include all four geometry terms, we get: T T T i i T M G t G = ) ( T T T T M G M S t s Q = ) , (
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So, With being the geometry matrix and M being the basis matrix. T
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Lecture23-Surfaces - CS 455 Computer Graphics Surfaces...

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