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OUTLINE
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- Quadratic Bezier Surfaces
- Cubic Bezier Surfaces
BEZIER SURFACES 1 OUTLINE Quadratic Bezier Surfaces Cubic - - PowerPoint PPT Presentation
BEZIER SURFACES 1 OUTLINE Quadratic Bezier Surfaces Cubic - - PowerPoint PPT Presentation
BEZIER SURFACES 1 OUTLINE Quadratic Bezier Surfaces Cubic Bezier Surfaces 2 DE CASTELJAU RECURSION REVISITED l 0 =p 0 , r 3 =p 3 l 1 =1/2(p 0 +p 1 ), r 2 =1/2(p 2 +p 3 ) l 2 =1/2(l 1 +1/2(p 1 +p 2 )), r 1 =1/2(r 2
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DE CASTELJAU RECURSION REVISITED
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- l0=p0, r3=p3
- l1=1/2(p0+p1), r2=1/2(p2+p3)
- l2=1/2(l1+1/2(p1+p2)), r1=1/2(r2+1/2(p1+p2))
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CURVED SURFACE
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CONTROL POINTS AND RESULTING SURFACE
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QUADRATIC BLENDING FUNCTIONS
- These are the same as the quadratic
Bezier curve blending functions
- Except that now we use them in two
dimensions
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BEZIER PATCHES
- Double summation used for surfaces as opposed to curves
- The set of points generated for the Bezier surface is called a Bezier patch
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CUBIC BEZIER PATCHES
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CUBIC BEZIER SURFACES
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CUBIC BEZIER SURFACE BLENDING FUNCTIONS
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SURFACES
- Can apply the recursive method to surfaces - a
Bezier patch curves of constant u (or v) are Bezier curves in u (or v)
- First subdivide in u
- Process creates new points
- Some of the original points are discarded
- riginal and kept
new
- riginal and discarded
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SECOND SUBDIVISION
16 final points for 1 of 4 patches created
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BASE CONDITION
- With Bezier curves, the base condition was whether the curve was “straight”
enough
- With surfaces, the base condition is whether the surface is “flat” enough
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UTAH TEAPOT
- Most famous data set in computer graphics
- Widely available as a list of 306 3D vertices and the
indices that define 32 Bezier patches
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TESSELLATION
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RECURSIVE SUBDIVISION
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ADDING SHADING
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GEOMETRY SHADER
- Basic limitation on rasterization is that each execution of a vertex shader is
triggered by one vertex and can output only one vertex
- Geometry shaders allow a single vertex and other data to produce many vertices
- Example: send four control points to a geometry shader and it can produce as
many points as needed for Bezier curve
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TESSELLATION SHADERS
- Can take many data points and produce triangles
- More complex since tessellation has to deal with inside/outside issues
and topological issues such as holes
- We’ll be looking at geometry and tessellation shaders in upcoming
topics
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SLIDE 20
SUMMARY
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- Quadratic Bezier Surfaces
- Cubic Bezier Surfaces