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1. 1
Polygon Mesh

2. 2
• We need smooth curves and surfaces in
many applications:
– model real world objects
– computer-aided design (CAD)
– high quality fonts
– data plots
– artists sketches

3. 3
Introduction
• Most common representation for surfaces:
– polygon mesh
– parametric surfaces
– quadric surfaces
• Solid modeling
– don’t miss the next episode...

4. 4
Introduction
• Polygon mesh:
– set of connected planar surfaces bounded by
polygons
– good for boxes, cabinets, building exteriors
– bad for curved surfaces
– errors can be made arbitrarily small at the cost
of space and execution time
– enlarged images show geometric aliasing

5. 5
Introduction
• Parametric polynomial curves:
– point on 3D curve = (x(t), y(t), z(t))
– x(t), y(t), and z(t) are polynomials
– usually cubic: cubic curves

6. 6
Introduction
• Parametric bivariate (two-variable)
polynomial surface patches:
– point on 3D surface = (x(u,v), y(u,v), z(u,v))
– boundaries of the patches are parametric
polynomial curves
– many fewer parametric patches than
polynomial patches are needed to approximate
a curved surface to a given accuracy
– more complex algorithms though

7. 7
Parametric cubic curves
• Polylines and polygons:
– large amounts of data to achieve good accuracy
– interactive manipulation of the data is tedious
• Higher-order curves:
– more compact (use less storage)
– easier to manipulate interactively
• Possible representations of curves:
– explicit, implicit, and parametric

8. 8
Parametric cubic curves
• Polylines and polygons:
– large amounts of data to achieve good accuracy
– interactive manipulation of the data is tedious
• Higher-order curves:
– more compact (use less storage)
– easier to manipulate interactively
• Possible representations of curves:
– explicit, implicit, and parametric