Parametric Surface Patches in Interactive Computer Graphics | CS 418, Study notes of Computer Graphics

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Natural way to think of a surface:

• curve is swept, and (possibly) deformed.

• Examples:

• ruled surface (line is swept),

• surface of revolution (circle is swept along line, grows and shrinks).

• Surface form:

Tensor product surfaces

x(u, v) = (x 1 (u)x 2 (v), y 1 (u)y 2 (v), z 1 (u)z 2 (v))

Tensor product surfaces

• Suggests form for surfaces:

ij

Xij fi(u)fj (v)

Tensor Product Bezier patches

Construct by de Casteljau algorithm

• repeated linear interpolation one way

• now go the other way

• OR

• repeated bilinear interpolation

P 589

Bilinear interpolation

Bilinear interpolation

Repeated bilinear interpolation yields a surface too

Tensor Product Bezier Patches

• It follows from the tensor product form that surface:

• interpolates four vertex points

• tangent plane at each vertex is given by three points at that vertex

• repeated de Casteljau (one direction, then the other) gives a point on the

surface, tangent plane to surface

Tensor product Bezier patches

Recall we wrote curves as:

• We can write surface as:

[

u 3 u 2 u 1

]

M

p 0 p 1 p 2 p 3     [ u 3 u 2 u 1

]

M

p 00 p 01 p 02 p 03 p 10 p 11 p 12 p 13 p 20 p 21 p 22 p 23 p 30 p 31 p 32 p 33

M

T

v 3 v 2 v 1

Tensor product Bezier patches

Applies also to surfaces