Title: Dr. Scott Schaefer
1Coons Patches and Gregory Patches
2Patches With Arbitrary Boundaries
- Given any 4 curves, f(s,0), f(s,1), f(0,t),
f(1,t) that meet continuously at the corners,
construct a smooth surface interpolating these
curves
3Patches With Arbitrary Boundaries
- Given any 4 curves, f(s,0), f(s,1), f(0,t),
f(1,t) that meet continuously at the corners,
construct a smooth surface interpolating these
curves
4Coons Patches
- Build a ruled surface between pairs of curves
5Coons Patches
- Build a ruled surface between pairs of curves
6Coons Patches
- Build a ruled surface between pairs of curves
7Coons Patches
- Build a ruled surface between pairs of curves
8Coons Patches
- Correct surface to make boundaries match
9Coons Patches
- Correct surface to make boundaries match
10Properties of Coons Patches
- Interpolate arbitrary boundaries
- Smoothness of surface equivalent to minimum
smoothness of boundary curves - Dont provide higher continuity across boundaries
11Hermite Coons Patches
- Given any 4 curves, f(s,0), f(s,1), f(0,t),
f(1,t) that meet continuously at the corners and
cross-boundary derivatives along these edges - , construct a smooth surface
interpolating these curves and derivatives
12Hermite Coons Patches
- Use Hermite interpolation!!!
13Hermite Coons Patches
- Use Hermite interpolation!!!
14Hermite Coons Patches
- Use Hermite interpolation!!!
15Hermite Coons Patches
- Use Hermite interpolation!!!
Requires mixed partials
16Problems With Bezier Patches
17Problems With Bezier Patches
18Problems With Bezier Patches
19Problems With Bezier Patches
Derivatives along edges not independent!!!
20Solution
21Solution
22Gregory Patches
23Gregory Patch Evaluation
24Gregory Patch Evaluation
Derivative along edge decoupled from adjacent
edge at interior points
25Gregory Patch Properties
- Rational patches
- Independent control of derivatives along edges
except at end-points - Dont have to specify mixed partial derivatives
- Interior derivatives more complicated due to
rational structure - Special care must be taken at corners (poles in
rational functions)
26Constructing Smooth Surfaces With Gregory Patches
- Assume a network of cubic curves forming quad
shapes with curves meeting with C1 continuity - Construct a C1 surface that interpolates these
curves
27Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
28Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
Fixed control points!!
29Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
30Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
31Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
Derivatives must be linearly dependent!!!
32Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
By construction, property holds at end-points!!!
33Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
Assume weights change linearly
34Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
Assume weights change linearly
A quartic function. Not possible!!!
35Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
Require v(t) to be quadratic
36Constructing Smooth Surfaces With Gregory Patches
- Need to specify interior points for
cross-boundary derivatives - Gregory patches allow us to consider each edge
independently!!!
37Constructing Smooth Surfaces With Gregory Patches
- Problem construction is not symmetric
- is quadratic
- is cubic
38Constructing Smooth Surfaces With Gregory Patches
- Solution assume v(t) is linear and use
-
to find - Same operation to find
39Constructing Smooth Surfaces With Gregory Patches
- Advantages
- Simple construction with finite set of (rational)
polynomials - Disadvantages
- Not very flexible since cross-boundary
derivatives are not full cubics - If cubic curves not available, can estimate
tangent planes and build hermite curves