Dr. Scott Schaefer

1
Coons Patches and Gregory Patches

• Dr. Scott Schaefer

2
Patches 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

3
Patches 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

4
Coons Patches

• Build a ruled surface between pairs of curves

5
Coons Patches

• Build a ruled surface between pairs of curves

6
Coons Patches

• Build a ruled surface between pairs of curves

7
Coons Patches

• Build a ruled surface between pairs of curves

8
Coons Patches

• Correct surface to make boundaries match

9
Coons Patches

• Correct surface to make boundaries match

10
Properties of Coons Patches

• Interpolate arbitrary boundaries
• Smoothness of surface equivalent to minimum
  smoothness of boundary curves
• Dont provide higher continuity across boundaries

11
Hermite 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

12
Hermite Coons Patches

• Use Hermite interpolation!!!

13
Hermite Coons Patches

• Use Hermite interpolation!!!

14
Hermite Coons Patches

• Use Hermite interpolation!!!

15
Hermite Coons Patches

• Use Hermite interpolation!!!

Requires mixed partials
16
Problems With Bezier Patches
17
Problems With Bezier Patches
18
Problems With Bezier Patches
19
Problems With Bezier Patches
Derivatives along edges not independent!!!
20
Solution
21
Solution
22
Gregory Patches
23
Gregory Patch Evaluation
24
Gregory Patch Evaluation
Derivative along edge decoupled from adjacent
edge at interior points
25
Gregory 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)

26
Constructing 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

27
Constructing Smooth Surfaces With Gregory Patches

• Need to specify interior points for
  cross-boundary derivatives
• Gregory patches allow us to consider each edge
  independently!!!

28
Constructing 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!!
29
Constructing Smooth Surfaces With Gregory Patches

• Need to specify interior points for
  cross-boundary derivatives
• Gregory patches allow us to consider each edge
  independently!!!

30
Constructing Smooth Surfaces With Gregory Patches

• Need to specify interior points for
  cross-boundary derivatives
• Gregory patches allow us to consider each edge
  independently!!!

31
Constructing 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!!!
32
Constructing 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!!!
33
Constructing 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
34
Constructing 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!!!
35
Constructing 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
36
Constructing Smooth Surfaces With Gregory Patches

• Need to specify interior points for
  cross-boundary derivatives
• Gregory patches allow us to consider each edge
  independently!!!

37
Constructing Smooth Surfaces With Gregory Patches

• Problem construction is not symmetric
• is quadratic
• is cubic

38
Constructing Smooth Surfaces With Gregory Patches

• Solution assume v(t) is linear and use
• 
  to find
• Same operation to find

39
Constructing 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