1 Dr. Scott Schaefer Coons Patches and Gregory Patches.

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Presentation transcript:

1 Dr. Scott Schaefer Coons Patches and Gregory Patches

2/39 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

3/39 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

4/39 Coons Patches Build a ruled surface between pairs of curves

5/39 Coons Patches Build a ruled surface between pairs of curves

6/39 Coons Patches Build a ruled surface between pairs of curves

7/39 Coons Patches Build a ruled surface between pairs of curves

8/39 Coons Patches “Correct” surface to make boundaries match

9/39 Coons Patches “Correct” surface to make boundaries match

10/39 Properties of Coons Patches Interpolate arbitrary boundaries Smoothness of surface equivalent to minimum smoothness of boundary curves Don’t provide higher continuity across boundaries

11/39 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

12/39 Hermite Coons Patches Use Hermite interpolation!!!

13/39 Hermite Coons Patches Use Hermite interpolation!!!

14/39 Hermite Coons Patches Use Hermite interpolation!!!

15/39 Hermite Coons Patches Use Hermite interpolation!!! Requires mixed partials

16/39 Problems With Bezier Patches

17/39 Problems With Bezier Patches

18/39 Problems With Bezier Patches

19/39 Problems With Bezier Patches Derivatives along edges not independent!!!

Solution 20/39

Solution 21/39

22/39 Gregory Patches

23/39 Gregory Patch Evaluation

24/39 Gregory Patch Evaluation Derivative along edge decoupled from adjacent edge at interior points

25/39 Gregory Patch Properties Rational patches Independent control of derivatives along edges except at end-points Don’t 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)

26/39 Constructing Smooth Surfaces With Gregory Patches Assume a network of cubic curves forming quad shapes with curves meeting with C 1 continuity Construct a C 1 surface that interpolates these curves

27/39 Constructing Smooth Surfaces With Gregory Patches Need to specify interior points for cross- boundary derivatives Gregory patches allow us to consider each edge independently!!!

28/39 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!!

29/39 Constructing Smooth Surfaces With Gregory Patches Need to specify interior points for cross- boundary derivatives Gregory patches allow us to consider each edge independently!!!

30/39 Constructing Smooth Surfaces With Gregory Patches Need to specify interior points for cross- boundary derivatives Gregory patches allow us to consider each edge independently!!!

31/39 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!!!

32/39 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!!!

33/39 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

34/39 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!!!

35/39 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

36/39 Constructing Smooth Surfaces With Gregory Patches Need to specify interior points for cross- boundary derivatives Gregory patches allow us to consider each edge independently!!!

37/39 Constructing Smooth Surfaces With Gregory Patches Problem: construction is not symmetric  is quadratic  is cubic

38/39 Constructing Smooth Surfaces With Gregory Patches Solution: assume v(t) is linear and use to find Same operation to find

39/39 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