Coons Patches and Gregory Patches

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Presentation transcript:

Coons Patches and Gregory Patches Dr. Scott Schaefer

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

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

Coons Patches Build a ruled surface between pairs of curves

Coons Patches Build a ruled surface between pairs of curves

Coons Patches Build a ruled surface between pairs of curves

Coons Patches Build a ruled surface between pairs of curves

Coons Patches “Correct” surface to make boundaries match

Coons Patches “Correct” surface to make boundaries match

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

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

Hermite Coons Patches Use Hermite interpolation!!!

Hermite Coons Patches Use Hermite interpolation!!!

Hermite Coons Patches Use Hermite interpolation!!!

Hermite Coons Patches Use Hermite interpolation!!! Requires mixed partials

Problems With Bezier Patches

Problems With Bezier Patches

Problems With Bezier Patches

Problems With Bezier Patches Derivatives along edges not independent!!!

Solution

Solution

Gregory Patches

Gregory Patch Evaluation

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

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)

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

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

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!!

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

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

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!!!

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!!!

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

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!!!

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

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

Constructing Smooth Surfaces With Gregory Patches Problem: construction is not symmetric is quadratic is cubic

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

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