1 dr. scott schaefer coons patches and gregory patches

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  • *Dr. Scott SchaeferCoons Patches and Gregory Patches

  • */39Patches With Arbitrary BoundariesGiven 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

  • */39Patches With Arbitrary BoundariesGiven 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

  • */39Coons PatchesBuild a ruled surface between pairs of curves

  • */39Coons PatchesBuild a ruled surface between pairs of curves

  • */39Coons PatchesBuild a ruled surface between pairs of curves

  • */39Coons PatchesBuild a ruled surface between pairs of curves

  • */39Coons PatchesCorrect surface to make boundaries match

  • */39Coons PatchesCorrect surface to make boundaries match

  • */39Properties of Coons PatchesInterpolate arbitrary boundariesSmoothness of surface equivalent to minimum smoothness of boundary curvesDont provide higher continuity across boundaries

  • */39Hermite Coons PatchesGiven 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

  • */39Hermite Coons PatchesUse Hermite interpolation!!!

  • */39Hermite Coons PatchesUse Hermite interpolation!!!

  • */39Hermite Coons PatchesUse Hermite interpolation!!!

  • */39Hermite Coons PatchesUse Hermite interpolation!!!Requires mixed partials

  • */39Problems With Bezier Patches

  • */39Problems With Bezier Patches

  • */39Problems With Bezier Patches

  • */39Problems With Bezier PatchesDerivatives along edges not independent!!!

  • Solution*/39

  • Solution*/39

  • */39Gregory Patches

  • */39Gregory Patch Evaluation

  • */39Gregory Patch EvaluationDerivative along edge decoupled from adjacent edge at interior points

  • */39Gregory Patch PropertiesRational patchesIndependent control of derivatives along edges except at end-pointsDont have to specify mixed partial derivativesInterior derivatives more complicated due to rational structureSpecial care must be taken at corners (poles in rational functions)

  • */39Constructing Smooth Surfaces With Gregory PatchesAssume a network of cubic curves forming quad shapes with curves meeting with C1 continuityConstruct a C1 surface that interpolates these curves

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!Fixed control points!!

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!Derivatives must be linearly dependent!!!

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!By construction, property holds at end-points!!!

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!Assume weights change linearly

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!Assume weights change linearlyA quartic function. Not possible!!!

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!Require v(t) to be quadratic

  • */39Constructing Smooth Surfaces With Gregory PatchesNeed to specify interior points for cross-boundary derivativesGregory patches allow us to consider each edge independently!!!

  • */39Constructing Smooth Surfaces With Gregory PatchesProblem: construction is not symmetric is quadratic is cubic

  • */39Constructing Smooth Surfaces With Gregory PatchesSolution: assume v(t) is linear and use to findSame operation to find

  • */39Constructing Smooth Surfaces With Gregory PatchesAdvantagesSimple construction with finite set of (rational) polynomialsDisadvantagesNot very flexible since cross-boundary derivatives are not full cubics

    If cubic curves not available, can estimate tangent planes and build hermite curves