The Goal

Blending transfinite interpolation surfacepatchesRichard SouthernUniversity of CambridgeThe GoalT o generate smooth surfaces from parametric curvenetworks.T o have a local andflexible surface definition.Designers use curves already fordefining surfaces.Lines of anisotropy are often drawnto convey maximum curvatureMost modeling kernels support thebounding curves(e.g.ACIS,Para-solid,etc.).Three different approaches to defining a surface frominput curves:A bounded parametric patch definition(Coo67;PC96).A variational,minimum energy approach(BX03;BS05).A parametric space definition(GR05).Only a parametric patch has local support,but cannothave“floating curves”(hybrid surfaces have yet to be explored.).We focus on smoothing over vertices of curve networksof quadrilateral parametric patches.It is very difficult to construct acurve network where patches meetneatly.Curves may not be continuousacross patches.Parametric curves may notevaluate to exactly the samepoint(rounding errors).It is difficult to determine surfacebehavior in the case of anirregular number of curvesmeeting at a point.MethodOverviewWe resolve the problem of smoothing over verticesby:extrapolating each patch over their neighbors at eachvertex,blending together all patches about each vertex,andblending together the relevant vertex patches over eachface.The method in .Talk OverviewParametric patch construction from boundary curvesCoons patchesMean Value InterpolationA little manifold theoryExtrapolating parametric patchesBlending the results togetherVertex patchesFace patchesResults and DiscussionOur parametric surface is an manifold embedded in.Each patch is parameterized in terms of some localparametric domain with coordinates.The boundary is defined by a curve,whichis only defined along the boundary of the patch.Our patch definition specifies some functionwhich defines the interior of the patch.f(t)v f(v )^Parametric domain Boundary function Surface patch p(t)CoonspatchesNamed after StevenCoons(Coo67). Simplest method to interpolate4curvesbounding a square parametric domain.Sum of the ruled blendsandminus an error term from the end points.Continuity is bounded by the conti-nuity of the curves only.b(v)b(v)01a(u)a(u)01a(u)a(u)010a(u)1a(u)011+-=b(v) ruled blenda(u) ruled blenderr o r t er mfi n a l s ur fac ePros:Fast tocompute.Simple.Continuity limited only by input curves.Cons:Surfaces do not minimize surfaceenergy.Folding can occur when extrapolating at singular or near singularvertices.Floater(Flo03)proposed Mean Value Coordinates forparameterization of points and giving smooth coordinates for general polygons.Ju et al.(JSW05)later defines Mean Value Interpolationproviding smooth coordinates for the interior and exterior of arbitrary(convex and concave)polyhedrons. Preserves linear precision.Wachspress Coordinates Mean Value CoordinatesThe point each point on’s boundary isprojected onto a unit sphere(centered at).The points associated value is then weightedby.This weighted function is then integrated over.Affine invariance is ensured by dividing the resultby the integral of the weight functiontaken over.Hence:S vp(t)This formulation can be simplified for closed polygons with function values defined at onlythe vertices.In this case the integration can be restricted to the circular arc from when the edgeis projected onto the circle such that:Grimm et al.(GH95)use an“atlas”analogy.A collection of atlas charts can be stitched togetheracross some shared overlap.A transition function tells us where we are in space if weare between charts.In our case,charts are our parameterized patches,andthe transition function is a method to construct thefinal patch.More formally a setwhere are one-to-one and coveris if a transition function from chart to chart,defined for where charts andintersect,are.This definition provides us with the theory with which wecan construct surfaces of arbitrary continuity.We need a smooth method to define how our charts stretch over the neighboring charts.We blend over each vertex,so we need to determine a representing how this patchstretches over neighboring patches.This is achieved by a conformal transform(with in complex notation)forvalence(valence:the number of faces meeting at a vertex).maps from the regular parametric space to conformal space.A rotation is applied to the coordinate such that we can deduce the position of from each neighboring patch.Applying the rotation about a vertex withWe now need a transform which would transform back to the parametric space of theoriginal polygon.Unfortunately the function is not injective,so a unique inverse does not exist for.We approximate this function with the functionwhere is a constant and is the phase of the point in conformal space.This function is only monotonically increasing for.An analytic solution to the forward transform does not exist,but can be solved fornumerically for a better geometryfit.Putting the whole expression together,we havewhere is the parametric position,and is the index of the patch anti-clockwise about the vertex with valence.Note that this can be precomputed for a regular sampling(i.e.are regularly sampled).Extrapolating a regular quad patch with.We would like to initially blend together theseextrapolated patches such that we createWe construct a smooth patch about a particular vertexby blending together patches about the vertex:A valence6vertex blend.Blending the results together:Face patchesFor each vertex in face blend together the parts of the vertex patch associated with face.A simple affine transformation is necessary to ensure that distortion is prevented.More complex blending is possible with differently blended patches(edge blends easy todefine).Valence 3,Mean Curvature Valence 5,Gauss CurvatureDiscussionThere are a number of potential optimizations:The expensive terms can be precomputed forall valid,,and for a given set ofsamples of.Similarly the weights can also be precomputed.The method lends itself easily to the SIMD GPUarchitecture.DiscussionVertices at surface discontinuities can be treated in anumber of ways:They be left to interpolate the boundary and blendedinto the remaining surface,orDummy vertex patches can be constructed using“ghost faces”by estimating the valence of the vertex.ConclusionWe have presented a method to blend around vertices ofparametric surface patches constructed from curvenetworks.The method is local to each patch and its immediateneighbors.The method is fast to compute.The methods continuity is only limited by the continuityof the input curves and its extrapolation.Self intersection is currently unavoidable.Future workThis method is a step towards my goal of“sketch basedmodeling”.Other methods,such as Attribute BasedModeling(GR05)give a method to quickly constructsurface geometry from an arbitrary collection of curvesin space(without variation design).I would like to explore Mean Value Coordinates for therapid design of surfaces from a sketch based interface.Questions?ReferencesAlexander Bobenko and Peter Schr¨oder.Discrete willmoreflow.In M.Desbrun and H.Pottmann,editors,Symposium of Ge-ometry Processing,pages101–110.Eurographics/ACM SIGGRAPH,July2005.Chandrajit Bajaj and Guoliang Xu.Anisotropic diffusion of sub-division surfaces and functions on surfaces.ACM T ransac-tions on Graphics,22(1):4–32,2003.Steven Coons.Surfaces for computer aided design of space forms.T echnical Report MAC-TR-41,Project MAC,Mas-sachusetts Institute of T echnology,1967.Michael S.Floater.Mean value puter Aided Geometric Design,20(1):19–27,2003.Cindy Grimm and John Hughes.Modeling surfaces of arbitrary puter Graphics(Proceedings of SIGGRAPH), pages359–369,August1995.Kun Gao and Alyn Rockwood.Multi-sided attribute based mod-eling.In Ralph Martin,Helmut Bez,and Malcolm Sabin, editors,Mathematics of Surfaces XI,pages219–232.IMA, September2005.T ao Ju,Scott Schaefer,and Joe Warren.Mean value coor-dinates for closed triangle meshes.In ACM SIGGRAPH, pages561–566,2005.D.Plowman and P.Charrot.A practical implementation ofvertex blend surfaces using an-sided patch.In GlenMullineux,editor,The Mathematics of Surfaces VI,pages 67–78.Clarendon Press,1996.。