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6. Surfaces and Surface Modeling

ME 521 Computer Aided Design. 6. Surfaces and Surface Modeling. Dr . Ahmet Zafer Şenalp e-mail: azsenalp@gyte.edu.tr Makine Mühendisliği Bölümü Gebze Yüksek Teknoloji Enstitüsü. Types od Surfaces. 6. Surfaces and Surface Modeling. Analytical Surfaces Primitive surfaces Plane surface

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6. Surfaces and Surface Modeling

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  1. ME 521 ComputerAidedDesign 6. SurfacesandSurfaceModeling Dr. Ahmet Zafer Şenalpe-mail: azsenalp@gyte.edu.tr Makine Mühendisliği Bölümü Gebze Yüksek Teknoloji Enstitüsü

  2. Types od Surfaces 6. Surfaces and Surface Modeling • AnalyticalSurfaces • Primitivesurfaces • Plane surface • Offsetsurface • Tabulated cylinder • Surface of revolution • Sweptsurface • Ruled surface • SyntheticSurfaces • Coons patches • Bilinearsurface • Bicubicsurface • Beziersurface • B-splinesurface • NURBS surface GYTE-Makine Mühendisliği Bölümü

  3. SurfacePatch 6. Surfaces and Surface Modeling A surface patch ⎯ a curved bounded collection ofpoints whose coordinates are given by continuous,two-parameter, single-valuedmathematicalexpression. Parametricrepresentation: p = p(u,v) x=x(u,v),y=y(u,v),z=z(u,v) p(u,v) = [x(u,v) y(u,v) z(u,v)]T GYTE-Makine Mühendisliği Bölümü

  4. SurfacePatch 6. Surfaces and Surface Modeling v Isoparametric curves u GYTE-Makine Mühendisliği Bölümü

  5. SurfacePatch 6. Surfaces and Surface Modeling v=1 u=ui v=vj - - p(ui,vj) n(ui,vj) v=0 GYTE-Makine Mühendisliği Bölümü

  6. AnalyticalSurfaces 6. Surfaces and Surface Modeling • Primitive surfaces • Plane surface • Offset surface • Tabulated cylinder • Surface of revolution • Swept surface • Ruled surface GYTE-Makine Mühendisliği Bölümü

  7. PrimitiveSurfaces 6. Surfaces and Surface Modeling Plane: P(u, v) = u i + v j + 0 k Cylinder: P(u, v) = R cos u i + R sin u j + v k GYTE-Makine Mühendisliği Bölümü

  8. PrimitiveSurfaces 6. Surfaces and Surface Modeling • Plane P(u, v) = u i + v j + 0 k • Cylinder P(u, v) = R cos u i + R sin u j + v k • Sphere P(u, v) = R cos u cos v i + R sin u cos v j+ R sin v k • Cone P(u, v) = m v cos u i + m v sin u j + v k • Torus P(u, v) = (R + r cos v) cos u i+ (R + r cos v) sin u j + r sin v k GYTE-Makine Mühendisliği Bölümü

  9. PlanarSurface 6. Surfaces and Surface Modeling Defined by 3 points and 3 vectors GYTE-Makine Mühendisliği Bölümü

  10. PlanarSurface 6. Surfaces and Surface Modeling GYTE-Makine Mühendisliği Bölümü

  11. OffsetSurface 6. Surfaces and Surface Modeling Offset yönü GYTE-Makine Mühendisliği Bölümü

  12. Tabulated Cylinder 6. Surfaces and Surface Modeling • Curve is projected along a vector • In most CAD software it is called as “extrusion” Vector Surfacegenerationcurve GYTE-Makine Mühendisliği Bölümü

  13. Surface of Revolution 6. Surfaces and Surface Modeling • Revolve curve about an axis Curve Axis GYTE-Makine Mühendisliği Bölümü

  14. Surface of Revolution 6. Surfaces and Surface Modeling When a planarcurve is revoledaroundtheaxiswith an angle v a circle is constructed (if v=360 ). Center is on therevolvingaxisandrz(u) is variable. GYTE-Makine Mühendisliği Bölümü

  15. Swept Surface 6. Surfaces and Surface Modeling • Defining curve swept along an arbitrary spine curve Spine Definingcurve GYTE-Makine Mühendisliği Bölümü

  16. Ruled Surface 6. Surfaces and Surface Modeling • Linear interpolation between two edge curves • Created by lofting through cross sections • Lines are used to connect edge curves • There is no restriction for edge curves • It is a linear surface Edge curve 2 Linear interpolation Edge curve 1 GYTE-Makine Mühendisliği Bölümü

  17. Ruled Surface 6. Surfaces and Surface Modeling Edge curves: G(u) ve Q(u) C1(u)=G(u) C2(u)=Q(u) Ruledsurceonlypermitsslope in thedirection of curves in u direction. Surface has zeroslope in v direction. Ruledsurfacecannot be usedto model surfacesthathaveslopes in 2 directions. GYTE-Makine Mühendisliği Bölümü

  18. SyntheticSurfaces 6. Surfaces and Surface Modeling • Coons patches • Bilinear surface • Bicubic surface • Bezier surface • B-spline surface • NURBS surface GYTE-Makine Mühendisliği Bölümü

  19. LinearlyBlendedCoons Surface 6. Surfaces and Surface Modeling p01 D1 C0 p11 C1 v p00 D0 u p10 GYTE-Makine Mühendisliği Bölümü

  20. LinearlyBlendedCoons Surface 6. Surfaces and Surface Modeling • Surface is defined by linearly interpolating between the boundary curves • Simple, but doesn’t allow adjacent patches to be joined smoothly GYTE-Makine Mühendisliği Bölümü

  21. LinearlyBlendedCoons Surface 6. Surfaces and Surface Modeling • Most of the surface algorithms use finite number of points to model surface. However Coons surface patch uses interpolation method with infinite number of points. • Coons surface seeks P(u,v) function that will fill between 4 edge curves. • Bilineer Coons patch form: GYTE-Makine Mühendisliği Bölümü

  22. LinearlyBlendedCoons Surface 6. Surfaces and Surface Modeling The fom given above does not satisfy the boundary conditions as shown below. Herebelow is a corrrectionsurface Withtheapplication of correctionsurface; elde edilir ve bu form sınır koşullarını sağlar. GYTE-Makine Mühendisliği Bölümü

  23. LinearlyBlendedCoons Surface 6. Surfaces and Surface Modeling –1, 1-u, u, 1-v, andv functionsarecalledblendingfunctions, becausetheyblendboundarycurvesto form onesurface. Forcubicblendingfunctionsthe form givenbelow is valid: Intheabovematrixleftcolumn is P1(u,v), middlecolumn is P2(u,v), rightcolumn is P3(u,v)’dir. GYTE-Makine Mühendisliği Bölümü

  24. LinearlyBlendedCoons Surface 6. Surfaces and Surface Modeling Coons surface can be used by using ruled surfaces. GYTE-Makine Mühendisliği Bölümü

  25. BilinearSurface 6. Surfaces and Surface Modeling A bilinear surface is derived by interpolating four data points, using linear equations in the parameters u and v so that the resulting surface has the four points at its corners, denoted; P00, P10, P01, ve P11. P0v = (1-v)P00 + vP01 P1v = (1-v)P10 + vP11 SimilarlyP(u, v) can be obtainedby usingP0vveP1v: P(u, v) = (1-u)P0v + uP1v ByreplacingP0vandP1vintoP(u, v): GYTE-Makine Mühendisliği Bölümü

  26. BilinearSurface 6. Surfaces and Surface Modeling • Advantage: • To supply 4 corner points is enough • Limitations: • Bilinear surface is flat • Surfaces generally form iin flat form GYTE-Makine Mühendisliği Bölümü

  27. Bicubic Patch 6. Surfaces and Surface Modeling • As blending functions are not linear unlike bilinear surfaces it is possible to model nonlinear surface forms • Extension of cubic curve • 16 unknown coefficients - 16 boundary conditions • Tangents and “twists” at corners of patch can be used • Like Lagrange and Hermite curves, difficult to work with GYTE-Makine Mühendisliği Bölümü

  28. Bicubic Patch 6. Surfaces and Surface Modeling GYTE-Makine Mühendisliği Bölümü

  29. Bicubic Patch • Tofind 16 coefficients in C matrix 16 boundaryconditionsarenecessary. Theseare: • 4 cornerpoints • 8 tangentvectors at cornerpoints (in u and v directions at eaachpoint) • 4 twistvectors at cornerpoints GYTE-Makine Mühendisliği Bölümü

  30. Bicubic Patch 6. Surfaces and Surface Modeling The twist vector at a point on a surface measures the twist in the surface at thepoint. It is the rate of change of the tangent vector Pu with respect to v or Pv withrespect to u or it is the cross (mixed) derivative vector at the point. The normal to a surface is another important analytical property. The surface normal at a point is avector which is perpendicular to both tangent vectors at the point. And the unit normal vector is given by: GYTE-Makine Mühendisliği Bölümü

  31. Bicubic Patch 6. Surfaces and Surface Modeling The Hermitebicubic surface can be written in terms of the 16 input vectors: ; Hermitematrix ; geometri ya da sınır koşulu matrisi GYTE-Makine Mühendisliği Bölümü

  32. Bicubic Patch 6. Surfaces and Surface Modeling P(u,v) equation can be further expressed as: The second order twist vectors Puvare difficult to define. The Ferguson surface (also called the F-surface patch) is a bicubic surface patch with zero twist vectors at the patch corners. Thus, the boundary matrix for the F-surfacepatchbecomes: GYTE-Makine Mühendisliği Bölümü

  33. Bicubic Patch 6. Surfaces and Surface Modeling F-yüzey yaması This special surface is useful in design and machining applications. GYTE-MakineMühendisliğiBölümü

  34. Bicubic Patch 6. Surfaces and Surface Modeling • Advantages – BoundarycurvesareHermitecurves – Interior points can be controlled • Disadvantages –Whatshould be thetwistfactor?It is not esayto sense theeffect of twistvector(Fergusonpacthtwistvector is 0). – Cannot be usedwithhighorderpolynomials. GYTE-Makine Mühendisliği Bölümü

  35. Bicubic PatchExample: 6. Surfaces and Surface Modeling Parametricbicubicsurface is defined in terms of cartesiancomponents: u=1/2, v=1 noktasındaki teğet vektörleri nelerdir? GYTE-Makine Mühendisliği Bölümü

  36. Bicubic PatchExample: 6. Surfaces and Surface Modeling Tofindthetangentvectors it is necesarytodifferentiatewithrespectto u and v: (s=1/2,t=1) noktasında GYTE-Makine Mühendisliği Bölümü

  37. Bezier Surfaces 6. Surfaces and Surface Modeling • Bezier curves can be extended to surfaces • Same problems as for Bezier curves: • no local modification possible • smooth transition between adjacent patches difficult to achieve ParametricspaceCartesianspace GYTE-Makine Mühendisliği Bölümü

  38. Bezier Surfaces 6. Surfaces and Surface Modeling BezierSurfaces: • Two sets of orthogonal Bezier curves can be used to design an object surface. • A tensor product Bezier surface is an extension for the Bezier curve in twoparametric directions u and v: • P(u, v) is any point on the surface and ij P are the control points. These points formthe vertices of the control or characteristic polyhedron. • Curvesareformed, when u is constant v changes in [0..1] when v is constant u changes in [0..1] • Like in BeziércurvesBin(u) andBjm(v) n. ve m. degreeBernsteinpolynomials. • Generallyn=m=3: cubicBeziérpatch is used.(4x4=16 controlpoints; Pi,jis necessary.) GYTE-Makine Mühendisliği Bölümü

  39. Bezier Surfaces 6. Surfaces and Surface Modeling P(u, v) is apoint on the surface and Pij are control points. These points form the control polygon’s vertex points. Below figure shows cubic Bezier patch. When n=3 and m=3 is placed in Bezier equation then Bezier patch equation becomes: ParametricspaceCartesianspace GYTE-Makine Mühendisliği Bölümü

  40. Bezier Surfaces 6. Surfaces and Surface Modeling GYTE-Makine Mühendisliği Bölümü

  41. Bezier Surfaces 6. Surfaces and Surface Modeling A 3rd degree Bezier surface defined with 16 control points: GYTE-Makine Mühendisliği Bölümü

  42. Bezier Surfaces 6. Surfaces and Surface Modeling Open and closed Bezier surface examples GYTE-Makine Mühendisliği Bölümü

  43. B-SplineSurfaces 6. Surfaces and Surface Modeling • As with curves, B-spline surfaces are a generalization of Bezier surfaces • The surface approximates a control polygon • Open and closed surfaces can be represented GYTE-Makine Mühendisliği Bölümü

  44. B-SplineSurfaces 6. Surfaces and Surface Modeling A tensor product B-spline surface is an extension for the B-spline curve in twoparametric directions u and v. For n=m=3, the equivalent bicubic formulation of an open and closed cubic B-spline surface can be derived as below. GYTE-Makine Mühendisliği Bölümü

  45. B-SplineSurfaces 6. Surfaces and Surface Modeling where [P] is an (n +1)×(m +1) matrix of the vertices of the characteristic polyhedron of the B-spline surface patch. For a 4×4 cubic B-spline patch: GYTE-Makine Mühendisliği Bölümü

  46. B-SplineSurfaces 6. Surfaces and Surface Modeling B-Spline surface example GYTE-Makine Mühendisliği Bölümü

  47. NURBS 6. Surfaces and Surface Modeling NURBS surface (Non-Uniform Rational B-Spline surface) is a generilization to Bézier and B-splines surfaces. NURBS is used widely in computer graphics in CAD applications. A NURBS surface is a parametric surface defined with its degree. GYTE-Makine Mühendisliği Bölümü

  48. NURBS 6. Surfaces and Surface Modeling GYTE-Makine Mühendisliği Bölümü

  49. TriangularPatches 6. Surfaces and Surface Modeling In triangulation techniques, three parameters u, v and w are used and the parametricdomain is defined by a symmetric unit triangle Cartesian space Parametric space The coordinates u, v and w are called “barycentric coordinates.” While the coordinate w is not independent of u and v(note that u+v+w=1 for any point in the domain) GYTE-Makine Mühendisliği Bölümü

  50. TriangularPatches 6. Surfaces and Surface Modeling A triangular Bezier patch is defined by: For example, a cubic triangular patch is; GYTE-Makine Mühendisliği Bölümü

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