1. 8. Surfaces and Surface Modeling e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gmail.comazsenalp@gmail.com Mechanical Engineering Department Gebze Technical University ME 521 Computer Aided Design

2.  Analytical Surfaces  Primitive surfaces  Plane surface  Offset surface  Tabulated cylinder  Surface of revolution  Swept surface  Ruled surface  Synthetic Surfaces  Coons patches  Bilinear surface  Bicubic surface  Bezier surface  B-spline surface  NURBS surface Dr. Ahmet Zafer Şenalp ME 521 2 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

3. A surface patch a curved bounded collection of points whose coordinates are given by continuous, two-parameter, single-valued mathematical expression. Parametric representation: 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 Dr. Ahmet Zafer Şenalp ME 521 3 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

4. v u Isoparametric curves Dr. Ahmet Zafer Şenalp ME 521 4 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

5. u=u i v=v j v=0 v=1 p(u i,v j ) - n(u i,v j ) - Dr. Ahmet Zafer Şenalp ME 521 5 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

6. Analytical Surfaces  Primitive surfaces  Plane surface  Offset surface  Tabulated cylinder  Surface of revolution  Swept surface  Ruled surface Dr. Ahmet Zafer Şenalp ME 521 6 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

7. Plane: P(u, v) = u i + v j + 0 k Cylinder: P(u, v) = R cos u i + R sin u j + v k Dr. Ahmet Zafer Şenalp ME 521 7 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

8. 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 Dr. Ahmet Zafer Şenalp ME 521 8 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

9. Defined by 3 points and 3 vectors Dr. Ahmet Zafer Şenalp ME 521 9 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

10. Dr. Ahmet Zafer Şenalp ME 521 10 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

11. Offset Surface Offset yönü Dr. Ahmet Zafer Şenalp ME 521 11 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

12. Tabulated Cylinder Curve is projected along a vector In most CAD software it is called as “extrusion” Surface generation curve Vector Dr. Ahmet Zafer Şenalp ME 521 12 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

13. Surface of Revolution Revolve curve about an axis Axis Curve Dr. Ahmet Zafer Şenalp ME 521 13 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

14. Surface of Revolution When a planar curve is revoled around the axis with an angle v a circle is constructed (if v=360 ). Center is on the revolving axis and r z (u) is variable. Dr. Ahmet Zafer Şenalp ME 521 14 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

15. Swept Surface Defining curve swept along an arbitrary spine curve Defining curve Spine Dr. Ahmet Zafer Şenalp ME 521 15 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

16. Ruled Surface 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 1 Edge curve 2 Linear interpolation Dr. Ahmet Zafer Şenalp ME 521 16 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

17. Ruled Surface Edge curves: G(u) and Q(u) Ruled surce only permits slope in the direction of curves in u direction. Surface has zero slope in v direction. Ruled surface cannot be used to model surfaces that have slopes in 2 directions. C 1 (u)=G(u) C 2 (u)=Q(u) Dr. Ahmet Zafer Şenalp ME 521 17 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

18. Synthetic Surfaces  Coons patches  Bilinear surface  Bicubic surface  Bezier surface  B-spline surface  NURBS surface Dr. Ahmet Zafer Şenalp ME 521 18 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

19. Linearly Blended Coons Surface p 00 p 11 p 01 p 10 v u D1D1 D0D0 C1C1 C0C0 Dr. Ahmet Zafer Şenalp ME 521 19 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

20. Linearly Blended Coons Surface Surface is defined by linearly interpolating between the boundary curves Simple, but doesn’t allow adjacent patches to be joined smoothly Dr. Ahmet Zafer Şenalp ME 521 20 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

21. Linearly Blended Coons Surface 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: Dr. Ahmet Zafer Şenalp ME 521 21 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

22. Linearly Blended Coons Surface The form given above does not satisfy the boundary conditions as shown below. Here below is a corrrection surface With the application of correction surface; elde edilir ve bu form sınır koşullarını sağlar. Dr. Ahmet Zafer Şenalp ME 521 22 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

23. Linearly Blended Coons Surface –1, 1-u, u, 1-v, and v functions are called blending functions, because they blend boundary curves to form one surface. For cubic blending functions the form given below is valid: In the above matrix left column is P 1 (u,v), middle column is P 2 (u,v), right column is P 3 (u,v). Dr. Ahmet Zafer Şenalp ME 521 23 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

24. Linearly Blended Coons Surface Coons surface can be formed by using ruled surfaces. Dr. Ahmet Zafer Şenalp ME 521 24 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

25. 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; P 00, P 10, P 01, and P 11. P 0v = (1-v)P 00 + vP 01 P 1v = (1-v)P 10 + vP 11 Similarly P(u, v) can be obtained by using P 0v ve P 1v : P(u, v) = (1-u)P 0v + uP 1v By replacing P 0v and P 1v into P(u, v): Dr. Ahmet Zafer Şenalp ME 521 25 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

26. Advantage: To supply 4 corner points is enough Limitations:  Bilinear surface is flat  Surfaces generally form in flat form Dr. Ahmet Zafer Şenalp ME 521 26 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

27. Bicubic Patch 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 Dr. Ahmet Zafer Şenalp ME 521 27 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

28. Bicubic Patch Dr. Ahmet Zafer Şenalp ME 521 28 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

29. Bicubic Patch To find 16 coefficients in C matrix 16 boundary conditions are necessary. These are:  4 corner points  8 tangent vectors at corner points (in u and v directions at each point )  4 twist vectors at corner points Dr. Ahmet Zafer Şenalp ME 521 29 Mechanical Engineering Department, GTU

30. Bicubic Patch The twist vector at a point on a surface measures the twist in the surface at the point. It is the rate of change of the tangent vector P u with respect to v or P v with respect 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 a vector which is perpendicular to both tangent vectors at the point. And the unit normal vector is given by: Dr. Ahmet Zafer Şenalp ME 521 30 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

31. Bicubic Patch The Hermite bicubic surface can be written in terms of the 16 input vectors: ; Hermite matrix ; geometri ya da sınır koşulu matrisi Dr. Ahmet Zafer Şenalp ME 521 31 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

32. Bicubic Patch P(u,v) equation can be further expressed as: The second order twist vectors P uv are 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-surface patch becomes: Dr. Ahmet Zafer Şenalp ME 521 32 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

33. Bicubic Patch F-surface patch This special surface is useful in design and machining applications. Dr. Ahmet Zafer Şenalp ME 521 33 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

34. Bicubic Patch Advantages – Boundary curves are Hermite curves – Interior points can be controlled Disadvantages –What should be the twist factor? It is not esay to sense the effect of twist vector(Ferguson pacth twist vector is 0). – Cannot be used with high order polynomials. Dr. Ahmet Zafer Şenalp ME 521 34 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

35. Bicubic Patch Example: Parametric bicubic surface is defined in terms of cartesian components for u=0.5 and v=1: u=1/2, v=1 noktasındaki teğet vektörleri nelerdir? Dr. Ahmet Zafer Şenalp ME 521 35 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

36. Bicubic Patch Example: To find the tangent vectors it is necesary to differentiate with respect to u and v: (s=1/2,t=1) noktasında Dr. Ahmet Zafer Şenalp ME 521 36 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

37. Bezier Surfaces 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 Parametric spaceCartesian space Dr. Ahmet Zafer Şenalp ME 521 37 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

38. Bezier Surfaces Bezier Surfaces: 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 two parametric directions u and v: P(u, v) is any point on the surface and P ij are the control points. These points form the vertices of the control or characteristic polyhedron. Curves are formed, when u is constant v changes in [0..1] when v is constant u changes in [0..1] Like in Beziér curves B i n (u) and B j m (v) n. and m. degree Bernstein polynomials. Generally n=m=3: cubic Beziér patch is used. (4x4=16 control points; P i,j is necessary.) Dr. Ahmet Zafer Şenalp ME 521 38 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

39. Bezier Surfaces P(u, v) is a point on the surface and P ij 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: Parametric space Cartesian space Dr. Ahmet Zafer Şenalp ME 521 39 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

40. Bezier Surfaces Dr. Ahmet Zafer Şenalp ME 521 40 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

41. Bezier Surfaces A 3 rd degree Bezier surface defined with 16 control points: Dr. Ahmet Zafer Şenalp ME 521 41 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

42. Bezier Surfaces Open and closed Bezier surface examples Dr. Ahmet Zafer Şenalp ME 521 42 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

43. B-Spline Surfaces 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 Dr. Ahmet Zafer Şenalp ME 521 43 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

44. B-Spline Surfaces A tensor product B-spline surface is an extension for the B-spline curve in two parametric 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. Dr. Ahmet Zafer Şenalp ME 521 44 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

45. B-Spline Surfaces 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: Dr. Ahmet Zafer Şenalp ME 521 45 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

46. B-Spline Surfaces B-Spline surface example Dr. Ahmet Zafer Şenalp ME 521 46 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

47. NURBS 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. Dr. Ahmet Zafer Şenalp ME 521 47 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

48. NURBS Dr. Ahmet Zafer Şenalp ME 521 48 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

49. Triangular Patches Cartesian space Parametric space In triangulation techniques, three parameters u, v and w are used and the parametric domain is defined by a symmetric unit triangle 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) Dr. Ahmet Zafer Şenalp ME 521 49 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

50. Triangular Patches A triangular Bezier patch is defined by: For example, a cubic triangular patch is; Dr. Ahmet Zafer Şenalp ME 521 50 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

51. Triangular Patches For n=4, the triangular patch is defined as; Dr. Ahmet Zafer Şenalp ME 521 51 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

52. Triangular Patches Dr. Ahmet Zafer Şenalp ME 521 52 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

53. Dr. Ahmet Zafer Şenalp ME 521 53 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

54. Sculptured Surface General surface form Composed of united surface pieces Dr. Ahmet Zafer Şenalp ME 521 54 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

55. Sculptured Surface Dr. Ahmet Zafer Şenalp ME 521 55 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

56. Subdivision Surface New points are added between control points by interpollation to obtain a fine surface Dr. Ahmet Zafer Şenalp ME 521 56 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling SUBDIVISION CURVES AND SURFACES

57. Subdivision Surface Mesh Subdivision A theory of subdivision surfaces has emerged that deals with both the theoretical and practical aspects of these ideas. We have two type of meshes: o triangles meshes. o quadrilaterals meshes. Dr. Ahmet Zafer Şenalp ME 521 57 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

58. Subdivision Surface Meshes methods Catmull Clark method: use to form a quadrilateral mesh. produces a smoother surface This method tends to move edge vertices at corners more than other outer vertices. Dr. Ahmet Zafer Şenalp ME 521 58 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

59. Subdivision Surface Catmull Clark method Dr. Ahmet Zafer Şenalp ME 521 59 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

60. Subdivision Surface Loop Subdivision Dr. Ahmet Zafer Şenalp ME 521 60 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling

61. Subdivision Surface New points are added between control points by interpollation to obtain a fine surface Dr. Ahmet Zafer Şenalp ME 521 61 Mechanical Engineering Department, GTU 8. Surfaces and Surface Modeling