1. Surface Modeling Parametric Surfaces Dr. S.M. Malaek Assistant: M. Younesi

2. Surface Modeling parametricimplicit There are two types of surfaces that are commonly used in modeling systems, parametric and implicit. Implicit Surface: f(x,y,z)=0 Example: (x-x 0 ) 2 +(y-y 0 ) 2 +(z-z 0 ) 2 -r 2 =0

3. Surface Modeling

4. Parametric Surfaces Parametric surfaces are defined by a set of three functions, one for each coordinate x=f(u,v), y=f(u,v), z=f(u,v)

5. Parametric Surfaces Parametric surfaces : f(u,v) = ( x(u,v), y(u,v), z(u,v) ) Assume both u and v are in the range of 0 and 1.

6. Parametric Surfaces Parametric surfaces or more precisely parametric surface patches are not used individually. Many parametric surface patches are joined together side- by-side to form a more complicated shape. Patch

7. Parametric Surface Patch Each patch is defined by control points net (Control Polyhedron).

8. Parametric Surface Patch A parametric surface patch can be considered as a union of (infinite number) of curves. Given a parametric surface f(u,v), if u is fixed to a value, and let v vary, this generates a curve on the surface whose u coordinate is a constant. This is the isoparametric curve in the v direction. Similarly, fixing v to a value and letting u vary, we obtain an isoparametric curve whose v direction is a constant.

9. Parametric Surface Patch Point Q(u,v) on the patch is the tensor product of parametric curves defined by the control points.

10. Bézier Surface Patch

11. Bézier Surface Patch A Bézier surface is defined by a two-dimensional set of control points p j,k, where j is in the range of 0 and m, and k is in the range of 0 and n.

12. Bézier Surface Patch Example: a Bézier surface defined by 3 rows and 3 columns (i.e., 9) control points and hence is a Bézier surface of degree (2,2).

13. Bézier Surface Patch The effect of “ lifting ” one of he control points of a Bézier patch.

14. Basis Functions of Bézier Surface Patches

15. Bézier Surface Patch Two-dimensional basis functions are the product of two one- dimensional Bézier basis functions. The basis functions for a Bézier surface are parametric surfaces of two variables u and v defined on the unit square. The basis functions for control points p 0,0 (left) and p 1,1 (right), respectively. For control point p 0,0, its basis function is the product of two one-dimensional Bézier basis functions B 2,0 (u) in the u direction and B 2,0 (v) in the v direction. In the left figure, both B 2,0 (u) and B 2,0 (v) are shown along with their product (shown in wireframe). The right figure shows the basis function for p 1,1, which is the product of B 2,1 (u) in the u direction and B 2,1 (v) in the v direction.

16. Joining Bézier Surface Patches

17. Joining Bézier Surface Patches C 0 continuity requires aligning boundary curves.

18. Joining Bézier Surface Patches C 1 continuity requires aligning boundary curves and derivatives.

19. Properties Of Bézier Surface Patches

20. Properties Of Bézier Surface Patches p(u,v) passes through the control points at the four corners of the control net: p 0,0, p m,0, p m,n and p 0,n. Nonnegativity: B m,i (u) B n,j (v) is nonnegative for all m, n, i, j and u and v in the range of 0 and 1. Partition of Unity: The sum of all B m,i (u) B n,j (v) is 1 for all u and v in the range of 0 and 1. Convex Hull Property: a Bézier surface p(u,v) lies in the convex hull defined by its control net. Affine Invariance

21. B-Spline Surface

22. A set of m+1 rows and n+1 control points p i,j, where 0 <= i <= m and 0 <= j <= n; A knot vector of h + 1 knots in the u-direction, U = { u 0, u 1,...., u h }; A knot vector of k + 1 knots in the v-direction, V = { v 0, v 1,...., v k }; The degree p in the u-direction; The degree q in the v-direction;

23. B-Spline Surface Bézier Surface B-Spline Surface B-Spline Surface patch is confined to the region nearer the central four control points (do not interpolate their control points).

24. Basis Functions of B-Spline Surface

25. Basis Functions of B-Spline Surface The coefficient of control point p i,j is the product of two one- dimensional B-spline basis functions, one in the u-direction, N i,p (u), and the other in the v-direction, N j,q (v). The basis functions of control points p 2,0, p 2,1, p 2,2, p 2,3, p 2,4 and p 2,5.The basis function in the u-direction is fixed while the basis functions in the v-direction change

26. Clamped, Closed and Open B-Spline Surface

27. Clamped, Closed and Open B-Spline Surface Clamped B-Spline Surface: If a B-spline is clamped in both directions, then this surface passes though control points p 0,0, p m,0, p 0,n and p m,n and is tangent to the eight legs of the control net at these four control points.

28. Clamped, Closed and Open B-Spline Surface Closed B-Spline Surface: If a B-spline surface is closed in one direction, then all isoparametric curves in this direction are closed curves and the surface becomes a tube.

29. Clamped, Closed and Open B-Spline Surface Open B-Spline Surface: If a B-spline surface is open in both directions, then the surface does not pass through control points p 0,0, p m,0, p 0,n and p m,n.

30. Clamped, Closed and Open B-Spline Surface Three B-spline surfaces clamped, closed and open in both directions. All three surfaces are defined on the same set of control points; but, as in B-spline curves, their knot vectors are different.

31. Properties Of B-Spline Surface

32. Properties Of B-Spline Surface Nonnegativity: N i,p (u) N j,q (v) is nonnegative for all p, q, i, j and u and v in the range of 0 and 1. Partition of Unity: The sum of all N i,p (u) N j,q (v) is 1 for all u and v in the range of 0 and 1. Strong Convex Hull Property Local Modification Scheme p(u,v) is C p-s (resp., C q-t ) continuous in the u (resp., v) direction if u (resp., v) is a knot of multiplicity s (resp., t). Affine Invariance