Surface Modeling Parametric Surfaces

1. Surface ModelingParametric Surfaces Dr. S.M. Malaek Assistant: M. Younesi

2. Surface Modeling • 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-x0)2+(y-y0)2+(z-z0)2-r2=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 vare in the range of 0 and 1.

6. Patch 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.

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 pointspj,k, where jis in the range of 0 and m, and k is in the range of 0and 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 twoone-dimensional Bézier basis functions. • The basis functions for a Bézier surface are parametric surfaces of two variablesuand vdefined on the unit square. The basis functions for control points p0,0(left) and p1,1 (right), respectively. For control point p0,0, its basis function is the product of two one-dimensional Bézier basis functions B2,0(u) in the udirection and B2,0(v) in thev direction. In the left figure, both B2,0(u) and B2,0(v) are shown along with their product (shown in wireframe). The right figure shows the basis function for p1,1, which is the product ofB2,1(u) in the udirection and B2,1(v) in the v direction.

16. Joining Bézier Surface Patches

17. Joining Bézier Surface Patches • C0 continuity requires aligning boundary curves.

18. Joining Bézier Surface Patches • C1 continuity requires aligning boundary curves and derivatives.

19. Properties Of Bézier Surface Patches

20. Properties Of Bézier Surface Patches • p(u,v)passesthrough the control points at the four corners of the control net: p0,0, pm,0, pm,n and p0,n. • Nonnegativity: Bm,i(u) Bn,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 Bm,i(u) Bn,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. B-Spline Surface • A set of m+1 rows andn+1 control pointspi,j, where 0 <= i <= m and 0 <= j <= n; • A knot vector of h + 1 knots in the u-direction, U = { u0, u1, ...., uh }; • A knot vector of k + 1 knots in the v-direction, V = { v0, v1, ...., vk }; • The degreep in the u-direction; The degree qin the v-direction;

23. Bézier Surface B-Spline 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 pi,jis the product of two one-dimensional B-spline basis functions, one in the u-direction, Ni,p(u), and the other in thev-direction, Nj,q(v). • The basis functions of control points p2,0, p2,1, p2,2, p2,3, p2,4 and p2,5.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 p0,0, pm,0, p0,n and pm,n and is tangent to the eightlegs 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 pointsp0,0, pm,0, p0,n and pm,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: Ni,p(u) Nj,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 Ni,p(u) Nj,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 Cp-s (resp.,Cq-t) continuous in theu (resp., v) direction if u (resp., v) is a knot of multiplicity s (resp.,t). • Affine Invariance