Surface ModelingParametric Surfaces

Dr. S.M. MalaekAssistant: M. Younesi

Surface ModelingThere are two types of surfaces that are commonly used in modeling systems,parametricparametric and implicitimplicit.

Implicit Surface: f(x,y,z)=0Example: (x-x0)2+(y-y0)2+(z-z0)2-r2=0

Surface Modeling

Parametric SurfacesParametric 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)

Parametric SurfacesParametric 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.

Parametric SurfacesParametric 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

Parametric Surface PatchEach patch is defined by control points net (Control Polyhedron).

Parametric Surface PatchA 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.

Parametric Surface PatchPoint Q(u,v) on the patch is the tensor productof parametric curves defined by the control points.

Bézier SurfacePatch

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

∑∑== =

m

j

n

knkmjkj uBvBvu

0 0,,, )()(),( pP

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).

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

Basis Functionsof

Bézier Surface Patches

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 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 u direction and B2,0(v) in the v 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 of B2,1(u) in the udirection and B2,1(v) in the v direction.

Joining Bézier SurfacePatches

Joining Bézier Surface PatchesC0 continuity requires aligning boundary curves. 10for ),0(),1( ≤≤= vvv RQ

0030

0131

0232

0333

RQRQRQRQ

====

Joining Bézier Surface PatchesC1 continuity requires aligning boundary curves and derivatives.

3,,0)()( 0123 L=−=− ik iiii RRQQ

PropertiesOf

Bézier Surface Patches

Properties Of Bézier Surface Patchesp(u,v) passes through 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

B-Spline Surface

B-Spline SurfaceA set of m+1 rows and n+1 control points pi,j, where 0 <= i <= mand 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 degree p in the u-direction; The degree q in the v-direction;

)()(),( ,0 0

,, vNuNvu qj

m

i

n

jpiji∑∑

= =

= pP

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).

Basis Functionsof

B-Spline Surface

Basis Functions of B-Spline SurfaceThe coefficient of control point pi,j is the product of two one-dimensional B-spline basis functions, one in the u-direction, Ni,p(u), and the other in the v-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

Clamped, Closed and Open

B-Spline Surface

Clamped, Closed and Open B-Spline SurfaceClamped 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 eight legsof the control net at these four control points.

Clamped, Closed and Open B-Spline SurfaceClosed 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.

Clamped, Closed and Open B-Spline SurfaceOpen B-Spline Surface: If a B-spline surface is open in both directions, then the surface does not pass through control points p0,0, pm,0, p0,n and pm,n.

Clamped, Closed and Open B-Spline SurfaceThree 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 vectorsare different.

PropertiesOf

B-Spline Surface

Properties Of B-Spline SurfaceNonnegativity: Ni,p(u) Nj,q(v) is nonnegative for all p, q, i, j and uand 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 vin the range of 0 and 1. Strong Convex Hull Property Local Modification Schemep(u,v) is Cp-s (resp., Cq-t) continuous in the u (resp., v) direction if u(resp., v) is a knot of multiplicity s (resp., t). Affine Invariance