B-Spline and NURBS Surfaces CS 525 Denbigh Starkey. 1. Background 2 2. B-Spline Surfaces 3 3. NURBS Surfaces 8 4. Surfaces of Rotation 9

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1 B-Spline and NURBS Surfaces CS 525 Denbigh Starkey 1. Background 2 2. B-Spline Surfaces 3 3. NURBS Surfaces 8 4. Surfaces of Rotation 9

2 1. Background In my preious notes I e discussed B-spline and NURBS cures. In these notes I m expanding those concepts into two parameters instead of one, which will map out surfaces instead of cures. Instead of a list of control points, a knot ector, an order, and weights for NURBS, we ll hae a rectangular array of control points, a knot ector in each direction, an order in each parametric direction, and weights associated with each control point for NURBS. E.g., we could hae a 3 5 array of 15 control points for a B-spline surface that is cubic in the long (row) direction and quadratic in the short (column) direction. In that case the two knot ectors would hae to hae lengths of 9 and 6, respectiely. In the notes I ll first describe B-spline surfaces with a simple example, I ll extend the definition to NURBS surfaces in the obious way, and will then show how to use NURBS to build surfaces of reolution. I ll finish up with an example of building a ery basic wine glass. 2

3 2. B-spline Surfaces Recall that the Cox-deBoor equations for B-spline cures were: P(u) = where d is the order of the cure and B k,1 (u) = B k,d (u) = B k,d-1 (u) + B k+1,d-1 (u), d > 1. For surfaces we need an extra parameter, which I ll call, and the equation becomes: S(u, ) = Where the p k,l are the grid of (n + 1) (m + 1) control points, and the order is d in the u direction and e in the direction. B-spline cures were defined in the parametric range which was under the control of d control points. For a surface we make the obious extension which requires that the surface is under the control of a rectangle of control points with edge sizes d and e in the appropriate directions. The best approach at this point is probably an example, een though it will quickly get ugly. Say that we hae a surface patch defined by a 4 4 grid of control points: p 03 p 13 p 23 p 33 p 02 p 12 p 22 p 32 p 01 p 11 p 21 p 31 p 00 p 10 p 20 p 30 and that we want to generate a surface that is quadratic in the u direction and cubic in the direction, and so d = 3 and e = 4. Using open uniform knot 3

4 ectors this gies U = (0, 0, 0, 1, 2, 2, 2) and V = (0, 0, 0, 0, 1, 1, 1, 1). First, as usual, we need the two blending function trees, shown below. B 0,3 B 1,3 B 2,3 B 3,3 1-u u 2-u u-1 B 1,2 B 2,2 B 3,2 1-u u 2-u u-1 B 2,1 B 3,1 0 u<1 1 u<2 B 0,4 B 1,4 0,3 B 2,4 B 3, B 1,3 B 2,3 B 3, B 2,2 B 3,2 1- B 3,1 Now look at the possible paths from the top blending functions down to the bottom blending functions in each tree. To keep things relatiely sane I ll gie the polynomials names, as shown below. In the first tree a, b, and c will be the paths from B 0,3, B 1,3, and B 2,3 to B 2,1, d, e, and f will be the paths from B 1,3, B 2,3, and B 3,3 to B 3,1. In the second tree g, h, i, and j will be the paths from B 0,4, B 1,4, B 2,4, and B 3,4 to B 3,1. I.e., a = (1 u) 2 b = u(1 u) + u(2 u) 0 <1 4

5 c = u 2 d = (2 u) 2 e = u(2 u) + (2 u)(u 1) f = (u 1) 2 g = (1 ) 3 h = 3(1 ) 2 i = 3 2 (1 ) j = 3. Now initially consider the ranges 0 u < 1 and 0 < 1. For these ranges the formula gies: S(u, ) = agp 00 + bgp 10 + cgp 20 + ahp 01 + bhp 11 + chp 21 + aip 02 + bip 12 + cip 22 + ajp 03 + bjgp 13 + cjp 23. For 1 u < 2 and 0 < 1, the other legal pair of ranges, we get: S(u, ) = dgp 10 + egp 20 + fgp 30 + dhp 11 + ehp 21 + fhp 31 + dip 12 + eip 22 + fip 32 + dajp 13 + ejgp 23 + fjp 33. These can be rewritten more coneniently in matrix form: S(u, ) = if 0 u < 1 and 0 < 1 S(u, ) = if 1 u < 2 and 0 < 1 We can use these to ealuate S at the knots (u = 0, 1, and 2, = 0 and 1): 5

6 When u = 0, a = 1, b = 0, and c = 0. When u = 1, d =, e =, and f = 0. When u = 2, d = 0, e = 0, and f = 1. When = 0, g = 1, h = 0, i = 0, and j = 0. When = 1, g = 0, h = 0, i = 0, and j = 1. Putting these into the appropriate formulae for S we get: S(0, 0) = p 00 S(0, 1) = p 03 S(1, 0) = (p 10 + p 20 ) S(1, 1) = (p 13 + p 23 ) S(2, 0) = p 31 S(2, 1) = p 31 Graphically, say that we hae the 16 control points shown below, p 03 p 13 p 23 p 33 p 02 p 12 p 22 p 32 p 21 p 31 p 0 p 1 p 00 p 20 p 30 p 10 where the X s show the six knot pair locations and the cures show the edges of the generated surface patch. An important thing to notice is the effect of using open uniform knot ectors in both parametric dimensions. With a cure this forced the cure to interpolate the first and last control points. With a surface, as we see in the example here, this forces the surface to interpolate the corner control points. 6

7 The easiest way to think about these B-spline patches is probably as a mesh of B-spline cures in both directions. I.e., if we fix a u alue we get a B- spline cure oer the parameter, and if we fix a alue we get a B-spline cure in the u direction. The four edge cures in the figure, aboe, are for u = 0 and 2 and for = 0 and 1. In the sketch below I e shown how a patch shape, in 3D, shows up fairly well when displaying just 18 of these u- and -direction B-spline cures. To render a patch like this we ll generate the points on the grid, and then diide the quadrilaterals into two (i.e., triangularize) to get planar pieces that can be rendered as usual. 7

8 3. NURBS Surfaces The definition of a NURBS surface is fairly obious, and is gien below: P(u) = As expected, I e modified the B-spline surface definition in the same way that the B-spline cure definition was modified to get NURBS cures. The main use of NURBS surfaces, as compared to B-spline surfaces, is to let us get objects like spheres. Howeer we can be more general than that. Gien a NURBS or B-spline cure we can rotate it 360 around an arbitrary axis, and get a surface of reolution. Obiously this includes generating circles, ellipsoids, etc., but it also lets us easily build cylinders, ases, and any other shape that is a surface of reolution. I ll look at how to do this in the next section. 8

9 4. Surfaces of Reolution Although it is easy to reole a NURBS or B-spline cure around any axis, to keep the math simpler I ll assume that the cure lies in the X/Z plane and will be rotated around the Z axis. Deeloping the general formula is, howeer, easy to do if you need it. The basic idea is that we start with a NURBS or B-spline cure, called the generatrix, defined oer the parameter. It will hae its control points, degree, knot ector, and weights (if it is a NURBS cure). We ll then use one of our standard circle generators to rotate each of the control points around the Z axis. I ll use the first circle generator that I showed, which was quadratic with nine control points (including the duplicate), weights (1,, 1,, 1,, 1,, 1), and knot ector (0, 0, 0, 1, 1, 2, 3, 3, 3, 4, 4, 4). Then transferring this into a surface will gie the desired surface of reolution. E.g., say that we want the simple wine glass shown below on the left: Z X Its right hand edge just consists of two lines, as shown on the right, and so the edge can be drawn as a linear (d = 2) B-spline with three control points and with the open uniform knot ector (0, 0, 1, 2, 2). Assume that the three control points lie in the X/Z plane, as discussed aboe, and so a typical control point has coordinates (x, 0, z). To generate our desired circle with radius x we ll need to rotate this around Z, giing the nine control points (x, 0, z), (x, x, z), (0, x, z), (-x, x, z), (-x, 0, z), (-x, -x, z), (0, -x, z), (x, -x, z), and (x, 0, z). We do this for eery control point in our cure. If our generatrix has m control points then this will lead to a grid of n 9 control points which we will use for our NURBS surface. To get the weights associated with these control points we multiply the weights on the control 9

10 points on the generatrix (using 1 s for weights if it was a B-spline) by the appropriate circle weights to get each mesh point weight. E.g., say that we want to display the wine glass shown aboe, and that its three control points were (6, 9), (1, 0) and (2, -4). Then our control point grid will be: (6,0,9) (6,6,9) (0,6,9) (-6,6,9) (-6,0,9) (-6,-6,9) (0,-6,9) (6,-6,9) (6,0,9) (1,0,0) (1,1,0) (0,1,0) (-1,1,0) (-1,0,0) (-1,-1,0) (0,-1,0) (1,-1,0) (1,0,0) (2,0,-4) (2,2,-4) (0,2,-4) (-2,2,-4) (-2,0,-4) (-2,-2,-4) (0,-2,-4) (2,-2,-4) (2,0,-4) The weights associated with these 27 control points will be: The surface will be quadratic (d = 3) in the u direction and linear (e = 2) in the direction with knot ectors (0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 4) in the u direction and (0, 0, 1, 2, 2) in the direction. If we want to remoe the restriction that the generatrix must lie in a plane and that it will be rotated around the Z axis, the only change is that we hae to put the extra control points on circles perpendicular to the axis of rotation with the axis through their centers and the generatrix control points on the circles. 10