Interpolating Splines, Implicit Descriptions

1. Interpolating Splines,Implicit Descriptions Glenn G. ChappellCHAPPELLG@member.ams.org U. of Alaska Fairbanks CS 481/681 Lecture Notes Friday, March 5, 2004

2. Review:Bézier Algorithms [1/3] • The de Casteljau Algorithm computes a point on a Bézier curve using this repeated lirping procedure. • Each lirp is done using the same value of t. • x, y, and z are computed separately. P0 P1 P2 P3 P4 lirp lirp lirp lirp lirp lirp lirp lirp lirp lirp Point on curve CS 481/681

3. Review:Bézier Algorithms [2/3] • Another way to draw a Bézier curve is to use the polynomial that describes it to compute coordinates of points. • If there are n control points (P0, P1, …, Pn–1), then a Bézier curve is parameterized by a polynomial of degree n–1. • For four control points the polynomial isP0·(1–t)3 + P1·3(1–t)2t +P2·3(1–t)t2 +P3·t3. • This is made up of pieces called Bernstein polynomials. • B30(t) = (1–t)3. • B31(t) = 3(1–t)2t. • B32(t) = 3(1–t)t2. • B33(t) = t3. • The pattern: • Descending powers of (1–t). • Ascending powers of t. • Coefficients are binomial coefficients. • Found in Pascal’s Triangle. CS 481/681

4. Review:Bézier Algorithms [3/3] • The way we described Bézier surfaces last time suggests how to extend curve-drawing algorithms to surfaces. • Say we have a grid of 9 control points, as shown. • The horizontal curves have the following formulas:P0,0·(1–t)2 + P0,1·2(1–t)t +P0,2·t2P1,0·(1–t)2 + P1,1·2(1–t)t +P1,2·t2P2,0·(1–t)2 + P2,1·2(1–t)t +P2,2·t2 • Now we use points on these curves as control points. Let our vertical parameter be s. We obtain the following formula for a Bézier patch:[ P0,0·(1–t)2 + P0,1·2(1–t)t +P0,2·t2 ]·(1–s)2 +[ P1,0·(1–t)2 + P1,1·2(1–t)t +P1,2·t2 ]·2(1–s)s +[ P2,0·(1–t)2 + P2,1·2(1–t)t +P2,2·t2 ]·s2 P0,0 P0,1 P0,2 P1,0 P1,1 P1,2 P2,0 P2,1 P2,2 CS 481/681

5. Review:Bézier Properties [1/3] • Two properties of Bézier curves are worth another look: • Affine Invariance • Convex-Hull Property • An affine transformation is a transformation that can be produced by doing a linear transformation followed by a translation. • All of the transformations we have dealt with, except for perspective projection, are affine. • Bézier curves have the property that applying an affine transformation to each control points results in the same transformation being applied to every point on the curve. • This property is called affine invariance. • For example, to rotate a Bézier curve, apply a rotation to the control points. • In short: Transformations act the way you want them to. CS 481/681

6. Review:Bézier Properties [2/3] • The convex hull of a set of points is the smallest convex region containing them. • Informally speaking, “lasso” (or shrink-wrap) the points; the region inside the lasso is the convex hull. A Bézier curve lies entirely in the convex hull of its control points. • This is called the convex-hull property. • This property makes it easy to specify where a curve will not go. • Smooth interpolating splines never have this property. P0 P2 P0 P2 Points Convex hull P1 P1 P0 P2 P1 CS 481/681

7. Review:Bézier Properties [3/3] • Bézier curves do not interpolate all their control points. • So we can easily specify where it does not go, but not where it does go. • Bézier curves also do not have “local control”. • A curve has local control if moving a single control point only changes a small part of the curve (the part near the control point). • Moving any control point on a Bézier curve changes the whole curve. • This is the main reason we do not use Bézier curves with a large number of control points. • Instead, we piece together several 3- or 4-point Bézier curves in a smooth way. • This multiple-Bézier curve does have local control. CS 481/681

8. Review:Building Better Curves • The biggest problems with Bézier curves are: • Lack of local control. • High degree, if there are a large number of control points. • Lack of interpolation (sometimes a problem, sometimes not). • Better curve #1: B-spline. • Parametric curve described by polynomials based on control points. • Has affine invariance & local control. • Has same degree regardless of number of control points. • Approximating, has convex-hull property. • Better curve #2: Catmull-Rom spline. • Parametric curve described by polynomials based on control points. • Has affine invariance & local control. • Has same degree (3) regardless of number of control points. • Interpolating, no convex-hull property, (and a bit tough to get “just right”). • Better curve #3: NURBS (Non-Uniform Rational B-Spline). • Parametric curve described by rational functions based on control points. • Has affine invariance & local control. • Very general & adaptable, but thus somewhat tougher to use. • Built into GLU. CS 481/681

9. Interpolating Splines:Overview • We would like to define an interpolating spline with lots of convenient properties. • We can get pretty much all the good properties, except for the convex-hull property. Remember that smooth, interpolating splines never have this property. • Some interpolating splines require additional information, besides their control points. Our goal is not to require this. • Our goal is called a Catmull-Rom spline. We will reach this goal in three steps. • Hermite Curves • Interpolating splines that require a vector to be specified at each control point. • Cardinal Splines • Nearly to the goal, but there is a “tension” parameter. • Catmull-Rom Splines • The goal. CS 481/681

10. Interpolating Splines:Hermite Curves — Idea [1/2] • The first step toward our goal is an Hermite curve. • French again. Say “air-MEET”. Clear your throat a little during the “r”, and you’ll have it perfect. • An Hermite curve is an interpolating spline whose formulas are cubic (degree  3) polynomials. • We look at the segment between two control points: Pi and Pi+1. • At each control point, we specify a velocity (or direction) vector. • Call the vectors vi and vi+1. vi Pi+1 Pi vi+1 CS 481/681

11. Interpolating Splines:Hermite Curves — Idea [2/2] • It turns out that there is a unique cubic polynomial that meets the following four constraints: • At t = 0, its value is Pi. • At t = 1, its value is Pi+1. • At t = 0, its derivative is vi. • At t = 1, its derivative is vi+1. • Remember that the value of this polynomial is a point in space. • So it will have the form f(t) = At3 + Bt2 + Ct + D, where A, B, C, and D are points/vectors. vi Pi+1 Pi vi+1 CS 481/681

12. Interpolating Splines:Hermite Curves — Computations • Again, there is a unique cubic polynomialf(t) = At3 + Bt2 + Ct + D that meets the following four constraints: • At t = 0, its value is Pi. • At t = 1, its value is Pi+1. • At t = 0, its derivative is vi. • At t = 1, its derivative is vi+1. • How do we find it? First, find the derivative: • f (t) = 3At2 + 2Bt + C • Next figure out what the four constraints mean: • Pi = f(0) = D • Pi+1 = f(1) = A + B + C + D • vi = f (0) = C • vi+1 = f (1) = 3A + 2B + C • This gives a system of 4 linear equations in 4 variables (A, B, C, D). Solve to obtain: • A = 2Pi – 2Pi+1 + vi + vi+1 • B = –3Pi + 3Pi+1 – 2vi – vi+1 • C = vi • D = Pi CS 481/681

13. Interpolating Splines:Hermite Curves — Code • The results: • A = 2Pi – 2Pi+1 + vi + vi+1 • B = –3Pi + 3Pi+1 – 2vi – vi+1 • C = vi • D = Pi • Here is some code using TRANSF: const int ncp; // Number of Control Points pos p[ncp]; // Array of control points vec v[ncp]; // Vector for each control point int i; // subscript, same as i above double t; // parameter value, in [0,1] vec A = 2.*(p[i] – p[i+1]) + v[i] + v[i+1]; vec B = -3.*(p[i] – p[i+1]) – 2*v[i] – v[i+1]; vec C = v[i]; pos D = p[i]; pos point_on_hermite_curve = t*t*t*A + t*t*B + t*C + D; CS 481/681

14. Interpolating Splines:Hermite Curves — Notes • In practice, Hermite curve segments join to make a smooth curve. • Hermite curves are relatively common. • Whenever you see an interpolating curve with direction handles on the control points, you are probably dealing with an Hermite curve. • Why did we use cubic polynomials? vi vi+2 Pi+1 Pi Pi+2 vi+1 CS 481/681

15. Interpolating Splines:Cardinal Splines • Hermite curves fall short of our goal, since they require direction vectors to be specified. • If we can compute these vectors automatically, based on the control points, then we can meet our goal using Hermite curves. • Idea • Let vi be parallel to the vector from Pi–1 to Pi+1. • So vi = k(Pi+1 – Pi–1). • The value k is called the tension. • The tension can be whatever you want, but usually it is between 0 and 1. • Tension 0 makes a polyline. • Problem: We cannot define the first & last curve segments. • We address this later. Pi vi Pi–1 Pi+1 CS 481/681

16. Interpolating Splines:Catmull-Rom Splines (The Goal) • Suppose the control points of a cardinal spline are equally spaced along a straight line. What should the tension be? • k = ½ seems natural, since it makes the speed constant. • A cardinal spline with tension ½ is called a Catmull-Rom spline. • Properties • Defined entirely in terms of control points. • Can use any number of control points; always defined by cubic polynomials. • Interpolating. • Smooth. • Local control. • Affine invariant. • NO convex-hull property • , but you can’t have everything. • Use the Hermite-curve code to draw it. CS 481/681

17. Interpolating Splines:Hitting the Endpoints • One small problem with a Catmull-Rom spline is that we cannot hit the first and last control points (unless they lie in a loop). • Each segment needs 4 constraints. • Normally, these are the two positions and the two vectors. • But we don’t have a vector at the first & last points. • Solution • Instead of constraining f to be a certain velocity vector, use the constraint f (0) = 0 or f (1) = 0 at the endpoints. • Then solve the system again. • Then, given any list of control points, we have a uniquely defined curve passing through all of them, in order. • There are any number of other proposed variations on the cardinal spline, higher-degree versions, etc. CS 481/681

18. Interpolating Splines:Surface Patches • We define a Catmull-Rom surface patch using the same ideas that we used to define a Bézier patch. • Start with a rectangular grid of control points. • Compute curves through each horizontal row of control points. • Use corresponding points on these curves as control points for vertical curves. • Put the vertical curves together to make the patch. • See the formula for a Bézier patch to get an idea of how the computation would look. CS 481/681

19. Implicit Descriptions:Introduction • Recall, a surface can be defined using: • A Polygon List • An Explicit description • Formulas based on parameters. • An Implicit Description • Equations to be solved. • Spline curves & surfaces are a way to create easy-to-define explicit descriptions. • Now, we look at implicit descriptions. CS 481/681

20. Implicit Descriptions:Why? • One problem with explicit descriptions is that they generally assume some kind of rectangular grid on a surface. • Can you put a rectangular grid on an arbitrary object? • What about a surface that can change topology? • Implicit descriptions do not have these problems. • Of course, they have other problems … CS 481/681

21. Implicit Descriptions:What They Look Like [1/2] • Strictly speaking, an implicit description is an equation to be solved. • In practice, however, things look a bit different. • Typically, we specify “potential” values at various points in space. • We want to approximate the surface consisting of all points at a particular “threshold” value. • This is called an isosurface. • Thus, we want solutions to this equation:potential(x, y, z) = threshold_value • However, we do not define the potential everywhere, so it might not be clear exactly what we are approximating. CS 481/681

22. Implicit Descriptions:What They Look Like [2/2] • Specifying potential values generally takes one of two forms: • Specify potentials at every point in a 3-D grid. • Specify potentials at a very limited set of points. • The former method is well studied, and somewhat reasonable algorithms exist to draw approximations of the surface. • The latter method is up-and-coming … • Time for a blackboard picture. CS 481/681