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Lecture 10 Curves and Surfaces I

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1 Lecture 10 Curves and Surfaces I
Computer Graphics Lecture 10 Curves and Surfaces I

2 Types of Curves Explicit: Implicit: Parametric:
y = mx + b r = Arx + Bry + Cr Implicit: Ax + By + C = (x – x0)2 + (y – y0)2 – r2 = 0 Parametric: x = x0 + (x1 – x0)t x = x0 + rcosq y = y0 + (y1 – y0)t y = y0 + rsinq 23/10/2007 Lecture 10

3 Why parametric? Parametric curves are very flexible.
                                                                  Parametric curves are very flexible. They are not required to be functions Curves can be multi-valued with respect to any coordinate system. Parameter count generally gives the object’s dimension. (x(u,v), y(u,v), z(u,v)) Coord functions independent. 23/10/2007 Lecture 10

4 Specifying curves Control Points: Knots: Interpolating spline:
A set of points that influence the curve’s shape. Knots: Control points that lie on the curve. Interpolating spline: Curve passes through the control points. Approximating spline: Control points merely influence shape. 23/10/2007 Lecture 10

5 Piecewise curve segments
We can represent an arbitrary length curve as a series of curves pieced together. But we will want to control how these curves fit together … 23/10/2007 Lecture 10

6 Parametric Cubic Curves
In order to assure C2 continuity our functions must be of at least degree 3. This is also the lowest order to describe a non-planar curve. Cubic has 4 degrees of freedom and can control 4 things. Use polynomials: x(t) of degree n is a function of t. - y(t) and z(t) are similar and each is handled independently (why parametric is easier to handle than implicit form). That is: 23/10/2007 Lecture 10

7 An Illustrative Example
The whole story of parametric splines is in deriving their coefficients. How we do that, by satisfying constraints set by the knots and continuity conditions, is what classifies the spline system and distinguishes the different systems. Example: Cubic Hermite Splines 23/10/2007 Lecture 10

8 Hermite splines 4 degrees of freedom, 2 at each end to control C0 and C1 continuity at each end. Polynomial can be specified by the position of, and gradient at, each endpoint of curve. Determine: x = X(t) in terms of x0, x0/, x1, x1/ Now: X(t) = a3t3 + a2t2 + a1t + a0 and X/(t) = 3a3t2 + 2a2t + a1 23/10/2007 Lecture 10

9 Finding Hermite coefficients
Substituting for t at each endpoint: x0 = X(0) = a x0/ = X/(0) = a1 x1 = X(1) = a3 + a2 + a1 + a x1/ = X/(1) = 3a3 + 2a2+ a1 And the solution is: a0 = x a1 = x0/ a2 = -3x0 – 2x0/ + 3x1 – x1/ a3 = 2x0 + x0/ - 2x1 + x1/ 23/10/2007 Lecture 10

10 The Hermite matrix: MH The resultant polynomial can be expressed in matrix form: X(t) = tTMHq ( q is the control vector) We can now define a parametric polynomial for each coordinate required independently, ie. X(t), Y(t) and Z(t) 23/10/2007 Lecture 10

11 Hermite Basis (Blending) Functions
The graph shows the shape of the four basis functions – often called blending functions. They are labelled with the elements of the control vector that they weight. Note that at each end only position is non-zero, so the curve must touch the endpoints x0 x1 x0/ x1/ 23/10/2007 Lecture 10

12 Family of Hermite curves.
y(t) Note : Start points on left. x(t) 23/10/2007 Lecture 10

13 Displaying Hermite curves.
Simple : Loop through t – select step size to suit. Plug x values into the geometry matrix. Evaluate P(t)  x value for current position. Repeat for y & z independently. Draw line segment between current and previous point. Joining curves: Coincident endpoints for C0 continuity Align tangent vectors for C1 continuity . 23/10/2007 Lecture 10

14 Bézier Curves Hermite cubic curves are difficult to model – need to specify point and gradient. More intuitive to only specify points. Pierre Bézier specified 2 endpoints and 2 additional control points to specify the gradient at the endpoints. Can be derived from Hermite matrix: Two end control points specify tangent 23/10/2007 Lecture 10

15 Bézier Curves P2 Note the Convex Hull has been shown as a dashed line – use as a bounding extent for intersection purposes. P4 P1 P3 P3 P4 P1 P2 23/10/2007 Lecture 10

16 Bézier Matrix We must first define the blending functions.
For a polynomial of order n (degree n-1), we have n control points with terms up to tn-1, as follows: fr is a blending function the coefficient of tr in the binomial expansion of (t+(1-t))n-1 23/10/2007 Lecture 10

17 Bézier Matrix The cubic form is the most popular X(t) = tTMBq (MB is the Bézier matrix) With n=4 and r=0,1,2,3 we get: Similarly for Y(t) and Z(t) 23/10/2007 Lecture 10

18 Bézier blending functions
This is how they look – Again the Convex Hull property holds, ie. The functions sum to 1 at any point along the curve. Endpoints have full weight The weights of each function is clear and the labels show the control points being weighted. q0 q3 q1 q2 23/10/2007 Lecture 10

19 Example Interpolation Spline.
Catmull-Rom spline interpolates control points. The gradient at each control point is the vector between adjacent control points. 23/10/2007 Lecture 10

20 Uniform Nonrational B-Splines
Cubic Bézier curves are joined together in the same way as Hermite curves for arbitrary length by joining short 4-point pieces with C0 and C1 order continuity. However cubic B-Splines can have any number of control points and any length of curve with C0, C1 and C2 order continuity. Segments of the B-Spline, joined at knots, depend on a few adjacent control points which are shared between segments for continuity. 23/10/2007 Lecture 10

21 What are B-Splines? Consider a bi-infinite set of knots {ti}
B-Splines are a set of local functions which together form a basis for certain classes of piecewise polynomial functions with these knots. Any piecewise polynomial of order k, degree k-1, with continuous (k-2)th derivative is a weighted sum of functions Bik from the set: 23/10/2007 Lecture 10

22 Parametric B-Spline curves
B-Spline functions are used as blending functions: fi(t) = Bi,k(t) Given control points: x1 … xn And B-Splines: Bi,k where i = 1 … n Then: Support of function X is tn < t < tn+k ie. the range over which X is non-zero. 23/10/2007 Lecture 10

23 Parametric B-Splines Also works for closed curve, ie. wrapped-round set of n knots. 23/10/2007 Lecture 10

24 What does a B-Spline look like?
1 They look rather like the shape above and there is one starting at each t. The one above is a cubic B-Spline with a support of 4 knot intervals. Thus within one interval there are portions of 4 splines which have influence. 23/10/2007 Lecture 10

25 B-Spline blending functions for one interval.
Here again the functions hold the Convex Hull property. They are non-negative. There are four of them and therefore another 4x4 matrix as we’ve seen. t 1 1/6 4/6 f(t) 23/10/2007 Lecture 10

26 Uniform Non-rational B-Splines.
For each i  4 , there is a knot between Qi-1 and Qi at t = ti. Initial points at t3 and tm+1 are also knots, so we have a total of m-1 knots. Knot. Control point. m = 9 (P0 ..P9) m-1 knots m+1 control points m-2 curve segments 23/10/2007 Lecture 10

27 Uniform Non-rational B-Splines.
First segment Q3 is defined by point P0 through P3 over the range t3 = 0 to t4 = 1. Knot. Control point. P1 P2 P3 P0 Q3 23/10/2007 Lecture 10

28 Uniform Non-rational B-Splines.
Second segment Q4 is defined by point P1 through P4 over the range t4 = 1 to t5 = 2. Knot. Control point. Q4 P1 P3 P4 P2 23/10/2007 Lecture 10

29 Reading for this lecture
Foley et al. Chapter 11, section 11.2 up to and including Introductory text Chapter 9, section 9.2 up to and including section 9.2.4 23/10/2007 Lecture 10

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