Modeling of curves

1. Modeling of curves Needs a ways of representing curves: • Reproducible - the representation should give the same curve every time; • Computationally Quick; • Easy to manipulate, especially important for design purposes; • Flexible; • Easy to combine with other segments of curve.

2. Bezier curve model

3. Bezier Curve

4. Bezier Curve from six points

5. Patching two Bezier Curves Together

6. Calculation of Bezier curves • Lets Р0=[1,1] Р1=[2,3] Р2=[4,3] P3=[3,1]

8. The coefficients of Bezier curve

9. The values of the points on the curve

10. Points of polygon and calculated Bezier curve

11. The properties of Bezier curves: • The curve passes through the start and finish points of the control polygon defining the curve. • The tangent to the curve at t = 0 lies in the direction of the line joining the first point to the second point. Also the tangent to the curve at t=1 is in the direction of the line joining the penultimate point to the last point. • Any point on the curve lies inside the convex hull of the control polygon. Also moving any control point will drag the curve towards that control point. • It can also be proved that any line/plane intersects the curve no more times than it intersects the open polygon formed by the control points. This means that no line can intersect the curve more than twice if the four control points form an open polygon as shown. Thus there can be no loops in the curve and it must be smooth. This is called the Variation Diminishing Property.

12. Disadvantages of Bezier curves • Their non-localness. Thus while a particular control point mainly influences the shape of the curve close to it, it also affects the entire curve to some extent. • The fact that the degree of the curve is related to the number of control points. Thus either high order polynomials have to be evaluated or multiple low-degree curve segments have to be used.

13. B-spline basis functions

14. Iterations scheme for cubic (k = 4) basis functions

16. Knot vectors Knot vectors are generally: uniform, open uniform and non-uniform. • Uniform knot vectors are the vectors for which   ti+1 - ti = const,   e.g. [0,1,2,3,4,5]. • Open Uniform knot vectors are uniform knot vectors which have k-equal knot values at each end:     ti = t0 ,   i < k     ti+1 - ti = const ,   k-1 <= i < n+1     ti = tk+n ,   i >= n+1 e.g. [0,0,0,1,2,3,4,4,4]   (k=3, n=5). • Non-uniform knot vectors. This is the general case, the only constraint is the standard ti <= ti+1

17. B-spline curve

18. B-spline curve

19. B-spline through 4 control points, with degree t = 2, 3, and 4. t = 2 is just linear interpolation, as the degree increases the smoother the curve becomes

20. Effect of the degree of B-spline curve on its shape

21. Multiple control points induce regions of high curvature of a B-spline curve

22. Advantagesof B-splines • Changes to a control point only affects the curve in that locality • Any number of points can be added without increasing the degree of the polynomial. • As with Bezier curves adding multiple points at or near a single position draws the curve towards that position. • Closed curves can be created by making the first and last points the same, although continuity will not be maintained automatically. • B-Splines lie in the convex hull of the control points