1. Vector Computer Graphic

2. Vector entities Line Circle, Ellipse, arc,… Curves: Spline, Bezier’s curve, … … Areas Solids Models

3. Interpolation The curve is passing through the control points

4. Polynomical interpolation Linear – 2 points Quadratic – 3 points Polynom n degree – n+1 points

5. Linear interpolation

6. Quadratic interpolation

7. 4 degree polynomical interpolation Control points: (-2,4) (-1,0) (0,3) (1,1) (2,-5) Equatations: 16a -8b +4c -2d + e = 4 a - b + c -d +e = -3 e = 3 a + b + c + d +e = 1 16a +8b +4c +2d +e =-5 Solution: a=0.458 b=-0.75 c=-2.95 d=1.25 e=3 Function: 0.458*x^4-0.75*x^3- 2.95*x^2+1.25*x+3

8. Spline curve The curve consists of segments expressed by polynom of lesser degree then the number of the points require. The curves in their border points have smooth continue.

9. Linear „spline“ Polynoms of first degree. In the border points the continuation is continuous. But the first derivation must not be continuous. So the curve must not be smooth. The simple term is polyline.

10. Quadratic spline The curve is formed by segments of parabolas. In the border points there is a smooth continuation, the first derivation is continuous. The following derivation must not be }and commonly are not) continuous. This is the most common version of spline curve. When only spline is said the quadratic spline is understood (AutoCAD).

11. Quadratic spline

12. Spline curves of higher degree Cubic – curve formed by segments of 3th degree functions (cubics), the continuation of first and second derivation is guarantee. General (n-th degree), the continuation of (n-1)th derivation is guarantee.

13. Approximation curves The curve does not necessary pass through the control points. Formally any curve is the aproximation curve. The main task is to find such an expression to be –Simple –To approximate the control points sufficiently well

14. Least squares approximation I choose the type of the function (commonly the polynomical function of lesser degree then the necessary degree for interpolation) I compute such parameters, so the summa of the squares of the deviations is minimal. ∑(y i -f(x i )) 2 → min

15. Least squares approximation

16. Bézier approximation (Bézier’s curve) Approximation by a polynom of n-th degree for n+1 control points P0,P1,…,Pn The curve pass through the first point P0 and the last point Pn The tangent in the first point P0 is parallel to the vector P0P1. The tangent in the last point Pn is paralle to the vector Pn-1 Pn The whole curve lies in the convex hull of the points P0, …,Pn

17. Pierre Ettiene Bézier (1910-1999)

18. The expression of the Bézier curve

19. Linear Bézier curve B(t) = (1-t).P0 + t.P1 The parametric expression of the abscissa.

20. Quadratic Bézier curve B(t) = (1-t) 2 P0 + 2t(1-t)P1 + t 2 P2

21. Cubic Bézier curve B(t) = (1-t) 3 P0 + 3t(1-t) 2 P1 + 3t 2 (1-t)P2 + t 3 P3

22. Bézier curve of higher degree Example of the expression for curve of 5 th degree

23. B-spline The segments of Bézier curves of lesser degree (commonly quadratic and cubic) are in their border points smoothly connected.

24. Example of the B spline curve 6 control points → 2 parabolas (2 Bézier curves of 2 nd degree)

25. Example of the B spline curve NURBS = Non Uniform rational Bezier Spline