Vector Computer Graphic

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. • ∑(yi-f(xi))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)2P0 + 2t(1-t)P1 + t2P2

21. Cubic Bézier curve B(t) = (1-t)3P0 + 3t(1-t)2P1 + 3t2(1-t)P2 + t3P3

22. Bézier curve of higher degree • Example of the expression for curve of 5th 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 2nd degree)

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