Vector Computer Graphic

Vector entities

• Line

• Circle, Ellipse, arc,…

• Curves: Spline, Bezier’s curve, …

• …

• Areas

• Solids

• Models

Interpolation

• The curve is passing through the control points

Polynomical interpolation

• Linear – 2 points

• Quadratic – 3 points

• Polynom n degree – n+1 points

Linear interpolation

Quadratic interpolation

4 degree polynomical interpolationControl 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

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.

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.

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).

Quadratic spline

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.

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

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

Least squares approximation

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

Pierre Ettiene Bézier (1910-1999)

The expression of the Bézier curve

Linear Bézier curve

• B(t) = (1-t).P0 + t.P1• The parametric

expression of the abscissa.

Quadratic Bézier curve

• B(t) = (1-t)2P0 + 2t(1-t)P1 + t2P2

Cubic Bézier curve

B(t) = (1-t)3P0 + 3t(1-t)2P1 + 3t2(1-t)P2 + t3P3

Bézier curve of higher degree

• Example of the expression for curve of 5th degree

B-spline

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

Example of the B spline curve6 control points → 2 parabolas (2 Bézier curves of 2nd degree)

Example of the B spline curveNURBS = Non Uniform rational Bezier Spline