Vector Computer Graphic
Vector entities
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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 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^32.95*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 curve
6 control points → 2 parabolas (2 Bézier
curves of 2nd degree)
Example of the B spline curve
NURBS = Non Uniform rational Bezier Spline