bézier curves: integrating math, arts and technology jomar f. rabajante uplb
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Bézier Curves: Integrating Math, Arts and Technology
Jomar F. Rabajante
UPLB
Parametric Curves
x10212
13
t0123
Parametric Curves
y1382
t0123
Parametric Curves
x y10 121 32 8
13 2
t0123
10464510
)2)(1(10)1(151110)(23
ttt
tttttttx
16
25
2
17
3
7
)2)(1(3
7)1(
2
321)(
23
ttt
ttttttty
Parametric Curves
Widely used in vector graphics and computer-aided designs
Example of Parametric Curve: Bézier curve
Affine transformations on the curve can be done by just manipulating the “control points”
Parametric Curves
Bézier Curves
Named after the French engineer Pierre Bézier of the Renault Automobile Company.
“Free form” curves Suppose we are given a set of control/Bézier
points:
i
ii y
xp
We can generate a curve using the parametric form (Bernstein representation):
n
0i
)1(
1t0 ,)(
)()(
iiin ptt
i
n
ty
txtP
Familiar?
Bézier Curves
For 3 points (Quadratic Bézier):
Notice that if t=0 we get (x0,y0). If t=1 we get (x2,y2).
As t takes on values between 0 & 1, a curve is traced but it may not pass through the central point.
10
))(1(2)1()(
))(1(2)1()(
22
102
22
102
t
ytyttytty
xtxttxttx
Bézier Curves
Source: Wikipedia
For 4 points (Cubic Bézier):
10
))(1(3
)()1(3)1()(
))(1(3
)()1(3)1()(
33
22
12
03
33
22
12
03
t
ytytt
yttytty
xtxtt
xttxttx
Bézier Curves
You can use MS Excel, GraphCalc or any graphing software…
TO DO:
1.5 2 2.5 3 3.5 4 4.5 5 5.50
0.5
1
1.5
2
2.5
3
3.5
The following control points are used: .
The Bézier curve lies entirely inside the convex hull containing all the control points.Convex hull of a set of points is the smallest convex set that contains the points. A set is convex iff the line segment between any two points in the set lies entirely in the set.
Examples of convex hull of four points:
Bézier Curves
Some curves that seem simple, such as the circle, cannot be described exactly by a Bézier or piecewise Bézier curve; RATIONAL BEZIER curves can do this.
Bézier Curves
de Casteljau’s Algorithm
Independently made by Paul de Faget de Casteljau to generate Bézier curves.
Uses barycenter coordinates.
Let’s use Geogebra
Bézier Curves: Integrating Math, Arts and Technology
Jomar F. Rabajante
UPLB