G43 – Geometric Modeling

Mathematical modeling of

Curves – Bezier curves

1

Dr.C. ParamasivamE-mail: cpmech@tce.edu

Curves – Bezier curves

1. Define the coordinate system for the development

of models based on input and geometry.

2. Develop and manipulate the curves and surfaces

using parametric equations.

3. Develop and manipulate the solid models using

Course Objectives

By the end of the course, student will be able to:

3. Develop and manipulate the solid models using

modeling techniques.4. Implement the transformation and projection over

the geometric models.5. Implement the neutral file formats over 2D

wireframe models.

• Bezier Curves were first developed in 1959 by

Paul de Casteljau.

• They were popularized in 1962 by French

engineer Pierre Bezier, who used them to

Bezier Curves

engineer Pierre Bezier, who used them to

design automobile bodies.

Approximation approaches to the representation of

curves provide a smooth shape that approximates

the original points, without exactly passing through

all of them.

Given n+1 control points, P0, P1, P2, ….., Pn, the Bezier

curve is defined by the following polynomial of

degree n:

Conditions required:

Set of (n+1) control points.

degree n:

10)()(0

, ≤≤=∑=

uuBPuP

n

i

nii

where

P(u) is any point on the curve

Pi is a control point, Pi = [xi yi zi]T

Bi,n are polynomials (serves as basis function for the Bezier Curve)

)!(!

!),(

)1(),()(,

ini

ninC

uuinCuBini

ni

−

=

−=−

where,

C(n, i) – Polynomial coefficientC(n, i) – Polynomial coefficient

The general equation can be expanded to give as:

n

n

n

n

nnn

uPuunnCP

uunCPuunCPuPuP

+−−+

+−+−+−=

−

−

−−

)1()1,(

..........)1()2,()1()1,()1()(

1

1

22

2

1

10

0 ≤ u ≤ 1

Quadratic Bezier curve

generation (DOC = 2)

Cubic Bezier curve generation (DOC = 3)

Some form of Bezier curves

Bezier curve with DOC 6

De Casteljau’s Algorithm

• Specify cubic curve with four control points

• Finding a point on a Bezier Curve.

Linear:

Pt = (1-t)P0 + tP1

De Casteljau Algorithm

3P

1P2

P

t

t−1

t−1 t−1

t−1t−1

t

tt

t

0P0P

t−10P

1P

2P

3P

t−1

t

t−1

t

t−1

t

10)1( tPPt +−

21)1( tPPt +−

32)1( tPPt +−

t−1

t

t−1

t

2

2

10

2)1(2)1(

Pt

tPtPt

+

−+−

3

2

21

2)1(2)1(

Pt

tPtPt

+

−+−

t−1

t

3

3

2

2

1

2

0

3

)1(3

)1(3

)1(

Pt

Ptt

tPt

Pt

+

−+

−+

−

ii

i

tti

−

=

−

∑ 3

3

0

)1(3

De Casteljau’s Algorithm (cont)

• Specify cubic curve with four control points

• Finding a point on a Bezier Curve.

Quadratic:

Pt = (1-t)[(1-t)P0 + tP1] +

t[(1-t)P1 + tP2]

= (1-t)2P0 +2t(1-t)P1 + t2P2

De Casteljau’s Algorithm (cont)

• Specify cubic curve with four control points

• Finding a point on a Bezier Curve.

Cubic:

Pt = (1-t)((1-t)2P0 + 2t(1-t)P1 + t2P2)+

t((1-t)2P1 + 2t(1-t)P2 + t2P3)

= (1-t)3P0 + 3t(1-t)2P1 +

3t2(1-t)P2 +t3P3

De Casteljau’s Algorithm

1. The first and last control points are interpolated.

2. The curve is tangent to the first and last segments ,of thecharacteristic polygon.

3. The reversing the direction of parameterization does notchange the curve shape.

4. The each control point is most influential on the curve

Characteristics of Bezier Curve

4. The each control point is most influential on the curveshape at u = i/n.

5. The curve shape can be modified by either changing one

or more vertices of its polygon or by keeping the polygonfixed and specifying multiple coincident points at a vertex.

6. A closed Bezier curve can simply be generated by closingits characteristic polygon or choosing Po and Pn to becoincident.

Closed Bezier curve

7. For any valid value of u, the sum of the Bi,n functions

associated with the control points is always equal to unity for

any degree of Bezier curve.

8. The curve lies entirely within the convex hull of its control

points.

0.6

0.8

1

1.2

P1

P3

Cubic Bezier Curve

0

0.2

0.4

0.6

0 0.5 1 1.5 2 2.5 3 3.5

P0

P2

0.8

1

1.2

B0

Shape of Bernstein function

0

0.2

0.4

0.6B1

B2

B3

Limitations

1. Lack of local control.

2. Not possible to add any control points

without modifying the degree of curves.

Continuity• Parametric continuity (CX)

1. Zero order continuity

» Positional continuity

2. First order continuity

» Tangential continuity

3. Second order continuity

» Curvature continuity

• Geometric continuity (Gx)

Only directions of the curve segments have to match.

(a)

)0(G continuity order th

0)0(C

Order of continuity

(b)

(c)

)1(G

)2(G

continuity order st1

continuity order nd2

)1(C

)2(C

Find the equation of the Bezier curve which passes through (0,0) and(-4,2) and controlled through (14,10) and (4,0).

Required Equation

Additional Problems

1. Find the points on a Bezier curve which hasstarting and ending points P0(2,3) and P3(4,-3)respectively, and is controlled by P1(6,6) andP2(8,1), for u=0.2 and 0.9.

2. A cubic Bezier curve segment is described bycontrol points P (1,2), P (2,7), P (10,9) andcontrol points P0(1,2), P1(2,7), P2(10,9) andP3(9,b). Another curve segment is described byQ0(a,6), Q1(8,c), Q2(13,2) and Q3(15,2).Determine the values of a, b and c, so that thetwo curve segments join smoothly.

3. Show that the Bezier curve always touches thestarting point (for u=0) and ending point (foru=1).