Bézier curve

Bézier curve

1

Introduction

Bézier curves are a particular family of curves in

ℝ

m

that parameterize a smooth path given points.
They are often used to model or approximate general curved shapes
because of their algebraic

simplicity

.

Bézier curves were originally developed by Paul de Casteljau in 1959. But it became a famous shape only when Pierre Bézier, a French engineer at Renault, used it to automobiles in the 1970’s.
They are now widely used in many such as industrial and computer-aided ,
vector-based drawing, font design, and 3D modeling.

2

Definition

Given

n

+

1

points

𝑷

i

in

ℝ

m

,
the

Bézier curve

for these

control points

is the curve

𝑩

:

[

0

,

1

]

→

ℝ

m

defined by

𝑩

⁢

(

t

)

=

∑

i

=

0

n

(

n

i

)

⁢

(

1

-

t

)

n

-

i

⁢

t

i

⁢

𝑷

i

.

The coefficients in this

expression

:

b

i

,

n

⁢

(

t

)

=

(

n

i

)

⁢

(

1

-

t

)

n

-

i

⁢

t

i

,

i

=

0

,

…

,

n

.

are the

Bernstein basis polynomials

.

3

Special cases

A Bézier curve between two control points is simply the

line segment

between the two points:

𝑩

⁢

(

t

)

=

(

1

-

t

)

⁢

𝑷

0

+

t

⁢

𝑷

1

.

For three control points, we have the quadratic Bézier curve:

𝑩

⁢

(

t

)

=

(

1

-

t

)

2

⁢

𝑷

0

+

2

⁢

t

⁢

(

1

-

t

)

⁢

𝑷

1

+

t

2

⁢

𝑷

2

.

For four control points, we have the cubic Bézier curve:

𝑩

⁢

(

t

)

=

(

1

-

t

)

3

⁢

𝑷

0

+

3

⁢

t

⁢

(

1

-

t

)

2

⁢

𝑷

1

+

3

⁢

t

2

⁢

(

1

-

t

)

⁢

𝑷

2

+

t

3

⁢

𝑷

3

.

These are the most commonly employed versions of Bézier curves.

4

Properties

•

Except for the first and the last points, where the curve starts and ends, the Bézier curve should not

pass through

any of the control points. i.e. all the controls points lie on the curve if and only if the curve is a

straight line

.

•

The curve is

tangent

to the segment of line

𝑷

0

⁢

𝑷

1

¯

at the start and

𝑷

n

-

1

⁢

𝑷

n

¯

at the end.

•

It is easy to keep until the

first order

continuity when chaining curves. Moreover, every Bézier curve can be cut at any point into two new Bézier curves. In fact, the industry uses series of Bézier curves with only 4 control points (a bicubic version of the equations) to draw complex shapes.

•

The curves never do fancy stuff and always remain within the convex hull of the control points.

•

For one and two points the

definition

is still valid and the result became a point and a segment of line, respectively.

•

But it is impossible to draw circles or arcs of circle.

•

It is also impossible to draw two

parallel curves

(except in

trivial

cases: straight line Béziers and

coincident

Béziers curves)

•

If the first and the last point are the same, the curve is closed. And if

𝑷

1

and

𝑷

n

-

1

are aligned with

𝑷

0

then the curve is closed with a first order continuity.

5

Curve Cut

Let

𝑩

p

⁢

(

t

)

be a

Bezier curve

, defined by its control points

𝑷

0

to

𝑷

n

for

t

∈

[

0

,

1

]

.

Let

a

,

b

be such as

[

a

,

b

]

⊂

[

0

,

1

]

Let

u

=

t

-

a

b

-

a

be such as

∀

u

∈

[

0

,

1

]

,

t

∈

[

a

,

b

]

𝑩

q

⁢

(

u

)

is a part of the initial bezier curve with new control points

𝑸

0

to

𝑸

n

whose will be determined.

For

n

control points

𝑩

⁢

(

t

)

is a

n

-

1

-order polynom.

𝑩

p

⁢

(

a

)

=

𝑩

q

⁢

(

0

)

𝑩

p

⁢

(

b

)

=

𝑩

q

⁢

(

1

)

Let take

n

-

2

more points, evenly spaced in [a,b]

u

i

=

k

/

n

-

1

∀

k

∈

[

1

.

.

n

-

1

]

𝑩

p

⁢

(

u

⁢

(

b

-

a

)

+

a

)

=

𝑩

q

⁢

(

u

i

)

for each

u

i

For a 4 control points bezier curve
If we take

u

in

(

0

,

1

3

,

2

3

,

1

)

, hence

t

0

,

t

1

,

t

2

,

t

3

respectively equal to

a

,

b

-

2

⁢

a

3

,

2

⁢

b

-

a

3

,

b

𝑸

0

=

𝑩

p

⁢

(

t

0

)

𝑸

1

=

-

15

⁢

𝑩

p

⁢

(

t

0

)

+

54

⁢

𝑩

p

⁢

(

t

1

)

-

27

⁢

𝑩

p

⁢

(

t

2

)

+

6

⁢

𝑩

p

⁢

(

t

3

)

/

18

𝑸

2

=

6

⁢

𝑩

p

⁢

(

t

0

)

-

27

⁢

𝑩

p

⁢

(

t

1

)

+

54

⁢

𝑩

p

⁢

(

t

2

)

-

15

⁢

𝑩

p

⁢

(

t

3

)

/

18

𝑸

3

=

𝑩

p

⁢

(

b

)

http://en.wikipedia.org/wiki/Bezier_curve

Wiki

Title

Bézier curve

Canonical

name

BezierCurve

Date of creation

2015-01-23 16:33:51

Last modified on

2015-01-23 16:33:51

Owner

yota (10184)

Last modified by

yota (10184)

Numerical id

29

Author

yota (10184)

Entry type

Definition

Classification

msc 51N05

Synonym

Bezier curve

Related topic

NURBS

Related topic

BSpline

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