Bézier Curves: Construction, Parametric Equation, and Coordinates, Slides of Mathematics

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Bézier Curves

Maurizio Vianello Politecnico di Milano

12 maggio 2018

Bézier Curves

“Bézier Curves” are used in Computer Graphics. The goals of this presentation are: Ò (^) introduction to the method of construction of Bézier curves; Ò (^) deduction of the parametric equation.

(from Wikipedia)

Bézier popularized but did not actually create the Bézier curve — using such curves to design automobile bodies. The curves were first developed in 1959 by Paul de Casteljau using de Casteljau’s algorithm, a numerically stable method to evaluate Bézier curves. The curves remain widely used in computer graphics to model smooth curves.

Bézier developed the notation, consisting of nodes with attached control handles, with which the curves are represented in computer software. The control handles define the shape of the curve on either side of the common node, and can be manipulated by the user, via the software.

A Bézier curve (of the third order) is built from 4 points:

Ò (^) Two extrema, here denoted as A and B; Ò (^) Two control points, here denoted as H and K.

One example

A B

H

K

Barycentric combinations

Given points A , B and a parameter (real number) t it is possible to define a point P ( t ) P ( t ) = ( 1 − t ) A + tB What is the meaning of this? Choose an origin O and let

P ( t ) = O + ( 1 − t )( A − O ) + t ( B − O )

Notice: A − O, B − O are vectors. This makes sense! Important: point P ( t ) does not depend on the choice of the origin!

Position vector: P ( t ) − O = ( 1 − t )( A − O ) + t ( B − O )

A

B

P (t)

A − O

B − O

P (t) − O = ( 1 − t)(A − O) + t(B − O)

O

Oriented segment AB

A

B

P (t) = ( 1 − t)A + tB

Ò (^) For t = 0 we have P ( 0 ) = A Ò (^) For t = 1 we have P ( 1 ) = B Ò (^) For t between 0 and 1 we have points P ( t ) in the interior of the segment Ò (^) For t > 1 we have points beyond B Ò (^) For t < 0 we have points before A

Oriented segment AB: P ( t ) = ( 1 − t ) A + tB

Ò (^) P ( 1 / 5 ) is at one fifth, from A towards B; Ò (^) P ( 1 / 3 ) is at one third; Ò (^) P ( 1 / 2 ) is at one half; Ò (^) P ( 3 / 5 ) is at three fifths; Ò (^) P ( 5 / 7 ) is at five fifths.

Barycentric coordinates for n points Pi

For n points Pi and n parameters α i such that

∑^ n

i= 1

α i = 1

it is possible to define

P =

∑^ n

i= 1

α i Pi

as P = O +

∑^ n

i= 1

α i ( Pi − O )

The definition does not depend on the choice of the origin!

Why the word “barycentric”?

If α i > 0 is the fraction of mass located at Pi then

P =

∑^ n

i= 1

α i Pi

is by definition the position of the center of mass of points Pi.

Curve construction

A B

H

K

R

S

T

Q

N

P (t)

Curve construction

For each value 0 ≤ t ≤ 1 we construct points (red) R, S, T : Ò (^) R = ( 1 − t ) A + tH Ò (^) S = ( 1 − t ) H + tK Ò (^) T = ( 1 − t ) K + tB

and then, from those, we get points (blue) Q e N: Ò (^) Q = ( 1 − t ) R + tS Ò (^) N = ( 1 − t ) S + tT

In the end, we obtain point P ( t ) (black) which corresponds to the value t of the parameter: Ò (^) P ( t ) = ( 1 − t ) Q + tN