Bézier Curves

1 / 28

# Bézier Curves - PowerPoint PPT Presentation

Bézier Curves. Representing free-form shape. CAD/CAM software requires shape to be represented mathematically Standard mathematical representations such as y=f(x) very problematic Overcome using parametric representations. The simplest parametric curve. The Line: L (p) = (1-p) L 0 + p L 1

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Representing free-form shape

• CAD/CAM software requires shape to be represented mathematically
• Standard mathematical representations such as y=f(x) very problematic
• Overcome using parametric representations

The simplest parametric curve

• The Line:
• L(p) = (1-p)L0 + pL1
• 0  p  1

L1

L(p)

L0

The simplest parametric curve

• The Line:
• L(p) = (1-p)L0 + pL1
• 0  p  1

L1

L0

L(0.0)

The simplest parametric curve

• The Line:
• L(p) = (1-p)L0 + pL1
• 0  p  1

L1

L(0.2)

L0

The simplest parametric curve

• The Line:
• L(p) = (1-p)L0 + pL1
• 0  p  1

L1

L(0.4)

L0

The simplest parametric curve

• The Line:
• L(p) = (1-p)L0 + pL1
• 0  p  1

L1

L(0.6)

L0

The simplest parametric curve

• The Line:
• L(p) = (1-p)L0 + pL1
• 0  p  1

L1

L(0.8)

L0

The simplest parametric curve

• The Line:
• L(p) = (1-p)L0 + pL1
• 0  p  1

L1

L(1.0)

L0

The simplest parametric curve

• The Line:
• L(p) = (1-p)L0 + pL1
• 0  p  1
• Note that every point we evaluate on the curve is constructed as a weighted average of the two end points

1

• L(p) = bi,1(p) Li

i=0

The simplest parametric curve

• We can reformulate the equation for a line:
• b0,1(p) = (1-p)
• b1,1(p) = p

0  p  1

1

• L(p) = bi,1(p) Li

i=0

The simplest parametric curve

• We can reformulate the equation for a line:
• b0,1(p) = (1-p)
• b1,1(p) = p

0  p  1

Weighing for

each point:

Basis Function (in p)

1

• L(p) = bi,1(p) Li

i=0

The simplest parametric curve

• We can reformulate the equation for a line:
• b0,1(p) = (1-p)
• b1,1(p) = p

0  p  1

Weighing for

each point:

Basis Function (in p)

Index

1

• L(p) = bi,1(p) Li

i=0

The simplest parametric curve

• We can reformulate the equation for a line:
• b0,1(p) = (1-p)
• b1,1(p) = p

0  p  1

Weighing for

each point:

Basis Function (in p)

Index

Degree

n

• C(u) = bi,n(u) Ci

i=0

General parametric curve

• Generalise the formulation:
• Ci = [xi, yi, zi]
• Shape depends on:
• Degree, n
• Position of Control Points,Ci
• Choice of Basis functions,bi,n(u)

0  u  1

Cubic Bezier curve

C3

C1

C(u)

C2

C0

Cubic Bezier curve

C3

C1

Cubic: n=3

(n+1) Control

Points

C(u)

C2

C0

Cubic Bézier curve

C3

C1

Control

Polygon

C(u)

C2

C0

Illustration of changing:

• Degree, n
• Position of Control Points,Ci
• Choice of Basis functions,bi,n(u)

n

• C(u) = bi,n(u) Ci

i=0

Bézier curve

0  u  1

• Each point on the curve is made up of a weighted average of the control points.
• Weightings given by the Basis functions,bi,n(u)

n

• C(u) = bi,n(u) Ci

i=0

Bézier basis functions

0  u  1

• bi,n(u) = nCi (1-u)n-i ui
• nCi = n!

(n-i)! i!

nCi = n!

(n-i)! i!

Cubic Bézier basis functions

bi,n(u) = nCi (1-u)n-i ui

bi,3(u)

1.0

b0,3(u)

b3,3(u)

0.666

b1,3(u)

b2,3(u)

0.333

0.0

u

0.0

0.333

0.666

1.0

Cubic Bézier basis functions

bi,3(u)

1.0

b0,3(u)

b3,3(u)

0.666

b1,3(u)

b2,3(u)

0.333

0.0

u

0.0

0.333

0.666

1.0

Cubic Bézier basis functions

• Sum of all basis functions is always equal to 1.0 for any u

bi,3(u)

1.0

b0,3(u)

b3,3(u)

0.666

b1,3(u)

b2,3(u)

0.333

0.0

u

0.0

0.333

0.666

1.0

Cubic Bézier basis functions

• C(0)C0
• C(1)Cn