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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

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PowerPoint Slideshow about 'Bézier Curves' - hanna-delgado


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slide2

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
slide3

The simplest parametric curve

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

L1

L(p)

L0

slide4

The simplest parametric curve

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

L1

L0

L(0.0)

slide5

The simplest parametric curve

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

L1

L(0.2)

L0

slide6

The simplest parametric curve

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

L1

L(0.4)

L0

slide7

The simplest parametric curve

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

L1

L(0.6)

L0

slide8

The simplest parametric curve

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

L1

L(0.8)

L0

slide9

The simplest parametric curve

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

L1

L(1.0)

L0

slide10

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
slide11

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

slide12

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)

slide13

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

slide14

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

slide15

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

slide16

Cubic Bezier curve

C3

C1

C(u)

C2

C0

slide17

Cubic Bezier curve

C3

C1

Cubic: n=3

(n+1) Control

Points

C(u)

C2

C0

slide18

Cubic Bézier curve

C3

C1

Control

Polygon

C(u)

C2

C0

slide19

Illustration of changing:

  • Degree, n
  • Position of Control Points,Ci
  • Choice of Basis functions,bi,n(u)
slide20

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)
slide21

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!

slide22

nCi = n!

(n-i)! i!

Cubic Bézier basis functions

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

slide23

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

slide24

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
slide25

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
slide26

Advantages of Bézier Curves:

  • Start and End on the first and last control points
  • Tangents defined by second and penultimate control points – can piece several curves together with tangent continuity
slide27

Advantages of Bézier Curves:

  • The shape of the curve roughly corresponds to the shape of the control polygon – not true of other basis functions
  • They are mathematically stable and well defined
  • They are invariant under control point transformations – to transform the curve just transform the control polygon
slide28

Advantages of Bézier Curves:

  • The maximum and minimum values in any direction are given by the bounding box. This is the smallest box that contains the control polygon.
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