Bzier Curves CPSC 453 Fall 2018 Sonny Chan Todays Outline - - PowerPoint PPT Presentation

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Bzier Curves CPSC 453 Fall 2018 Sonny Chan Todays Outline - - PowerPoint PPT Presentation

Jul 03, 2023 314 likes •690 views

Bzier Curves CPSC 453 Fall 2018 Sonny Chan Todays Outline Quadratic Bzier curves de Casteljau formulation Bernstein polynomial form Bzier splines Cubic Bzier curves Drawing Bzier curves freeform shape?

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Bézier Curves

CPSC 453 – Fall 2018 Sonny Chan

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Today’s Outline

  • Quadratic Bézier curves
  • de Casteljau formulation
  • Bernstein polynomial form
  • Bézier splines
  • Cubic Bézier curves
  • Drawing Bézier curves
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How might we represent a

freeform shape?𝄟

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The Goal:

Create a system that provides an accurate, complete, and indisputable definition of freeform shapes.

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Parametric Segment by Linear Interpolation

p(u) = lerp(p0, p1, u) = (1 − u)p0 + up1 p0 p1

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Can we use the same machinery for

defining curves?

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de Casteljau’s Algorithm (quadratic)

p1

0 = lerp(p0, p1, u)

p1

1 = lerp(p1, p2, u)

p(u) = lerp(p1

0, p1 1, u)

p0 p1 p2

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Bernstein Polynomials

0.25 0.5 0.75 1 0.25 0.5 0.75 1

b0,2(u) = (1 − u)2 b1,2(u) = 2u(1 − u) b2,2(u) = u2

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Bernstein Form of a Quadratic Bézier

p0 p1 p2 p(u) = (1 − u)2p0 + 2u(1 − u)p1 + u2p2 = b0,2(u)p0 + b1,2(u)p1 + b2,2(u)p2 =

2

X

i=0

bi,2(u)pi

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Property #1:

endpoint interpolation

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Endpoint Interpolation

p0 p1 p2 p(0) = p0 p(1) = p2

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Property #2:

endpoint tangents

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Endpoint Tangents

p0 p1 p2 p0(0) = 2(p1 − p0) p0(1) = 2(p2 − p1)

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Property #3:

convex hull

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Convex Hull

p0 p1 p2

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Bézier Splines

  • f the 2nd degree
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Splines are just curve segments joined together:

p0 p1 p2 = q0 q1 q2

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Is it continuous?

C0 continuity: p(1) = q(0)

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Is it smooth?

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Is it smooth?

C1 continuity: p0(1) = q0(0)

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Is it smooth?

G1 continuity: p0(1) = s q0(0), s ∈ R+

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Using a Bézier spline to represent a

freeform shape

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Bézier Curves

  • f the 3rd degree
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de Casteljau Construction

u = 1

2

u = 1

4

u = 3

4

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0.25 0.5 0.75 1 0.25 0.5 0.75 1

Third Degree Bernstein Polynomials

b0,3(u) = (1 − u)3 b1,3(u) = 3u(1 − u)2 b2,3(u) = 3u2(1 − u) b3,3(u) = u3

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Bernstein form of a Cubic Bézier

p(u) = (1 − u)3p0 + 3u(1 − u)2p1 + 3u2(1 − u)p2 + u3p3 = b0,3(u)p0 + b1,3(u)p1 + b2,3(u)p2 + b3,3(u)p3 =

3

X

i=0

bi,3(u)pi p0 p1 p2 p3

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Do our properties still apply?

Endpoints Tangents Convex Hull

✓ ✓ ✓

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Property #4:

curve subdivision

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Curve Subdivision

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Drawing Curves

Or how to approximate them with straight line segments.

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Two methods for evaluating your curve

p0 p1 p2 p3 p1 p1

1

p1

2

p2 p2

1

p(u) u 1 − u p(u) =

n

X

i=0

bi,n(u)pi bi,n(u) = ✓n i ◆ ui(1 − u)n−i de Casteljau Bernstein Which one do you think is more efficient?

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How many line segments do you need?

The key question:

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Things to Remember

  • Bézier curves and splines can provide an accurate,

complete, and indisputable definition of freeform shapes

  • defined by de Casteljau construction or Bernstein polynomials
  • quadratic (3 points), cubic (4 points), or higher order
  • Splines are just curve segments joined together
  • can have various degrees of parametric/geometric continuity
  • Draw splines by approximating them with line segments,

just like parametric curves! (same considerations apply)