Curves II

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Curves II

• Blending Functions
• Bernstein Polynomials
• Bézier Surface Patches

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A Bézier Curve
p1
p2
p3
p0
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Bézier Curve Geometry Matrix

• The matrix we computed for the Bézier Curve is
  called a Geometry Matrix

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How a Geometry Matrix is used
or
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Blending Polynomials
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So
b(u)
These are called blending polynomials
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The Bézier Blending Polynomials
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Remember Linear Interpolation?
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The Bézier Blending Polynomials
p0
p3
p1
p2
End of the curve
Beginning of the curve
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Bernstein Polynomials
Bernstein Polynomials
d is degree, 3 in our case for a cubic. k is
the polynomial number 0-3
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Blending functions and our curve
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Extended to a surface

• Suppose the controls points were points on a
  curve?
• Specify each control point for a curve with a
  Bézier curve
• We need 4 sets of 4 points 16 points.

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Example
p3 (u)
p2 (u)
p1 (u)
p0(u)
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Bézier Surface Patches
Curve
Extended to 2D
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Example
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Questions

• Continuity?
• Which points are on the surface?