Splines for interpolation

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Splines for interpolation

• Bezier polynomialsUsing Splines for Surface
  Modeling

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Hermite Curves

• One way to design a curve segment
• End points (P1 and P4)
• End tangent vectors (derivatives)
• R1 and R4
• 4 parameters and is sufficient to define an n3
  polynomial.
• This is the Hermite curve.

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Specifying a Hermite Curve
4 segments P1 and P4 for each segment R1 and R4
for each segment
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Blending function form for Hermite polynomials
B0(t)
B1(t)
B2(t)
B3(t)
Hermite Basis polynomials
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Bézier Curves

• Hermite curves are difficult to specify
• Derivatives are not natural
• Tangent vectors overlap
• Tangent vectors are a bit long for display
• Bézier Curves
• Well use end points of 1/3 and 1/3 along the
  tangent vectors as control points.

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A Bézier Curve
p1
p2
p3
p0
4 segments p0, p1, p2, and p3 for each segment
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A Bézier Curve
p1
p2
p3
p0
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The Control Points

• The control points
• p0 p(0), First end point
• p3 p(1), Second end point
• p1c0 1/3 c1
• p2 c0c1c2c3 - 1/3 (c1 2c2 3c3) c0
  2/3 c1 1/3c2

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Blending function form for Bezier polynomials
B0(t)
B1(t)
B2(t)
B3(t)
Bezier Blending polynomials Bernstein polnomials
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Affine Transformations

• Affine transformations translation, rotation,
  scale
• Applying the affine transformation to the control
  points and recalculating the curve is equivalent
  to applying the transformation to the 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
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Patches, control point polygon
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Effect of control points
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Effect of control points
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Effect of control points
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Effect of control points
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Utah teapot -8 patches on the body
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Utah teapot
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Utah teapot 4 patches on the handle
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Utah teapot 4 patches on the spout