Dr. Scott Schaefer

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The Bernstein Basis and Bezier Curves

• Dr. Scott Schaefer

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Problems with Interpolation
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Problems with Interpolation
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Bezier Curves

• Polynomial curves that seek to approximate rather
  than to interpolate

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Bernstein Polynomials

• Degree 1 (1-t), t
• Degree 2 (1-t)2, 2(1-t)t, t2
• Degree 3 (1-t)3, 3(1-t)2t, 3(1-t)t2, t3

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Bernstein Polynomials

• Degree 1 (1-t), t
• Degree 2 (1-t)2, 2(1-t)t, t2
• Degree 3 (1-t)3, 3(1-t)2t, 3(1-t)t2, t3
• Degree 4 (1-t)4, 4(1-t)3t, 6(1-t)2t2,
  4(1-t)t3,t4

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Bernstein Polynomials

• Degree 1 (1-t), t
• Degree 2 (1-t)2, 2(1-t)t, t2
• Degree 3 (1-t)3, 3(1-t)2t, 3(1-t)t2, t3
• Degree 4 (1-t)4, 4(1-t)3t, 6(1-t)2t2,
  4(1-t)t3,t4
• Degree 5 (1-t)5, 5(1-t)4t, 10(1-t)3t2,
  10(1-t)2t3 ,5(1-t)t4,t5
• Degree n for

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Properties of Bernstein Polynomials

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Properties of Bernstein Polynomials

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Properties of Bernstein Polynomials

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Properties of Bernstein Polynomials

Binomial Theorem
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Properties of Bernstein Polynomials

Binomial Theorem
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Properties of Bernstein Polynomials

Binomial Theorem
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Properties of Bernstein Polynomials

Binomial Theorem
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Properties of Bernstein Polynomials

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Properties of Bernstein Polynomials

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Properties of Bernstein Polynomials

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Properties of Bernstein Polynomials

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More Properties of Bernstein Polynomials

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More Properties of Bernstein Polynomials

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More Properties of Bernstein Polynomials

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More Properties of Bernstein Polynomials

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Properties of Bernstein Polynomials
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Properties of Bernstein Polynomials

• Base case

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Properties of Bernstein Polynomials

• Base case

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Properties of Bernstein Polynomials

• Base case
• Inductive Step Assume

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Properties of Bernstein Polynomials

• Base case
• Inductive Step Assume

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Properties of Bernstein Polynomials

• Base case
• Inductive Step Assume

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Properties of Bernstein Polynomials

• Base case
• Inductive Step Assume

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Bezier Curves
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Bezier Curves
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Bezier Curves
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Bezier Curve Properties

• Interpolate end-points

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Bezier Curve Properties

• Interpolate end-points

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Bezier Curve Properties

• Interpolate end-points

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Bezier Curve Properties

• Interpolate end-points

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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge

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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge

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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge

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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge

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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge

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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge

Another Bezier curve of vectors!!!
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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge

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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge
• Curve lies within the convex hull of the control
  points

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Bezier Curve Properties

• Interpolate end-points
• Tangent at end-points in direction of first/last
  edge
• Curve lies within the convex hull of the control
  points

Bezier
Lagrange
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Matrix Form of Bezier Curves
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Matrix Form of Bezier Curves
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Matrix Form of Bezier Curves
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Matrix Form of Bezier Curves
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Matrix Form of Bezier Curves
Computation in monomial basis is unstable!!!
Most proofs/computations are easier in Bernstein
basis!!!
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Change of Basis
Bezier coefficients
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Change of Basis
Coefficients in monomial basis!!!
Bezier coefficients
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Change of Basis
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Change of Basis
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Change of Basis
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Change of Basis
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Change of Basis
monomial coefficients
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Change of Basis
Coefficients in Bezier basis!!!
monomial coefficients
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Degree Elevation

• Power basis is trivial add 0 tn1
• What about Bezier basis?

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Degree Elevation
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Degree Elevation
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Degree Elevation
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Degree Elevation
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Degree Elevation
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Degree Elevation
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Degree Elevation
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Degree Elevation
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Degree Elevation
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Degree Elevation
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Degree Elevation
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Pyramid Algorithms for Bezier Curves

• Polynomials arent pretty
• Is there an easier way to evaluate the equation
  of a Bezier curve?

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Pyramid Algorithms forBezier Curves
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Pyramid Algorithms forBezier Curves
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Pyramid Algorithms forBezier Curves
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Pyramid Algorithms forDerivatives of Bezier
Curves
Take derivative of any level of pyramid!!! (up to
constant multiple)
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Pyramid Algorithms forDerivatives of Bezier
Curves
Take derivative of any level of pyramid!!! (up to
constant multiple)
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Subdividing Bezier Curves

• Given a single Bezier curve, construct two
  smaller Bezier curves whose union is exactly the
  original curve

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Subdividing Bezier Curves

• Given a single Bezier curve, construct two
  smaller Bezier curves whose union is exactly the
  original curve

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Subdividing Bezier Curves

• Given a single Bezier curve, construct two
  smaller Bezier curves whose union is exactly the
  original curve

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Subdividing Bezier Curves

• Given a single Bezier curve, construct two
  smaller Bezier curves whose union is exactly the
  original curve

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Subdividing Bezier Curves

• Given a single Bezier curve, construct two
  smaller Bezier curves whose union is exactly the
  original curve

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Subdividing Bezier Curves
Control points for left curve!!!
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Subdividing Bezier Curves
Control points for right curve!!!
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Subdividing Bezier Curves
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Variation Diminishing

• Intuitively means that the curve wiggles no
  more than its control polygon

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Variation Diminishing

• Intuitively means that the curve wiggles no
  more than its control polygon

Lagrange Interpolation
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Variation Diminishing

• Intuitively means that the curve wiggles no
  more than its control polygon

Bezier Curve
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Variation Diminishing

• For any line, the number of intersections with
  the control polygon is greater than or equal to
  the number of intersections with the curve

Bezier Curve
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Variation Diminishing

• For any line, the number of intersections with
  the control polygon is greater than or equal to
  the number of intersections with the curve

Bezier Curve
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Variation Diminishing

• For any line, the number of intersections with
  the control polygon is greater than or equal to
  the number of intersections with the curve

Bezier Curve
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Variation Diminishing

• For any line, the number of intersections with
  the control polygon is greater than or equal to
  the number of intersections with the curve

Bezier Curve
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Variation Diminishing

• For any line, the number of intersections with
  the control polygon is greater than or equal to
  the number of intersections with the curve

Bezier Curve
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Variation Diminishing

• For any line, the number of intersections with
  the control polygon is greater than or equal to
  the number of intersections with the curve

Lagrange Interpolation
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Applications Intersection

• Given two Bezier curves, determine if and where
  they intersect

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Applications Intersection

• Check if convex hulls intersect
• If not, return no intersection
• If both convex hulls can be approximated with a
  straight line, intersect lines and return
  intersection
• Otherwise subdivide and recur on subdivided pieces

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Applications Intersection
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Applications Intersection
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Applications Intersection
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Applications Intersection
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Applications Intersection
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Applications Intersection
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Applications Intersection
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Applications Intersection
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Applications Intersection
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Applications Intersection
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Applications Intersection
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Application Font Rendering
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Application Font Rendering