Rational Curve

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Rational Curve
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Rational curve

• Parametric representations using polynomials are
  simply not powerful enough, because many curves
  (e.g., circles, ellipses and hyperbolas) can not
  be obtained this way.
• to overcome use rational curve
• What is rational curve?

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Rational curve

• Rational curve is defined by rational function.
• Rational function ? ratio of two polynomial
  function.
• Example
• Parametric cubic Polynomial
• - x(u) au3 bu2 cu d
• Rational parametric cubic polynomial
• x(u) axu3 bxu2 cxu dx
• ahu3 bhu2 chu dh

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Rational curve

• Use homogenous coordinate
• E.g
• Curve in 3D space is represented by 4 coord (x,
  y, z, h).
• Curve in 2D plane is represented by 3 coord.(x,
  y, h).
• Example (parametric quadratic polynomial in 2D)
• P UA
• x(u) axu2 bxu cx
• y(u) ayu2 byu cy
• P x, y U u2 ,u, 1 A ax ay
• bx by
• cx cy

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Rational curve

• Rational parametric quadratic polynomial in 2D
• Ph UAh h homogenous coordinates
• Ph hx, hy, h
• Matrix A (3 x 2) is now expand to 3 x 3
• Ah
• hx axu2 bxu cx
• hy ayu2 byu cy
• h ahu2 bhu ch

ax ay ah bx by bh cx cy ch
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Rational curve

• If h 1 Ph x, y, 1
• 1 h/h , x hx/h, y yh/h
• x(u) axu2 bxu cx
• ahu2 bhu ch
• y(u) ayu2 byu cy
• ahu2 bhu ch
• h ahu2 bhu ch 1
• ahu2 bhu ch

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Rational B-Spline

• B-Spline ?P(u) ? Ni,k(u)pi
• Rational B-Spline
• P(u) ? wiNi,k(u)pi
• ? wiNi,k(u)
• w ? weight factor ? shape parameters ? usually
  set by the designer to be nonnegative to ensure
  that the denominator is never zero.

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Rational B-Spline

• B-Spline ?P(u) ? Ni,k(u)pi
• Rational B-Spline
• P(u) ? wiNi,k(u)pi
• ? wiNi,k(u)
• The greater the value of a particular wi, the
  closer the curve is pulled toward the control
  point pi.
• If all wi are set to the value 1 or all wi have
  the same value ? we have the standard B-Spline
  curve

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Rational B-Spline

• Example
• To plot conic-section with rational B-spline,
  degree 2 and 3 control points.
• Knot vector 0, 0, 0, 1, 1, 1
• Set weighting function
• ? w0 w2 1
• ? w1 r/ (1-r) 0lt r lt 1

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Rational B-Spline

• Example (cont)
• Rational B-Spline representation is
• ?P(u) p0N0,3r/(1-r) p1N1,3 p2N2,3
• N0,3r/(1-r) N1,3 N2,3
• We obtain the various conic with the following
  valued for parameter r
• rgt1/2, w1 gt 1 ? hyperbola section
• r1/2, w1 1 ? parabola section
• rlt1/2, w1 lt 1 ? ellipse section
• r0, w1 0 ?straight line section

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Rational B-Spline
P1
w1 gt 1
w1 1
P0
w1 lt 1
w1 0
P2
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Rational B-Spline advantages

• Can provide an exact representation for quadric
  curves (conic) such as circle and ellipse.
• Invariant with respect to a perspective viewing
  transformation.?we can apply a perspective
  viewing transformation to the control points and
  we will obtain the correct view of the curve.