Rational curve
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Rational Curve. 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?. Rational curve.

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Rational Curve

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Rational curve

Rational Curve


Rational curve1

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?


Rational curve

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


Rational curve

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


Rational curve

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


Rational curve

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


Rational b spline

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.


Rational curve

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


Rational curve

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) 0<= r <= 1


Rational curve

Rational B-Spline

  • Example (cont)

    • Rational B-Spline representation is

    • P(u) = p0N0,3+[r/(1-r)] p1N1,3+ p2N2,3

      • N0,3+[r/(1-r)] N1,3+ N2,3

  • We obtain the various conic with the following valued for parameter r

  • r>1/2, w1 > 1  hyperbola section

  • r=1/2, w1 = 1  parabola section

  • r<1/2, w1 < 1  ellipse section

  • r=0, w1 = 0 straight line section


  • Rational curve

    Rational B-Spline

    P1

    w1 > 1

    w1 = 1

    P0

    w1 < 1

    w1 = 0

    P2


    Rational b spline advantages

    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.


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