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Rational Curve

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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  1. Rational Curve

  2. 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?

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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.

  8. 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

  9. 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

  10. 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

  11. Rational B-Spline P1 w1 > 1 w1 = 1 P0 w1 < 1 w1 = 0 P2

  12. 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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