Rational Curve

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# Rational Curve - PowerPoint PPT Presentation

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

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 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

• 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 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

• 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
• 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 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 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 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 B-Spline

P1

w1 > 1

w1 = 1

P0

w1 < 1

w1 = 0

P2