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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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## PowerPoint Slideshow about ' Rational Curve' - murray

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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?

• 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

• 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 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

• 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

• 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.

• 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

• 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

• 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

• P1

w1 > 1

w1 = 1

P0

w1 < 1

w1 = 0

P2

• 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.