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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?

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

- x(u) = axu3 + bxu2 + cxu + dx

- Parametric cubic Polynomial

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

- P = UA

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

- 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

Rational B-Spline

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.