rational curve n.
Download
Skip this Video
Download Presentation
Rational Curve

Loading in 2 Seconds...

play fullscreen
1 / 12

Rational Curve - PowerPoint PPT Presentation


  • 217 Views
  • Uploaded on

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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Rational Curve' - murray


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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?
slide3

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
slide4

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
slide5

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

slide6

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.
slide8

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
slide9

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
slide10

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
slide11

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.