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) = au 3 + bu 2 + cu + d Rational parametric cubic polynomial -x(u) = a x u 3 + b x u 2 + c x u + d x a h u 3 + b h u 2 + c h u + d h

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) = a x u 2 + b x u + c x y(u) = a y u 2 + b y u + c y P = [x, y] U = [u 2,u, 1] A = a x a y b x b y c x c y

Rational curve Rational parametric quadratic polynomial in 2D P h = UA h h – homogenous coordinates P h = [hx, hy, h] Matrix A (3 x 2) is now expand to 3 x 3 A h = hx = a x u 2 + b x u + c x hy = a y u 2 + b y u + c y h = a h u 2 + b h u + c h a x a y a h b x b y b h c x c y c h

Rational curve If h = 1 P h = [x, y, 1] 1 = h/h, x = hx/h, y = yh/h x(u) = a x u 2 + b x u + c x a h u 2 + b h u + c h y(u) = a y u 2 + b y u + c y a h u 2 + b h u + c h h = a h u 2 + b h u + c h = 1 a h u 2 + b h u + c h

Rational B-Spline B-Spline  P(u) =  N i,k (u)p i Rational B-Spline –P(u) =  w i N i,k (u)p i –  w i N i,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) =  N i,k (u)p i Rational B-Spline –P(u) =  w i N i,k (u)p i –  w i N i,k (u) –The greater the value of a particular w i, the closer the curve is pulled toward the control point p i. –If all w i are set to the value 1 or all w i 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  w 0 = w 2 = 1  w 1 = r/ (1-r) 0<= r <= 1

Rational B-Spline Example (cont) Rational B-Spline representation is  P(u) = p 0 N 0,3 +[r/(1-r)] p 1 N 1,3 + p 2 N 2,3 N 0,3 +[r/(1-r)] N 1,3 + N 2,3 We obtain the various conic with the following valued for parameter r r>1/2, w 1 > 1  hyperbola section r=1/2, w 1 = 1  parabola section r<1/2, w 1 < 1  ellipse section r=0, w 1 = 0  straight line section

P0P0 Rational B-Spline P1P1 P2P2 w 1 = 0 w 1 < 1 w 1 = 1 w 1 > 1

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. Rational B-Spline : advantages