Compute the position of the car on the road, Computer Graphics

An animation demonstrates a car driving along a road that is given by a Bezier curve along with the subsequent control points: 

Xk

0

5

40

50

Yk

0

40

5

15

The animation finals 10 seconds and the key frames are to be calculated at 1 second intervals. Compute the position of the car on the road at the beginning of the 6th second of the animation.

Solution: By using similar process in the previous exercise we can compute the blending functions as:

1.   B03 = 3!/(0! x (3-0)!) u0(1 - u)(3-0) = 1u0(1 - u)3 = (1 - u)3

2.   B13 = 3!/(1! x (3-1)!) u1(1 - u)(3-1) = 3u1(1 - u)2 = 3u(1 - u)2

3.   B23 = 3!/(2! x (3-2)!) u2(1 - u)(3-2) = 3u2(1 - u)1 = 3u2(1 - u)

4.   B33 = 3!/(3! x (3-3)!) u3(1 - u)(3-3) = 1u3(1 - u)0 = u3

The function x(u) is equivalent to x(u) =   ∑xkBk; here k=0,1,2,3

x(u)       = ∑ xkBk =  x0B03   +   x1B13 + x2B23 +  x2B33

=  (0)(1  -  u)3    +  5  [3u(1  -  u)2]  +  40  [3u2(1  -  u)]  +  50  u3

= 15u(1 - u)2 + 120u2(1 - u) + 50u3

As the same y(u) = y0B03 + y1B13+ y2B23+  y2B33

=  (0)(1  -  u)3    +  40  [3u(1  -  u)2]  +  5  [3u2(1  -  u)]  +  15  u3

= 120u(1 - u)2 + 15u2(1 - u) + 15u3

At the beginning of the sixth second of the animation, that is, when u=0.6, we can utilize these equations to work out such x(0.6) = 29.52 and y(0.6) = 16.92.

The way of the car looks like as:

887_Compute the position of the car on the road.png

Posted Date: 4/4/2013 6:02:30 AM | Location : United States







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