DESIGN OF CURVES 129rP
P
P1 2222
111
0000
1
2( ) = [uuu 1]abc
abc
abc⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥rP
P
P2 2222
111
0001
2
3( ) = [uuu 1]abc
abc
abc⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥[^2 1]222
111
000uuabc
abc
abc⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥= (a 2 u^2 + a 1 u + a 0 ,b 2 u^2 + b 1 u + b 0 ,c 2 u^2 + c 1 u + c 0 ) = {a(u),b(u),c(u)}The nine elements of the 3 × 3 matrix are unknowns and are to be calculated from the following
conditions:
(a) The two tangents are to meet at the common point Q 2 with C^1 continuity, that is
r 1 (u = 1) = r 2 (u = 0) and ̇r 12 ( = 1) = ( = 0)uu ̇r(b) The entire curve should be independent of the coordinate system used which means that the weights
should sum to unity, that is, a(u) + b(u) + c(u) = 1.
Show that the matrix is given by
1
21–21
–2 2 0
110⎡⎣⎢
⎢
⎢⎤⎦⎥
⎥
⎥Determine the start and end points Q 1 ,Q 2 ,Q 3. Draw the curve with the control points given as (1, 2),
(3, 6), (7, 10), (12, 3).