The question is given as:
A cubic Bezier curve segment is described by control points P0(2,2), P1(4,8), P2(8,8) and P3(9,5). Another curve segment is described by Q0(a,b), Q1(c,2), Q2(15,2) and Q3(18,2). Find the values of a, b, and 10. (ac such that the curve segments join smoothly and C1 continuity exists between them.
Now we are given that C1 continuity exists in the two curve segments. For C1 continuity, the tangents of the curve are equal in magnitute and direction. But before that if two curves are continous, then the endpoint of the first curve is the same as the start point of the next curve.
So, Q0(a,b) = P3(9,5). Therefore,
a = 9
b = 5
Now we know that tangent vector of a cubic bezier curve is given by:
Now either we can use this monstrous equation or we have a simple trick to solve this question.
The tangent vector at the end point of the first curve segment (which is also Q0) is given by the difference between the last two control points of the first curve:
T1 = P3 - P2 = (9,5) - (8,8) = (1,-3)
T2 = Q1 - Q0 = (c,2) - (9,5) = (c-9,-3)
c-9 = 1
c = 10
Hence we have found the value of a, b and c. We can of course use the equation to find the value of c too. But that would be far too complex and we should try to avoid it.