Final answer:
The equations for the composite cubic Bezier curve use the control points to define two segments with shared continuity. The curve has C⁰ continuity because control points at the segment transition are identical, and C¹ continuity is confirmed through proportional directional vectors at the transition.
Step-by-step explanation:
To find the equations for a composite cubic Bezier curve, we need to use the given control points to define each cubic Bezier segment. The given control points are (3, 10), (4, 7), (6, 6), (7.5, 7.5), (8.2, 8.2), (11, 7), and (14, 6). Assuming that we're creating a composite curve from these, we can create two cubic Bezier segments with the first four and the last four control points respectively.
- The first segment is defined by the points P0 = (3, 10), P1 = (4, 7), P2 = (6, 6), and P3 = (7.5, 7.5).
- The second segment is defined by the points Q0 = (7.5, 7.5), Q1 = (8.2, 8.2), Q2 = (11, 7), and Q3 = (14, 6).
To ensure C⁰ continuity, we verify that the end of the first segment P3 is the same as the start of the second segment Q0 -- which it is. For C¹ continuity, we check that the vectors (P3 - P2) and (Q1 - Q0) are collinear. This means the direction from the second to last point of the first curve and the direction from the first to the second point of the second curve must be the same. If they are proportional, the curves are C¹ continuous.
Mathematically, the two conditions for C¹ continuity are (7.5 - 6, 7.5 - 6) must be proportional to (8.2 - 7.5, 8.2 - 7.5). Simplifying the vectors, we get (1.5, 1.5) and (0.7, 0.7), which are indeed proportional; so, we confirm that the composite curve has C¹ continuity.