|
|
 |
 |
 | | | | |
 |
|||
 |
||||||
          |
 |
 |
||||
|
Connect the Curves Usually one curve is not enough to describe what you want. Because Bézier curves pass through their end control points, a continuous sequence of curves can be created by having them share endpoints. The curves will just physically connect, but not necessarily smoothly. This is called geometric continuity of order 0, often referred to as G0. Often it is desirable to have the curves appear to connect smoothly. This is referred to as geometric continuity of order 1, or G1. To have the curves connect smoothly, the tangent lines to the two curves at the shared point should have the same slope. This can be guaranteed by having the shared endpoint and the adjacent control points be collinear. Geometric continuity addresses how the curve appears, but there is more rigorous form of continuity, known as analytical continuity, that addresses how the curve behaves as well as appears. This type of continuity is defined in terms of derivatives and is represented by C instead of G in discussions. In general, C0 is equal to G0. C1 means that the first derivatives of the two curves are equal at the shared point. G1 is equal to C1 when the tangent lines not only have the same slope, but the same magnitude as well. The first derivative is often interpreted as the velocity, so it is important for a curve to have C1 continuity when the curve is used to represent a path that a camera might take or else there will be a noticeable jump in the velocity as the camera passes through the shared point. If the second derivatives match, implying C2 continuity, the acceleration through the shared point matches as well. For modelers, maintaining geometric continuity is usually sufficient, but often the more strict analytical continuity will be required when describing paths. Some Uses for Curves We examined Bézier curves as a step along our path to using them to generate curved surfaces, but they do have some uses of their own. Primary among these uses is to have them define paths for things like cameras and entities. Now your creatures will not have to move only along straight lines, but can move along curves as well. |
|
|
Features
