Interpolating values between interval, interpolation as per Bezier curve

Question

To implement a 2D animation I am looking for interpolating values between two key frames with the velocity of change defined by a Bezier curve. The problem is Bezier curve is re

Answer 1

You have a few options:

Let's say your curve function F(t) takes a parameter t that ranges from 0 to 1 where F(0) is the beginning of the curve and F(1) is the end of the curve.

You could animate motion along the curve by incrementing t at a constant change per unit of time. So t is defined by function T(time) = Constant*time

For example, if your frame is 1/24th of a second, and you want to move along the curve at a rate of 0.1 units of t per second, then each frame you increment t by 0.1 (t/s) * 1/24 (sec/frame).

A drawback here is that your actual speed or distance traveled per unit time will not be constant. It will depends on the positions of your control points.

If you want to scale speed along the curve uniformly you can modify the constant change in t per unit time. However, if you want speeds to vary dramatically you will find it difficult to control the shape of the curve. If you want the velocity at one endpoint to be much larger, you must move the control point further away, which in turn pulls the shape of the curve towards that point. If this is a problem, you may consider using a non constant function for t. There are a variety of approaches with different trade-offs, and we need to know more details about your problem to suggest a solution. For example, in the past I have allowed users to define the speed at each keyframe and used a lookup table to translate from time to parameter t such that there is a linear change in speed between keyframe speeds (it's complicated).

Another common hangup: If you are animating by connecting several Bezier curves, and you want the velocity to be continuous when moving between curves, then you will need to constrain your control points so they are symmetrical with the adjacent curve. Catmull-Rom splines are a common approach.