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Approach to achieving non-linear interpolation?

I need to implement a non-linear interpolation method between values, lightness, ease, general simplifying curves, and also user-defined curves.

I have a basic idea on how to do this, but I'm not sure if this will be the most effective solution. My idea is basically this:

Use a two-dimensional cubic, quadratic, or nth lowercase Bezier curve to control interpolation. Walk the curve linearly to get the nonlinear component Y, and use it to calculate a simple linear interpolation method:

value = v1 + (v2 - v1) * t;

Where t is the nonlinear Y-component of the control curve.

This allows you to use custom, user-defined methods for interpolation, but at the same time it costs, the cost of one non-linear interpolation is:

1 + 2 * (n-1)

full interpolations, where n is the order or number of control points of the control curve.

I am NOT MATH, this is the best I could come up with, so my question is, is there a better solution?

EDIT: I probably don't explain this correctly, I'm not a native speaker, so here is something, I hope everyone will understand: control curve interpolation

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From what I understand, yours tis actually a family of f functions i (u), where both u and f i (u) are between 0 and 1. If so, it does not get better than you already suggested.

It seems you are worried about evaluating these f values i (u) . , . , , f i (u j) u j (, 100 1000 0 1) , , .

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