Cubic polynomials and cubic Beziers

So it turns out that for cubic Bezier curves, t values of 0, 1/3, 2/3 and 1 have special meanings. A general cubic polynomial is of the form

y = a0 + a1 * x + a2 * x^2 + a3 * x^3

where ai’s are real constants.

If the variable x is limited to the interval x0 <= x <= x0 + χ (that's the Greek letter Chi), where χ > 0, then it’s equivalent to a special case of cubic Bezier curves. Namely, when the t values are 0, 1/3, 2/3 and 1.

In fact, there’s a mathematical proof of it. Thanks to Professor Samuel Dagan of Tel-Aviv University for writing in and letting me know of his work. Here’s more of his work.

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