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Re:Calculus for Beginners and Artists (2003) by jacobolus on Sunday July 14, 2019 @07:06PM (#20436845)
It’s simpler if you use 0 and 1 instead of a and b. If we want a linear function such that

f(0) = A, f(1) = B,

then the unique solution is

f(t) = (1-t)A + tB

The more general version with a and b is just composing this with a transformation of [a, b] into [0, 1]

Relevant keywords: linear interpolation, affine combination, barycentric coordinates.

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