Bezier Curve parameter and the distance taken

Question

For a cubic bezier curve, if its parameter t is 0.5 then does this split the curve equal pieces lengthwise? Reasoning is appreciated.

Answer 1

In general, no, unless in some special cases, for example all control points of the cubic Bezier curve are collinear and evenly-spaced. The reason is simple: a Bezier curve is in general not arc-length parameterized.

Answer 2

Simple example: take three 2D control points (1,0), (2,0) and (4,0)

The point t=0.5 is (2,0). But this point does not have equal distances to the first and last point.

Well, and this should not depend on the cubic or linear case.