Cubic Bézier curve
A cubic Bézier curve is drawn, as well as the steps of its construction: Four control points A, B, C, D can be moved, as can a point AB on the segment [AB]. Points BC on [BC], CD on [CD] are then calculated, with the same ratio of lengths (the aligned red on green segments are all in the same proportions). Then points ABC and BCD are calculated in the same way (they each define a quadratic Bézier curve with three control points). Finally, between these last two points, point M travels the Bézier curve.
These points can be constructed geometrically, or analytically by barycentric coordinates, or by using the associated Bernstein polynomials.