An interesting note is you could use these to make pretty lines for noodle graphs. Placing the tangents or phantom points in the right places will result in a line that curves to the entry and exit of the nodes.

Mathy notation is in this card. Points are vectors in practice but presented as symbols in the formulas.

Beziers

A bezier curve has Points P1 and P2. There are also tangent points T1 and T2.

bezier(t) = P1 * (1-t)^3 + T1 * 3 * (1-t)^2 * t + T2 * 3 * (1-t) * t^2 + P2 * t^3

Spline (four points) to tangents

Catmull-Rom curve is a spline defined by four points. This can be solved to convert it to a bezier.

T1 = (d_1^2 * P2 - d_2^2 * P0 + (2*d_1^2 + 3 * d1 * d2 + d_2^2) * P1) / (3 * d1 * (d1 + d2))

T2 = (d_3^2 * P1 - d_2^2 * P3 + (2*d_3^2 + 3 * d3 * d2 + d_2^2) * P2) / (3 * d3 * (d3 + d2))

di = |Pi - P_i-1|^a

Where 'a' is the constant alpha determining the tightness of a curve.

a = 0: uniform

a = 0.5: centripetal

a = 1: chordal

The spline formula accepts four points and produces tangents only for the middle points. You have to use "phantom" points (or previous points on the line.)