Ode to Cubic Béziers

Should 'twixt four points you find a need
For smooth interpolation,
But one method guarantees
Absolute placation:

Cubic Hermites, you will see
Shan't fit each situation,
For singularities guarantee
A loss of information.

Though Catmull-Rom's controllable
And passes through each point with ease,
Higher order splines shall pull
Your CPU down to its knees.

de Boor's may seem approachable,
With short equations in pairs of threes,
But B-Splines are impossible
(Your frontal cortex surely agrees).

No, one alone can satisfy
All manner of requirements:
De Casteljau's, with cubic splines,
Unlike all such aspirants.

Simplicity it offers you,
And easy mutability,
Simplification rewards you, too
With fast computability.

The algorithm's but one step
Repeated thrice recursively,
One shortened to a nickname, prepped
For memorizability:

just
lerp()s
all
the
way

down.