Why a curve is not "a line with points"

Every vector tool you have used — Illustrator, Figma, Inkscape — is built on one primitive: the Bézier curve. And the single most useful thing to understand about it is that a curve is not stored as a set of points along its length. It is stored as a few control points, plus a rule that turns them into any point you ask for.

That "ask and it computes" operation is called evaluation. It matters because it means:

· You can zoom to 10000% and it stays smooth — there are no sample points to run out of.
· You can ask for a point at *t* = 0.37 without having computed 0..0.36.
· A CAD kernel can solve curve intersections from the curve equations, using robust algorithms and a chosen numerical tolerance, instead of treating a sampled polyline as the curve itself.
· Animation can interpolate along the curve with guaranteed continuity.

Compare that to a "polyline" (a list of points joined by straight segments). A polyline is easy but has two fatal properties: smoothing it changes its shape, and it never gets better than the points you gave it.

// A polyline: 5 points, permanently 5 straight segments
P = [(0,0), (1,2), (2,1), (3,3), (4,0)]

// A cubic Bézier: 4 points, but you can ask for ANY t
B(t) = (1-t)^3 P0 + 3(1-t)^2 t P1 + 3(1-t) t^2 P2 + t^3 P3

That formula is the whole of the simplest useful curve. `t` runs 0→1. `P0` and `P3` are the endpoints — the curve passes through them. `P1` and `P2` are control points — the curve is pulled toward them but does not generally touch them. That asymmetry is the first thing that surprises everyone.

So the mental model is: a Bézier is a smooth way to blend four points. Move a control point and the whole blend shifts. That is exactly why dragging handles in a design tool feels the way it does.