Splines: joining curves without a visible seam

A single cubic can only make a simple S or arc. Any real outline — a letter "S", a car body, a font glyph — is many cubic segments joined end to end. If you just place them next to each other you get a visible kink. The whole game is controlling the smoothness at those joints.

Smoothness has grades, and it is worth knowing the names because they are used constantly:

· C0 — the endpoints meet. The curve is connected but may kink.
· G1 — the endpoints meet and the tangent directions match. There is no corner, though the curve may speed up or slow down at the join.
· C1 — the endpoints and first-derivative vectors match. This implies G1 and also keeps the parameterised speed continuous.
· C2 — the first and second derivatives match, so the curve’s bending changes continuously too.

A0 ---------- A1  [J]  B1 ---------- B2
                  ^         ^
                  |         |
            control before  control after
   G1  <=>  A1, J, B1 are collinear, with matching tangent direction
   C1  <=>  3(J - A1) == 3(B1 - J)  (same parameter interval)

   C2 also requires the second-derivative vectors to match
   (equal handle lengths alone are not enough)

When you drag a handle and the opposite handle stays in line, a design tool is enforcing tangent alignment (G1). If it also keeps the handle vectors equal under the same parameter scaling, it enforces C1. Once you know the distinction, you can deliberately break tangent alignment to make a sharp corner — what "corner" mode does.