Polynomials

Core Idea

The gold curve above is one expression — P(t) = a·t³ + b·t² + c·t + d — and the four knobs are its entire personality. Set them one way and it’s a straight line; another way and it’s smoothstep; another and it wiggles. A polynomial is just that: a variable raised to whole-number powers, each scaled by a coefficient, summed. It is the most computable shape in mathematics — evaluating one takes a handful of multiplies and adds — which is why it became the raw material of graphics: every ease curve, every font glyph outline, every camera path segment is a polynomial being paid to bend.

Four Numbers, One Shape

P(t) = aₙtⁿ + … + a₂t² + a₁t + a₀

Only whole-number powers; only add, subtract, multiply. (No 1/t, no √t — those break the rules and lose the guarantees below.) The degree is the highest power, and in graphics t almost always runs 0 → 1 across a segment. The cubic’s knobs even have readable jobs: d sets where the curve starts, c its launch slope, while b and a spend the curve’s two bends. That’s the deal the hero makes visible — the coefficients are the interface.

The Derivative Comes Free

Differentiate a polynomial and you get another polynomial, one degree down — the dashed sea curve in the hero, updating live from the same four numbers:

P(t)  = a·t³ + b·t² + c·t + d
P′(t) = 3a·t² + 2b·t + c

On a curve, P′(t) is the tangent — the velocity of the moving point, the arrow riding the hero. The second derivative gives curvature. This is why polynomials dominate motion: the smoothness you feel is a statement about derivatives, and polynomial derivatives are free to compute and easy to control. smoothstep(t) = 3t² − 2t³ is the canonical example — its derivative 6t − 6t² is zero at both t = 0 and t = 1, so motion leaves and arrives with zero velocity. That endpoint argument is the entire trick; how the different eases feel against each other lives in interpolation.

What a Polynomial Can’t Do

The guarantees cut both ways:

• Always smooth — infinitely differentiable, no corners, no breaks. (Corners must be built from pieces.)
• Infinite extent — as t → ±∞ the curve runs to ±∞. A polynomial can’t close on itself.
• One value per t — so no exact circles, ever. A circle needs two y values for one x; the fix is rational curves (ratios of polynomials), which is NURBS territory — covered in curves.

Degree Is a Budget

Each extra degree buys roughly one more bend. That sounds like an invitation to buy a big polynomial for any shape — and this is the experiment that says no:

A line misses almost everything; a parabola bends once; a cubic bends twice; degree 5 finally passes through all six points — and overshoots between them, because a single polynomial is global: every coefficient owns the whole curve, so forcing it through one point disturbs it everywhere. Graphics’ answer is the last button: stop at degree 3 and chain small cubics, each controlling only its own stretch, glued with continuity conditions (C0 touch, C1 matched velocity, C2 matched curvature). That decision — cubic, piecewise, joined carefully — is the founding move of splines, and curves takes it from there: Bézier is exactly a cubic polynomial rewritten so its coefficients become draggable control points.

Where They Hide

• Easing — CSS cubic-bezier(…), ease-in t³, smoothstep: cubics from 0 to 1.
• Shader math — polynomial approximations replace expensive functions in per-pixel code; smoothstep is the workhorse of every soft edge in shading.
• Fonts — TrueType glyphs are quadratic Bézier segments; PostScript/CFF outlines are cubic. Every letter on this page is piecewise polynomials.
• Paths — camera and object motion through keyframes: chained cubics, again.

Connections

Curves is this article’s direct sequel — Bernstein blending rewrites the cubic so control points replace raw coefficients, and de Casteljau evaluates it with nothing but lerps. The lerp itself is the degree-1 polynomial, which makes interpolation the family’s simplest member. Trigonometry is the sibling material: sin is an infinite polynomial (its Taylor series), polynomials approximate it well over short ranges, but for motion that must loop forever the periodic functions win — a polynomial always escapes to infinity. And in shading, cheap polynomial stand-ins for expensive math are a standing GPU idiom: the shape is close enough, and the multiplies are nearly free.