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Bezier Curve Point Calculator

Enter the three control points and t (0–1) to find the curve point.

How it is calculated

The tool evaluates a quadratic Bezier curve with three control points P0, P1 and P2. For a parameter t from 0 to 1 it forms the weights (1−t)², 2(1−t)t and t², which always add up to 1. Each coordinate is then blended separately: x = (1−t)² x₀ + 2(1−t)t x₁ + t² x₂, and the same for y. At t = 0 the point is P0, at t = 1 it is P2, and in between it is pulled toward P1 without touching it. The result is shown to three decimals. Coordinates can be any numbers, including negative ones, but t must be between 0 and 1. Cubic curves with four points are not handled.

Worked example

Let P0 = (0, 0), P1 = (50, 100), P2 = (100, 0) and t = 0.25. The weights are (1 − 0.25)² = 0.5625, 2 × 0.75 × 0.25 = 0.375 and 0.25² = 0.0625. For x: 0.5625 × 0 + 0.375 × 50 + 0.0625 × 100 = 18.75 + 6.25 = 25. For y: 0.5625 × 0 + 0.375 × 100 + 0.0625 × 0 = 37.5. The curve point is (25.000, 37.500). A quarter of the way along the parameter, the point has already risen to 37.5, although the control point sits at height 100.

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Common mistakes

  • Expecting the curve to pass through P1. Only P0 and P2 lie on the curve; P1 only pulls it, so at t = 0.5 the point is halfway between P1 and the midpoint of P0 and P2, here (50, 50), not (50, 100).
  • Treating t as a distance fraction along the arc. It is a parameter, and equal steps in t do not give equal lengths of curve.
  • Entering t as a percentage (25) instead of a fraction (0.25). A value above 1 returns no result, as does a negative t.
  • Using this for a cubic Bezier from a design tool, which has two control points. A cubic curve needs a different formula, and this tool will not match it.
  • Swapping the order of the points. The curve runs from P0 to P2, so exchanging P0 and P2 reverses the direction, and t = 0.25 becomes t = 0.75 of the original.

About this calculator

A quadratic Bezier curve is defined by three control points — a start point P0, a control point P1 that pulls the curve toward it without the curve actually touching it, and an end point P2 — and a parameter t that sweeps from 0 to 1 along the curve. The point at any t is found with B(t) = (1−t)² × P0 + 2(1−t)t × P1 + t² × P2, a weighted blend where the weights are the terms of a binomial expansion, which is why the curve always stays within the triangle (convex hull) formed by the three control points.

This exact math underlies vector graphics formats like SVG and PostScript/PDF paths, font outline definitions (TrueType uses quadratic Beziers directly), CAD curve modeling, and animation easing curves, because designers and engineers need a compact way to describe smooth curves with just a few numbers. The parameter t runs from 0 at the first control point to 1 at the last, so values in between trace the curve.

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