Find a point on a cubic Bézier curve for numeric checks in drawing and path design. Enter the coordinates of P0, P1, P2 and P3, followed by a parameter t from 0 to 1. P0 is the start point, P3 is the end point, and P1 and P2 control the curve’s shape.
Enter all four points in their curve order: start P0, first control point P1, second control point P2, and end P3. Use one consistent coordinate system and scale. Negative and fractional coordinates are allowed. Enter t between 0 and 1, then read the two outputs together as the point (x, y).
The inputs use the dimensionless unit 1; no physical-unit conversion is performed. In a screen coordinate system where y increases downward, the output follows that same convention.
For each coordinate, interpolate between values a and b using L(a, b, t) = a + t(b − a). Apply the same t at every stage:
- Form A0 from P0 and P1, A1 from P1 and P2, and A2 from P2 and P3.
- Form B0 from A0 and A1, and B1 from A1 and A2.
- Interpolate between B0 and B1 to obtain the curve point.
This is the de Casteljau construction. Each coordinate uses only subtraction, multiplication and addition. At t = 0.5, each interpolation is a midpoint. Northeastern University’s Bézier Cubics reference illustrates that midpoint construction.
With P0 = (0, 0), P1 = (0, 2), P2 = (2, 2), P3 = (2, 0), and t = 0.5, the first stage gives A0 = (0, 1), A1 = (1, 2), and A2 = (2, 1). The second stage gives B0 = (0.5, 1.5) and B1 = (1.5, 1.5). The final interpolation gives C(0.5) = (1, 1.5).
| Parameter t | Curve x | Curve y |
|---|---|---|
| 0 | 0 | 0 |
| 0.25 | 0.3125 | 1.125 |
| 0.5 | 1 | 1.5 |
| 0.75 | 1.6875 | 1.125 |
| 1 | 2 | 0 |
Does t = 0.5 mean halfway along the curve's length?
Not in general. The parameter controls interpolation, and equal changes in t need not produce equal travel distances. This calculator evaluates a position at a supplied parameter; it does not calculate arc length.
Does the curve pass through all four control points?
It starts at P0 and ends at P3. It generally does not pass through P1 or P2, which influence its shape.
Can I enter repeated or collinear control points?
Yes. The interpolation remains defined. If all four points are identical, every parameter produces that point. Collinear control points keep the result on their common line.
Can this find t from a desired x coordinate?
No. You supply t and receive x and y. Finding t from a coordinate requires solving the curve equation and may have multiple solutions.
How should I interpret rounding or an unavailable result?
Outputs are displayed to eight decimal places and use finite-precision arithmetic. Very large coordinates or intermediate differences can exceed the numeric range. An unavailable result is not zero; try expressing the coordinates at a smaller consistent scale.