Bézier Curve Point Calculator

How to use the calculator

1. Enter control points in the text area, one point per line, as x,y. You may include optional spaces, for example 0,0 or 1.5, -2.75.
2. Provide at least two points. With two points you get a straight line segment (a linear Bézier); with three points, a quadratic curve; with four, a cubic curve, and so on.
3. Enter a parameter value t between 0 and 1. At t = 0 you get the first control point; at t = 1 you get the last control point; intermediate values move smoothly along the curve.
4. Click the button to compute the point. The calculator runs De Casteljau’s algorithm on your control points and returns a single 2D coordinate.
5. Copy the resulting point into your own graphics code, animation system, or design tool as needed.

Example input you can paste directly:

• Quadratic curve: 0,0 on the first line, 1,2 on the second, 2,0 on the third.
• Cubic curve: 0,0, 1,3, 3,3, 4,0 (one pair per line).

What is a Bézier curve?

A Bézier curve is a parametric curve defined by a sequence of control points . The curve starts at and ends at . The intermediate control points influence the shape of the curve but do not, in general, lie on the curve itself.

The parameter runs from 0 to 1. As increases, the corresponding point on the curve moves smoothly from the first to the last control point. Bézier curves are ubiquitous in vector graphics software, font outlines, motion paths for animation, and modern UI frameworks.

In the most common “Bernstein polynomial” form, an -degree Bézier curve is written as:

where the Bernstein basis polynomials are

This formula is mathematically compact but not always the most convenient way to compute the point in floating-point arithmetic, especially for higher degrees. That is where De Casteljau’s algorithm comes in.

How De Casteljau’s algorithm works

De Casteljau’s algorithm evaluates a Bézier curve through repeated linear interpolation between neighboring control points. It is numerically stable and closely matches the geometric intuition designers use when manipulating control handles.

Suppose you have control points . For a given parameter in , define the first level of interpolated points by

for .

This takes each pair of neighboring points and finds the point between them at a fraction of the way from the first to the second. Then you repeat the same process on the new list of points:

, and so on.

After such interpolation steps, you end up with a single point $P_0^{\left(n\right)}$. That point is exactly , the value of the Bézier curve at parameter . The calculator implements exactly this iterative interpolation procedure for your control points.

Because each step only performs simple linear interpolation in 2D, the algorithm is easy to implement and resistant to numerical issues that can appear if you evaluate the Bernstein polynomial form directly for higher degrees or extreme coordinates.

Geometric intuition and use cases

The control points of a Bézier curve form what is called the control polygon. As you move these points, the overall shape of the curve changes smoothly. The curve always remains within the convex hull of its control points, which gives designers predictable control over the shape.

When you apply De Casteljau’s algorithm, each stage of interpolation creates a new, smaller polygon that “pulls tight” toward the curve. If you were to draw all intermediate segments for many values of , you would see the curve traced out as the limit of these nested polygons.

Typical places where you might use this calculator include:

• Checking the numeric coordinates of points along a motion path in an animation system.
• Inspecting a glyph outline in a font and sampling points on a cubic curve defined by font control points.
• Debugging custom easing functions or timing curves that are represented as cubic Béziers.
• Verifying that a point lies on a curve created in a vector illustration or CAD tool.

Worked example: quadratic Bézier curve

Consider a quadratic Bézier curve with three control points:

Suppose we want the point at . The first level of De Casteljau interpolation gives two new points:

Interpolating once more between these two new points, we get:

.

So the point halfway along this quadratic Bézier curve is . If you enter the three control points and set t = 0.5 in the calculator, you should see this same result.

The same algorithm works for any degree:

• For a linear Bézier defined by two points, De Casteljau’s algorithm simply gives linear interpolation between them.
• For a cubic Bézier with four control points, there are three points at the first level, two at the second level, and one final point at the third level.
• For higher-order curves, the number of interpolation stages increases, but the procedure remains identical.

Interpreting the results

The calculator returns a single 2D coordinate, typically written as . How you interpret that pair depends on your application:

• In screen space, the values may represent pixel coordinates, often with the origin at the top-left corner.
• In mathematical plots, they may be in a Cartesian coordinate system, with increasing upward.
• In normalized coordinates, both and may lie in a range like or , which you then map to physical units.

For animation, changing t from 0 to 1 over time lets you move an object smoothly along the curve. For design work, sampling the curve at several values of t (for example, 0, 0.25, 0.5, 0.75, 1) can give you a sense of its shape or help you approximate it with straight segments.

Comparison: different Bézier degrees

Bézier curves of different degrees have different flexibility and typical uses. The calculator treats them all uniformly, as long as you provide the appropriate number of control points.

Assumptions, limitations, and notes

• 2D points only: This tool assumes each control point is a 2D coordinate in the form x,y. It does not handle 3D or higher-dimensional control points.
• Input format: Use a comma between x and y. Spaces around the comma are allowed, but additional text on the same line is not.
• Parameter range: The intended range for t is [0, 1]. Values outside this interval correspond to extrapolation beyond the usual Bézier segment and may produce points far away from the control polygon.
• Single point output: The calculator returns one point on the curve for the specific t you choose. It does not draw or sample the entire curve automatically; to trace the curve, you can evaluate multiple values of t and plot the resulting points yourself.
• Numeric precision: Very large or very small coordinate values can magnify floating-point rounding errors. For most graphics-scale coordinates (for example, within a few thousand units of the origin), the De Casteljau evaluation used here is typically accurate enough for practical work.
• Order of points: The sequence in which you enter control points matters. Reordering them will usually change the shape of the curve and the resulting point for a given t.
• Continuity between segments: If you chain multiple Bézier segments together in your own code, continuity (for example, matching end points and tangents) is your responsibility. This calculator evaluates each segment independently based on the control points you provide.

Within these assumptions, the calculator gives a direct, implementation-friendly way to answer the question: “Where exactly is the point on this Bézier curve when the parameter is t?”