Barycentric Coordinate Calculator

What Are Barycentric Coordinates?

Barycentric coordinates provide a way to describe the position of a point relative to the vertices of a triangle instead of using the usual Cartesian coordinates. For a triangle with vertices A, B, and C, any point P in the plane can be written as a weighted combination of these three vertices. The weights are the barycentric coordinates.

Suppose the triangle has vertices

• A = (x₁, y₁)
• B = (x₂, y₂)
• C = (x₃, y₃)

and P = (x, y) is any point in the plane. Then there exist numbers λ₁, λ₂, λ₃ such that

P = λ₁A + λ₂B + λ₃C,

with the constraint

λ₁ + λ₂ + λ₃ = 1.

The triple (λ₁, λ₂, λ₃) is called the barycentric coordinates of P with respect to triangle ABC.

Key Properties and Interpretation

• The sum λ₁ + λ₂ + λ₃ is always exactly 1 (up to numerical rounding).
• P lies inside the triangle if and only if all three barycentric coordinates are strictly positive.
• P lies on an edge of the triangle if one coordinate is zero and the other two are between 0 and 1.
• P coincides with a vertex if one coordinate is 1 and the other two are 0.
• P lies outside the triangle if at least one barycentric coordinate is negative.

Geometrically, you can think of barycentric coordinates as expressing P as a “center of mass” of three point masses placed at A, B, and C, with masses proportional to λ₁, λ₂, and λ₃.

Formulas for Barycentric Coordinates

There are several equivalent formulas. One common approach uses determinants (or signed areas). Define the denominator Δ as

In expanded 2D form, this is usually written as

Δ = (x₂ - x₁)(y₃ - y₁) - (x₃ - x₁)(y₂ - y₁).

If Δ = 0, the three vertices are collinear and do not form a valid triangle, so barycentric coordinates are not defined (the calculator will treat this as an error condition).

Given P = (x, y), one convenient formula for the barycentric coordinates is

λ₁ = ((x₂ - x)(y₃ - y) - (x₃ - x)(y₂ - y)) / Δ,

λ₂ = ((x₃ - x)(y₁ - y) - (x₁ - x)(y₃ - y)) / Δ,

λ₃ = 1 - λ₁ - λ₂.

Algebraically equivalent variants using 3×3 determinants or raw coordinate products are also common, and the calculator uses one of these determinant-based formulations internally.

Introduction: How the Calculator Works

The calculator takes as input the coordinates of the triangle vertices A, B, C and a point P. It then:

1. Computes the denominator Δ from A, B, and C to ensure the triangle is not degenerate (collinear vertices).
2. Computes three sub-determinants or area-like expressions that correspond to areas of sub-triangles PBC, APC, and ABP.
3. Divides each sub-determinant by Δ to obtain λ₁, λ₂, and λ₃.
4. Checks for numerical rounding and presents the coordinates, usually to a fixed number of decimal places.

Because these formulas use standard floating-point arithmetic, extremely large or small coordinate values may introduce small rounding errors. However, the sum of the three barycentric coordinates should still be very close to 1.

Interpreting the Calculator Output

After you click the compute button, you will see three numbers corresponding to (λ₁, λ₂, λ₃). To read these values:

• Check the sum: λ₁ + λ₂ + λ₃ should be 1 (up to rounding).
• All positive: If each λ_i > 0, P is strictly inside the triangle.
• One zero: If one λ_i is 0 and the other two are between 0 and 1, P lies on the edge opposite vertex i.
• One equal to 1: If a single λ_i = 1 and the others are 0, P coincides with vertex i.
• Negative value: If any λ_i is negative, P lies outside the triangle on the side opposite that vertex.

This makes barycentric coordinates a natural tool for point-in-triangle tests, interpolation, and geometric reasoning on triangles.

Worked Example

Consider a triangle with vertices

• A = (0, 0)
• B = (4, 0)
• C = (0, 3)

and let us compute the barycentric coordinates of point

P = (1, 1).

Step 1: Compute Δ

Using the denominator formula

Δ = (x₂ - x₁)(y₃ - y₁) - (x₃ - x₁)(y₂ - y₁),

we substitute

• x₁ = 0, y₁ = 0
• x₂ = 4, y₂ = 0
• x₃ = 0, y₃ = 3

Then

Δ = (4 - 0)(3 - 0) - (0 - 0)(0 - 0) = 4 × 3 - 0 = 12.

Step 2: Compute λ₁

Using the area/determinant-based formula (one common variant) for λ₁:

λ₁ = ((x₂ - x)(y₃ - y) - (x₃ - x)(y₂ - y)) / Δ.

Here x = 1, y = 1, so

• x₂ - x = 4 - 1 = 3
• y₃ - y = 3 - 1 = 2
• x₃ - x = 0 - 1 = -1
• y₂ - y = 0 - 1 = -1

Thus

(x₂ - x)(y₃ - y) = 3 × 2 = 6,

(x₃ - x)(y₂ - y) = (-1) × (-1) = 1,

so

λ₁ = (6 - 1) / 12 = 5 / 12 ≈ 0.4167.

Step 3: Compute λ₂

A symmetric formula for λ₂ is

λ₂ = ((x₃ - x)(y₁ - y) - (x₁ - x)(y₃ - y)) / Δ.

Compute the differences:

• x₃ - x = 0 - 1 = -1
• y₁ - y = 0 - 1 = -1
• x₁ - x = 0 - 1 = -1
• y₃ - y = 3 - 1 = 2

Then

(x₃ - x)(y₁ - y) = (-1) × (-1) = 1,

(x₁ - x)(y₃ - y) = (-1) × 2 = -2,

so

λ₂ = (1 - (-2)) / 12 = 3 / 12 = 0.25.

Step 4: Compute λ₃

Finally, use the fact that the three coordinates sum to 1:

λ₃ = 1 - λ₁ - λ₂ = 1 - 5/12 - 1/4.

Note that 1/4 = 3/12, so

λ₃ = 1 - 5/12 - 3/12 = 1 - 8/12 = 4/12 = 1/3 ≈ 0.3333.

Step 5: Interpret the Result

The barycentric coordinates of P with respect to triangle ABC are approximately

(λ₁, λ₂, λ₃) ≈ (0.4167, 0.25, 0.3333).

All three values are positive and less than 1, and they sum to 1, so P lies inside the triangle. In terms of the vertices, P is a weighted average of A, B, and C with those weights.

You can enter these same coordinates into the calculator to verify that you obtain essentially the same barycentric values (small differences may occur due to rounding of intermediate steps).

How to use: Applications and Use Cases

Barycentric coordinates appear in many areas of mathematics and applied science:

• Computer graphics and rendering: Used to interpolate colors, normals, textures, and other per-vertex attributes across a triangle when rasterizing 3D models.
• Point-in-triangle testing: Efficiently check whether a point belongs to a triangle, lies on an edge, or is outside.
• Finite element methods (FEM): Act as shape functions for triangular elements when approximating solutions of partial differential equations.
• Geometric modeling: Help define coordinates that are independent of specific Cartesian axes, making many constructions more natural.

Comparison with Other Coordinate Systems

Limitations and Assumptions

• Non-collinear vertices: The three points A, B, and C must not lie on a single straight line. If they are collinear, Δ = 0 and barycentric coordinates with respect to a triangle are not defined.
• 2D only: This calculator works in 2D Cartesian coordinates. Extensions to 3D (for tetrahedra or projected triangles) are not covered here.
• Floating-point precision: Extremely large, small, or nearly collinear configurations can lead to rounding errors. You may see sums of barycentric coordinates like 0.9999 or 1.0001 instead of exactly 1.
• Orientation and sign conventions: Signed area and determinant formulas depend on vertex ordering (clockwise vs counterclockwise). The calculator uses a consistent convention internally, but reversing vertex order will change the sign of Δ while leaving the geometric interpretation (inside vs outside) unchanged.
• Degenerate and boundary cases: Points exactly on edges or vertices may be sensitive to tiny numerical noise, especially if coordinates are large. When interpreting such cases, allow for small tolerances when deciding whether a value is effectively zero or one.

As long as you supply three non-collinear vertices in 2D and reasonable coordinate magnitudes, the barycentric coordinates produced by this tool will give a clear and reliable description of where your point lies relative to the triangle.