Barycentric Coordinate Calculator

Barycentric Coordinates Relative to a Triangle

Barycentric coordinates describe where a point sits relative to the vertices of one specific triangle rather than relative to fixed Cartesian axes. For a triangle with vertices A, B, and C, any point P in the plane can be written as a weighted combination of those three vertices. The weights are the barycentric coordinates.

For this barycentric-coordinate calculation, let the triangle vertices be

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

and P = (x, y) is a point in the same 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) gives the barycentric coordinates of P with respect to triangle ABC. Each coordinate is paired with its corresponding vertex: λ₁ with A, λ₂ with B, and λ₃ with C.

Barycentric Coordinate Properties and Triangle Position

• For barycentric coordinates returned by this triangle calculator, the sum λ₁ + λ₂ + λ₃ is 1 apart from 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, barycentric coordinates express P as a “center of mass” of point masses placed at A, B, and C, with masses proportional to λ₁, λ₂, and λ₃. This interpretation is useful because it ties the numerical weights directly to the triangle’s vertices.

Determinant Formulas for Barycentric Coordinates

This barycentric coordinate calculator uses a signed determinant to measure the triangle’s orientation and area-like denominator. Define the denominator Δ as

In expanded 2D form, this denominator is

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 reports this as a triangle-area error condition.

Given P = (x, y), the calculator’s determinant ratios give the barycentric coordinates as

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

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

λ₃ = 1 - λ₁ - λ₂.

Equivalent 3×3 determinant and signed-area forms produce the same barycentric weights when the vertex order is kept consistent. The calculator evaluates this determinant-based relationship directly from the eight entered coordinates.

Introduction: Barycentric Coordinate Calculation Steps

The barycentric coordinate calculator accepts the Cartesian coordinates of triangle vertices A, B, and C together with the Cartesian coordinates of 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 the determinant expressions associated with the sub-triangles PBC and APC.
3. Divides those expressions by Δ to obtain λ₁ and λ₂, then obtains λ₃ from the requirement that the three weights sum to 1.
4. Formats the three resulting weights and their displayed sum to four decimal places.

Because this barycentric computation uses standard floating-point arithmetic, extremely large or very small coordinate values may introduce small rounding errors. Even so, the unrounded weights are constructed so their sum is 1.

Interpreting Barycentric Coordinate Calculator Output

After you compute a triangle point’s barycentric coordinates, the result lists (λ₁, λ₂, λ₃) and their sum. Read the output as follows:

• Check the sum: λ₁ + λ₂ + λ₃ should be 1, subject to the displayed 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.

These triangle-relative weights make barycentric coordinates useful for point-in-triangle tests, interpolation, and geometric reasoning. The calculator’s inside/outside message follows the signs of the three calculated weights.

Worked Example: Barycentric Weights in a Right Triangle

This barycentric coordinate example uses a triangle with vertices

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

and calculates the weights of point

P = (1, 1).

Step 1: Compute the Triangle Determinant Δ

Using the barycentric denominator formula

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

substitute the triangle’s vertex coordinates:

• 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 the Weight at Vertex A, λ₁

For this triangle, the determinant formula for the weight associated with A is

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

Here x = 1 and 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 the Weight at Vertex B, λ₂

The corresponding barycentric formula for the weight at B is

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

Compute the coordinate 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 the Weight at Vertex C, λ₃

The final barycentric weight follows from the requirement that all three weights sum to 1:

λ₃ = 1 - λ₁ - λ₂ = 1 - 5/12 - 1/4.

Since 1/4 = 3/12,

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

Step 5: Interpret the Triangle Position

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

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

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

Entering these coordinates in the calculator produces the same barycentric weights, with the displayed values rounded to four decimal places.

How to Use Barycentric Coordinates: Applications and Use Cases

Barycentric coordinates are especially valuable whenever a calculation must relate a point to the vertices of a triangle:

• 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.

Barycentric Coordinates Compared with Other Coordinate Systems

Limitations and Assumptions for 2D Barycentric Coordinates

• Non-collinear vertices: Barycentric coordinates for a triangle require A, B, and C not to lie on one straight line. If they are collinear, Δ = 0 and the triangle-relative weights are undefined.
• 2D only: This calculator works in 2D Cartesian coordinates. Extensions to 3D 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 displayed sums such as 0.9999 or 1.0001 rather than exactly 1.
• Orientation and sign conventions: Signed-area and determinant formulas depend on vertex ordering (clockwise versus counterclockwise). The calculator uses one consistent ordering; reversing the vertex order changes the sign of Δ while preserving the same geometric inside/outside classification.
• Degenerate and boundary cases: Points exactly on edges or vertices may be sensitive to tiny numerical noise, especially if coordinates are large. For boundary decisions, allow a small tolerance when judging whether a weight is effectively zero or one.

With three non-collinear 2D vertices and reasonably scaled coordinate values, this barycentric coordinate tool provides a direct description of P’s location relative to triangle ABC.