Triangle Point Barycentric Coordinates

Description

Invert planar barycentric coordinates on a nondegenerate supplied triangle; floating arithmetic and signed exterior weights are retained.

Planar geometry operates on two-coordinate Cartesian points and explicit shape boundaries. A closed rectangle includes its edges, while a segment parameter locates a point along a supporting line. Signed cross products, areas and axis gaps preserve orientation or direction information that a magnitude alone would lose.

Invert planar barycentric coordinates on a nondegenerate supplied triangle; floating arithmetic and signed exterior weights are retained.

When to use this tool

Use this tool to invert planar barycentric coordinates on a nondegenerate supplied triangle; floating arithmetic and signed exterior weights are retained.

How to use the tool and how it works

Enter vertices, point using the input definitions below.
Run the operation. Resolve malformed-input errors before interpreting the result.
Read result.weights. Interpret signs, units, ordering and null values using the rules below.

Let den = cross(A,B,C). Reject den = 0; calculate u = cross(A,point,C)/den and v = cross(A,B,point)/den, returning [1−u−v,u,v] for vertices [A,B,C].^[1]

Input field	Meaning and units	Accepted input
vertices	Exactly three planar vertices [[Ax,Ay],[Bx,By],[Cx,Cy]] in weight order.	Dense list of at most 128 rows of at most 128 finite numbers; the operation’s shape constraints also apply.
point	Planar point [x,y] in the shape’s coordinate units.	Dense list of at most 128 finite numbers, subject to the stated integer/shape rules.

Output field	Interpretation
result	Object containing the fields below.
result.weights	Three signed dimensionless weights in the input vertex order.

json

{
  "input": {
    "vertices": [
      [
        0,
        0
      ],
      [
        2,
        0
      ],
      [
        0,
        2
      ]
    ],
    "point": [
      0.5,
      0.5
    ]
  },
  "output": {
    "result": {
      "weights": [
        0.5,
        0.25,
        0.25
      ]
    }
  }
}

Limitations and assumptions

Negative weights describe exterior positions. Near-degenerate triangles can amplify floating-point error; no numerical tolerance is inferred.
Point components and rectangle bounds are finite within ±1,000,000,000. Rectangles use [minX,minY,maxX,maxY] with ordered inclusive bounds. Floating arithmetic is used without an implicit robustness tolerance.
Every declared top-level input is required and unknown fields are rejected. Ordinary strings are limited to 8,192 UTF-16 code units and lists to 128 entries, with tighter operation-specific bounds where stated. JSON text has bounded nesting and node counts.
Numeric outputs must be finite and within the absolute safe-integer magnitude; exact larger integers use decimal text. Source positions are zero-based UTF-16 offsets and span ends are exclusive unless a different convention is explicitly stated.

Alternative or Complementary methods

Choose the boundary and coordinate convention explicitly. Use robust geometric predicates or a topology-aware library when near-coincidence, holes, invalid polygons or geographic coordinates matter.

References

Tool behavior and example: Triangle Point Barycentric Coordinates — tool.io

Similar or alternative tools

Triangle Barycentric Point
Map three supplied barycentric weights summing to one to a planar triangle point; weights may be negative, no inside-triangle claim.
Triangle Barycentric Membership
Audit supplied three-coordinate barycentric closed-triangle membership with an explicit nonnegative absolute tolerance; does not validate a triangle’s geometry.