Barycentric Coordinates Calculator

Barycentric Coordinates (α, β, γ):

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Area Coordinates:

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Understanding Barycentric Coordinates

What are Barycentric Coordinates?

Barycentric coordinates are a powerful mathematical tool used to describe the position of a point within a triangle (or a simplex in higher dimensions) relative to its vertices. Instead of using standard (x, y) coordinates, barycentric coordinates express a point as a "weighted average" of the triangle's corners. This concept is fundamental in various fields, including computer graphics, engineering, and physics, because it simplifies many geometric calculations and provides intuitive insights into a point's location.

Imagine a triangle with vertices A, B, and C. Barycentric coordinates (α, β, γ) for any point P inside or outside this triangle tell us how much "influence" each vertex has on the point P. These three values are always non-negative for points inside the triangle and always sum up to 1 (α + β + γ = 1). Think of them as weights: if P is very close to vertex A, then α will be close to 1, and β and γ will be close to 0.

The Fundamental Equation:

The position of point P can be expressed as a linear combination of the triangle's vertices:

P = αA + βB + γC

Where A, B, C are the position vectors of the vertices, and α, β, γ are the barycentric coordinates.

A crucial property is that the sum of the coordinates must equal one:

α + β + γ = 1

Area Method for Calculation:

For a point P within a triangle ABC, its barycentric coordinates can be calculated using the ratios of areas of smaller triangles formed by P and the vertices:

  • α = Area(PBC) / Area(ABC) (The ratio of the area of the triangle formed by P, B, and C to the total area of triangle ABC)
  • β = Area(PCA) / Area(ABC) (The ratio of the area of the triangle formed by P, C, and A to the total area of triangle ABC)
  • γ = Area(PAB) / Area(ABC) (The ratio of the area of the triangle formed by P, A, and B to the total area of triangle ABC)
  • Weighted Average: Barycentric coordinates represent a point as a weighted average of the triangle's vertices.
  • Sum to One: The sum of the three coordinates (α + β + γ) always equals 1.
  • Geometric Interpretation: They can be thought of as "mass coordinates," where masses placed at the vertices balance the triangle at point P.
  • Location Indicator:
    • If all coordinates are positive (α, β, γ > 0), the point P is strictly inside the triangle.
    • If one coordinate is zero and two are positive, the point lies on an edge.
    • If two coordinates are zero and one is one, the point is a vertex.
    • If any coordinate is negative, the point lies outside the triangle.

Advanced Properties and Applications

Barycentric coordinates offer deep insights into geometric relationships and are widely applied in various technical fields due to their elegant properties.

  • Geometric Interpretation:
    • Mass Distribution: Historically, barycentric coordinates were linked to the center of mass. If masses α, β, γ are placed at vertices A, B, C respectively, then P is the center of mass of this system.
    • Center of Mass: This connection makes them intuitive for understanding physical balance and equilibrium within a triangular region.
    • Affine Combinations: They are a special case of affine combinations, where the sum of coefficients is 1. This ensures that the transformation preserves collinearity and ratios of distances.
    • Areal Coordinates: When calculated using the ratio of areas, they are often referred to as areal coordinates, emphasizing their connection to the sub-triangles formed.
  • Special Points within a Triangle:
    • Centroid (1/3, 1/3, 1/3): The geometric center of the triangle, where all medians intersect. This is the point where the triangle would perfectly balance.
    • Vertices (1,0,0), (0,1,0), (0,0,1): Each vertex itself has barycentric coordinates where its own weight is 1 and others are 0.
    • Midpoints of Edges: For example, the midpoint of AB would be (1/2, 1/2, 0).
    • Incenter: The center of the inscribed circle, with coordinates proportional to the side lengths.
    • Circumcenter: The center of the circumscribed circle, with coordinates related to the angles and side lengths.
  • Applications:
    • Computer Graphics: Essential for interpolating colors, textures, and normals across triangle surfaces in 3D rendering. They allow smooth transitions of properties from vertices to any point within the triangle.
    • Finite Element Analysis (FEA): Used in engineering simulations to approximate solutions to complex problems by dividing a larger domain into smaller, simpler triangular (or tetrahedral) elements.
    • Interpolation: Provide a natural way to interpolate values (like temperature, pressure, or elevation) within a triangle, given values at its vertices.
    • Physical Modeling: Used in physics for problems involving centers of mass, force distribution, and stability analysis.
    • Robotics and Kinematics: Can be used to describe the position of a robot's end-effector relative to a triangular workspace.

Key Properties and Theorems

Barycentric coordinates are deeply connected to fundamental geometric theorems and possess properties that make them incredibly versatile.

Ceva's Theorem

Ceva's Theorem provides a condition for three cevians (lines from a vertex to the opposite side) of a triangle to be concurrent (intersect at a single point). When expressed using barycentric coordinates, the theorem becomes elegantly simple, often involving the product of ratios of segment lengths along the sides, which equals 1.

(BD/DC) * (CE/EA) * (AF/FB) = 1

Affine Invariance

Barycentric coordinates are "affine invariant." This means that if you apply an affine transformation (like translation, rotation, scaling, or shearing) to the triangle and the point P, the barycentric coordinates of P relative to the transformed triangle remain the same. This property is extremely useful in computer graphics and geometry processing.

Convex Combinations

For a point P to be strictly inside the triangle ABC, all its barycentric coordinates (α, β, γ) must be positive. If they are all non-negative, the point lies within or on the boundary of the triangle. This is a specific type of convex combination, where the weights sum to 1 and are non-negative, ensuring the point stays within the convex hull of the vertices.

Homogeneous Form

Sometimes, barycentric coordinates are used in a "homogeneous" form (α:β:γ) where their sum is not necessarily 1, but their ratios are preserved. This is particularly useful in projective geometry, where points at infinity can also be represented. To convert homogeneous coordinates back to standard barycentric coordinates, you simply divide each component by their sum.