Barycentric Coordinates

Barycentric coordinates are a coordinate system used to express the position of a point within a triangle relative to its vertices. In computer graphics, they are particularly useful for tasks like interpolation, as they allow values (such as color, texture coordinates, or depth) to be smoothly distributed across a triangle’s surface. By representing a point as a weighted combination of a triangle’s vertices, barycentric coordinates enable efficient calculations of attributes at any point inside the triangle. This method is widely used in rasterization for shading and texture mapping, as it ensures that interpolated values appear smooth and realistic across 3D surfaces.

Calculating Barycentric Coordinates

Barycentric coordinates are useful for expressing a point’s position inside a triangle relative to its vertices. Here’s how to calculate them step by step.

Step 1: Define the Triangle and the Point

Let’s start with a triangle defined by its vertices v0, v1, and v2, and a point P for which we want to find the barycentric coordinates (u, v, w).

• Vertices: v0(x0, y0), v1(x1, y1), v2(x2, y2)
• Point: P(px, py)

Step 2: Calculate Edge Vectors

Calculate the vectors representing the edges of the triangle:

• Edge from v0 to v1:

  v0v1 = v1 - v0

• Edge from v0 to v2:

  v0v2 = v2 - v0

• Vector from v0 to P:

  v0p = P - v0

Step 3: Compute Dot Products

Using these vectors, calculate the following dot products:

• d00: Dot product of v0v1 with itself
  d00 = v0v1 · v0v1

• d01: Dot product of v0v1 with v0v2
d01 = v0v1 · v0v2

• d11: Dot product of v0v2 with itself
  d11 = v0v2 · v0v2

• d20: Dot product of v0p with v0v1
d20 = v0p · v0v1

• d21: Dot product of v0p with v0v2
d21 = v0p · v0v2

Step 4: Calculate the Barycentric Coordinates

With the dot products, calculate the determinant to find the weights for each vertex. The determinant represents the “area” scaling factor of the triangle in 2D space (or volume in higher dimensions). This value is important because it allows us to normalize the calculations so that we can express the relative weights (barycentric coordinates) of a point inside the triangle.

1. Determinant
   denom = (d00 * d11) - (d01 * d01)

2. Coordinates:

   • v:
     v = ((d11 * d20) - (d01 * d21)) / denom

   • w:
     w = ((d00 * d21) - (d01 * d20)) / denom

   • u: Since u + v + w = 1, calculate u as
     u = 1 - v - w

Step 5: Interpret the Coordinates

The resulting (u, v, w) values represent the barycentric coordinates of point P relative to the vertices:

• If all coordinates are between 0 and 1, point P lies inside the triangle.
• The values u, v, and w correspond to the weights of each vertex (v0, v1, v2) in determining the position of P.

This method provides an efficient way to calculate barycentric coordinates, which are useful in computer graphics for interpolation and shading.