10. Triangles and Barycentric Coordinates

Spheres and infinite planes are excellent for exposing ray equations, but most real-time scene geometry is delivered as triangles. A triangle is finite, planar and defined by three vertices. Intersecting its supporting plane is not enough; the resulting point must also lie inside the three edges. A successful test should tell us more than distance because the triangle's three vertex weights will later interpolate normals, texture coordinates and other attributes.

We shall use the Möller-Trumbore form of the ray-triangle test. Its compact code should not be mistaken for an incantation. The operations solve for three unknowns: ray distance and two barycentric coordinates. The third barycentric coordinate follows because the three weights sum to one.

From vertices to a finite surface

Let the vertices be A, B and C. Two edges from A span the triangle's plane:

E1 = B − A, E2 = C − A

The cross product E1 × E2 gives a face normal whose sign depends on vertex order. Reversing two vertices reverses the normal. This order is called winding. The intersection below can either cull one orientation or accept both. Our manual tracer accepts both and uses the oriented hit-record rule from Lesson 6.

Solve distance and weights together

function hitTriangle(ray, triangle, tMin, tMax) {
    const edge1 = subtract(triangle.b, triangle.a);
    const edge2 = subtract(triangle.c, triangle.a);
    const p = cross(ray.direction, edge2);
    const determinant = dot(edge1, p);

    if (Math.abs(determinant) < 1e-7) {
        return null;
    }

    const inverse = 1 / determinant;
    const fromA = subtract(ray.origin, triangle.a);
    const beta = dot(fromA, p) * inverse;
    if (beta < 0 || beta > 1) {
        return null;
    }

    const q = cross(fromA, edge1);
    const gamma = dot(ray.direction, q) * inverse;
    if (gamma < 0 || beta + gamma > 1) {
        return null;
    }

    const t = dot(edge2, q) * inverse;
    if (t < tMin || t > tMax) {
        return null;
    }

    const alpha = 1 - beta - gamma;
    const outward = normalise(cross(edge1, edge2));
    return makeHitRecord(ray, t, pointOnRay(ray, t), outward,
                         triangle.material, [alpha, beta, gamma]);
}

A near-zero determinant means the ray is parallel to the supporting plane or the triangle is degenerate. The beta and gamma tests constrain the point to the finite triangular region. Non-negative alpha is enforced by beta + gamma <= 1. The usual ray interval then decides whether the otherwise valid triangle point lies along the useful part of the ray.

Barycentric coordinates describe location

A point on the triangle can be written as a weighted combination of its vertices:

P = αA + βB + γC, where α + β + γ = 1

At vertex A the weights are (1, 0, 0). Halfway along edge AB they are (0.5, 0.5, 0). At the centroid they are one third each. Inside the triangle all three are non-negative. This gives both an inside test and a reusable coordinate system over the surface.

Join the existing scene contract

The laboratory adds a triangle to the sphere-and-plane scene. Its hit record still contains t, position, oriented normal and material, with barycentric coordinates as useful additional data. Closest-hit traversal compares it without knowing it is a triangle. Shadow rays can be blocked by it. The light calculation reads its normal.

Keep the rendered triangle fixed while the diagram interrogates one point using barycentric weights. Otherwise, visible triangle movement becomes a tempting proxy for the weight calculation. Choose α and β; the laboratory derives γ = 1 - α - β. Non-negative weights whose sum is one describe a point inside or on the triangle. A negative derived weight places the reconstructed point outside. Meanwhile, the scene still uses the closest-hit criterion from Lesson 5. A new primitive has added a new intersection equation, not a new definition of visibility.

Degenerate triangles are not small triangles

If A, B and C are collinear or repeated, edge1 × edge2 has zero area and no stable surface normal. Treating the determinant threshold as a complete repair would conceal invalid input. The test can reject the primitive safely, but a mesh-building or loading stage should also identify and report degeneracy.

Very small valid triangles introduce scale and precision questions similar to the shadow bias. A single fixed determinant threshold may reject a triangle after a scene-scale change. Robust intersection code considers relative error and representation. Our threshold is suitable for the present coordinate range and should be challenged, not worshipped.

Our rays can now meet the primitive used by general meshes. Lesson 11 lets a surface create a reflected ray, turning one visibility question into a bounded recursive chain.