Overview
Ray-Triangle Intersection is a core geometric test for rendering, collision picking, and spatial queries.
ExtendedSource checked
Ray-Triangle Intersection
Finds whether and where a ray hits a triangle.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Geometry
- Difficulty
- Intermediate
- Family
- Geometric Intersection
- Tier
- Extended
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 1
- Relationships
- 5
- Code example
- Ready
- Visualization
- Ready
- Aliases
- Moller-Trumbore Intersection, Möller-Trumbore Intersection
Core Concepts
ray equationbarycentric coordinatesplane intersectioninside-triangle test
Quick facts
Complexity
Runtime and memory behavior at a glance.
Best
O(1)
Average
O(1)
Worst
O(1)
Space
O(1)
When to Use
- Use when a ray needs an exact hit test against triangle geometry.
- Use when ray tracing is a central requirement.
When Not to Use
- Avoid when a bounding volume can reject the ray before triangle-level tests are needed.
- Avoid when floating-point precision or degenerate geometry cannot be handled robustly.
What It Does
- Tests a ray against a triangle plane
- Checks whether the hit lies inside the triangle
- Returns hit distance and barycentric coordinates
Use Cases
- Ray tracing
- Mouse picking in 3D
- Collision queries
Inputs
- Ray origin and direction
- Triangle vertices
Outputs
- Hit flag
- Ray distance
- Barycentric coordinates
Key Formulas (1)
Ray equation
p(t) = origin + t · direction
p(t) = origin + t * directionRepresents every point along the ray.
Pseudocode
- 1.Compute triangle edge vectors
- 2.Solve for ray distance and barycentric coordinates
- 3.Reject if the ray is parallel or behind the origin
- 4.Reject if barycentric coordinates fall outside the triangle
- 5.Return the hit
Canonical Example
- Input
- ray starts in front of a triangle and points toward it
- Output
- hit = true, t = 4.2
- Explanation
- The test solves the ray position and verifies it lies inside the triangle.
Implementation Notes
- Use an epsilon for determinant checks and document backface-culling behavior.
- Use robust predicates or tolerances when near-degenerate inputs are expected.
- Keep a tiny example with expected output beside the implementation.
Common Pitfalls
- Not handling nearly parallel rays can create unstable hits.
- Collinear, overlapping, and duplicate points need explicit tie-breaking rules.
- Not defining boundary, empty, or degenerate inputs before implementation.
Interactive Mechanism Walkthrough
Spatial Primitive & Sweep-Line CanvasShow a ray piercing a triangle and the hit point expressed with barycentric weights.
Step 1 of 5: Compute triangle edge vectors
geometry-diagramNotes
Moller-Trumbore is a common fast formulation of this test.
Sources (2)
What each reference supports for this entry.
Real-Time Rendering
BookRendering reference for ray-geometry intersection tests including ray-triangle intersection.
Open sourceReal-Time Collision Detection
BookGeometric collision reference for Ray-Triangle Intersection's primitive tests, bounding volumes, and numerical edge cases.
Explore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageRay-triangle hit points are often represented with barycentric coordinates.
Often paired withBVH Traversal
BVH traversal often ends in ray-triangle intersection tests.
Used byRay Tracing
Ray tracers use ray-triangle tests at primitive intersections.
Related Algorithms
Tags
GeometryRay IntersectionComputational GeometryGraphics