Overview
Barycentric coordinates are used for triangle interpolation, point-in-triangle tests, rasterization, and mesh processing.
CoreSource checked
Barycentric Coordinates
Represents a point inside a triangle as weights on the triangle's vertices.
Quick Facts
Compact metadata for implementation and discovery.
- Category
- Geometry
- Difficulty
- Intermediate
- Family
- Coordinate System
- Tier
- Core
- Status
- Verified
- Review
- Source Checked
- Sources
- 2
- Formulas
- 1
- Relationships
- 4
- Code example
- Ready
- Visualization
- Ready
Core Concepts
triangle weightsaffine combinationpoint containmentattribute interpolation
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 collision and intersection tests needs triangle weights.
- Use when the input matches the assumptions behind coordinate System.
When Not to Use
- Avoid when floating-point robustness is critical and no tolerance strategy is defined.
- Avoid when a simpler algorithm gives the required result with less setup.
What It Does
- Represents a point inside a triangle as weights on the triangle's vertices
- Uses triangle weights and affine combination to guide the work.
- Produces geometric relation, simplified shape, or constructed geometry.
Use Cases
- Collision and intersection tests
- Shape analysis
- Geometry processing
Inputs
- Points, segments, polygons, or meshes
- Query shape or tolerance
Outputs
- Geometric relation, simplified shape, or constructed geometry
Key Formulas (1)
Barycentric combination
p = u · a + v · b + w · c, with u + v + w = 1
p = u * a + v * b + w * c, with u + v + w = 1Represents a point using triangle vertex weights.
Pseudocode
- 1.Prepare the inputs and state needed for Barycentric Coordinates.
- 2.Apply the triangle weights step while the stopping condition has not been met.
- 3.Update intermediate state and discard invalid candidates.
- 4.Return geometric relation, simplified shape, or constructed geometry.
Canonical Example
- Input
- Barycentric Coordinates with a minimal inspectable input
- Output
- Geometric relation, simplified shape, or constructed geometry
- Explanation
- Barycentric Coordinates applies triangle weights to produce geometric relation, simplified shape, or constructed geometry.
Implementation Notes
- Use robust orientation and tolerance checks for boundary cases.
- Keep the triangle weights step separate enough to test independently.
Common Pitfalls
- Degenerate cases such as collinear points or shared edges often need explicit handling.
- Forgetting to check edge cases can make Barycentric Coordinates look correct on only ideal inputs.
Interactive Mechanism Walkthrough
Spatial Primitive & Sweep-Line CanvasDraw the input shapes, active tests, and generated geometric result.
Step 1 of 4: Prepare the inputs and state needed for Barycentric Coordinates.
geometry-mechanismNotes
Barycentric Coordinates should be used with the assumptions and tradeoffs of coordinate System in mind.
Sources (2)
What each reference supports for this entry.
Computational Geometry: Algorithms and Applications
BookComputational geometry textbook support for Barycentric Coordinates's geometric model, data structures, and robustness concerns.
Open sourceThe Algorithm Design Manual
BookAlgorithm catalog support for Barycentric Coordinates's problem context, known alternatives, and practical selection tradeoffs.
Open sourceExplore Next
Good next jumps from the relationship graph, ranked for useful discovery.
Relationships
View full relationship pageUsed byEar Clipping Triangulation
Triangulation and interpolation workflows commonly use barycentric coordinates.
Used byRay-Triangle Intersection
Ray-triangle tests often compute barycentric coordinates for hit points.
Related Algorithms
Tags
Computational GeometryInterpolationTriangulationLinear Algebra