triangleheat

Triangleheat is a term used in computational geometry and numerical simulations to describe the diffusion of heat over a surface discretized into triangular elements. In this framework, the surface is represented as a triangular mesh, and heat flow is modeled by solving a discrete heat equation on the mesh. The term emphasizes the role of triangular elements in shaping the diffusion process.

In the triangleheat model, the temperature at mesh vertices or over triangles evolves in time according to a discrete approximation of the heat equation. Common approaches use finite element or finite volume methods; the discrete Laplacian is defined using the mesh topology and geometry, often via cotangent weights for robustness on irregular meshes. Boundary conditions and source terms can be incorporated to simulate heating, cooling, or localized heat sources.

The model respects conservation of total heat in the absence of sources or sinks and produces smoothing of initial temperature distributions over time. The rate of diffusion depends on the mesh density and the chosen time step, with finer meshes offering higher fidelity at increased computational cost.

Applications include real-time shading and texture mapping in computer graphics, thermal analysis in engineering, and digital fabrication simulations. Variants of triangleheat may consider anisotropic diffusion, nonuniform material properties, or adaptive mesh refinement to balance accuracy and efficiency.

See also: heat equation, discrete Laplacian, finite element method, mesh processing.