pointdepend

Pointdepend is a concept used in computational geometry, computer-aided design, and geometric constraint systems to describe how the position of a point is determined by other points in a model. It formalizes the idea that a point can be constrained to move only in relation to a predefined set of anchor points, so that changing one control point automatically propagates to dependent points.

Formally, let P be a set of points and A ⊆ P an anchor subset that influence a dependent point p ∈ P. p is point-dependent on A if there exist weights w_a for a ∈ A such that p = Σ_{a∈A} w_a a, with Σ w_a = 1 (an affine or barycentric combination) and the weights are fixed by the constraint system. The dependencies are represented by a directed graph where an edge a → p indicates a influences p. Under this model, the graph is typically acyclic to ensure deterministic propagation.

Algorithmically, pointdepend enables incremental constraint propagation. When an anchor moves, updates traverse the graph in topological order, recomputing affected points efficiently. This supports interactive editing and real-time feedback in modeling environments.

Applications include CAD constraint solvers, animation rigs, and geometric modeling tools that require maintaining relationships between points during manipulation. Examples include a point inside a triangle expressed as a barycentric combination of the triangle’s vertices, or a point on a line defined as a weighted average of endpoints.

See also: affine dependence, barycentric coordinates, geometric constraints, constraint solver.