gequation

Gequation is a term used to describe the representation of geometric constraints as algebraic equations. In practice, a gequation expresses relationships among coordinates, distances, angles, and incidences in a coordinate system, enabling the use of algebraic methods to analyze and solve geometric problems. The concept appears in discussions of computational geometry, computer-aided design, and computer graphics, where preserving and manipulating geometric constraints is central.

Gequations are typically implicit: instead of describing points parametrically along a curve, a gequation asserts a relation that must hold for all points in the solution set. Common forms include distance constraints, such as (x - x0)^2 + (y - y0)^2 = r^2, which defines a circle; line constraints, such as n_x x + n_y y + c = 0; and angle constraints, such as (v1 · v2) / (|v1||v2|) = cos theta. In 3D, similar relations extend to planes, spheres, and dihedral or angle constraints.

Solving a system of gequations typically requires algebraic or numeric methods. Symbolic techniques from algebraic geometry, such as elimination and Groebner bases, can yield all solutions, while numerical solvers handle large, nonlinear systems common in robotics and CAD. Gequations can be used for constraint solving, kinematic analysis, and implicit modeling.

Relation to other concepts: gequation overlaps with implicit equations in mathematics and with constraint programming. It emphasizes the encoding of geometry into algebraic form, enabling the application of standard solver tools and geometric reasoning. Limitations include nonlinearity, multiple or degenerate solutions, and sensitivity to coordinate choices.

See also: algebraic geometry, implicitization, constraint satisfaction problem, computational geometry, computer-aided design, robotics.