fixedeangle

Fixedeangle is a term used in geometry, computer graphics, robotics, and related fields to describe a constraint that preserves a specified angle between two features. Although not a standard mathematical term, it is commonly interpreted as enforcing a fixed angular relationship between two direction vectors, lines, or joints during configuration, optimization, or transformation.

Definition and formulation

Consider two direction vectors u and v in a Euclidean space. A fixedeangle constraint fixes the angle theta0 between them, so that arccos((u · v) / (||u|| ||v||)) = theta0, or, equivalently, their dot product satisfies u · v = ||u|| ||v|| cos(theta0). In many practical settings, vectors are normalized, reducing the constraint to u · v = cos(theta0). This constraint can be applied as a hard constraint in a solver or as a soft constraint via a penalty term in an objective function.

Applications

Fixedeangle constraints appear in animation rigs to stabilize joint articulation, in robotics to maintain a desired limb angle, and in computer-aided design to enforce specific linkage geometries. They are also used in shape matching, pose estimation, and mechanical design where a precise angular relationship must be maintained under movement or deformation.

Implementation notes

Hard constraints can be enforced with Lagrange multipliers or by solving a constrained optimization problem. Soft constraints use penalty terms that discourage deviation from theta0. Numerical stability is important, especially when vectors are near zero length or when theta0 approaches 0 or pi. Degenerate configurations should be avoided in the design phase.

Relation to related concepts

Fixedeangle relates to angle constraints and hinge-like joints but emphasizes maintaining a particular angle between two features rather than restricting motion to a single axis. It overlaps with similarity transforms in the sense that some transformations preserve fixed angles. See also angle constraint, hinge constraint, and rotation invariance.