Closertotangential

Closertotangential is an informal term used in geometry and related disciplines to describe a quantity or procedure that aligns more closely with the tangent direction of a smooth curve at a given point. It is not a standardized notion, and definitions vary by context; authors typically quantify “closeness” to tangential alignment with a directional measure derived from projection onto the tangent.

In differential geometry, consider a smooth curve r(t) with unit tangent T(t) = r'(t)/|r'(t)|. A vector field V(t) along the curve is said to be closer to tangential if its projection onto T has greater magnitude relative to its normal component. A common scalar measure is (V·T)^2/|V|^2, which approaches 1 when V is perfectly tangent.

In numerical path-following, computer graphics stroke approximation, or motion planning, strategies that select directions to minimize deviation from the tangent are described as closer to tangential, because they reduce lateral curvature error and improve stability or continuity.

Because there is no universal convention, the term often appears with explicit definitions or metrics in the corresponding literature. Typical indicators include the angle theta between V and T, with cos(theta) or cos^2(theta) used as a tangential alignment score; 0 degrees indicates exact tangency, 90 degrees indicates orthogonality.

See also: tangent line, curvature, projection (linear algebra), tangential continuity. Notes: Not a standard term; usage is context-dependent.