curviligne

Curviligne is a term used in geometry and related disciplines to describe lines, shapes, or regions that are formed by curves rather than by straight lines. In English, the equivalent term is curvilinear. The designation emphasizes bending or curvature as opposed to rectilinearity, and it is applied to angles, boundaries, coordinates, and paths that deviate from a straight line.

Originating from Latin curvus “bent” and linea “line,” via French usage, curviline or curviligne entered mathematical and design vocabularies to denote curved geometry.

In mathematics, curvilinear describes objects in a plane or space defined by continuous curves. A curvilinear coordinate system uses coordinates that run along curves; examples include polar coordinates (r, θ) in the plane and cylindrical or spherical coordinates in space. Curvature measures how sharply a curve bends; straight lines have zero curvature, circles have constant curvature.

Applications appear in cartography (curvilinear grids on maps), architecture and industrial design (smooth, organic forms), computer graphics (rendering of curved surfaces), and robotics or path planning (curvilinear trajectories).

See also curvilinear coordinates, curve, rectilinear, differential geometry.