semispiral

Semispiral is a geometric term for a spiral arc that spans a half turn around a central point. In polar coordinates, a semispiral is typically described by a spiral equation with the angular parameter θ restricted to a semicircle, usually θ ∈ [0, π]. This yields a curve that begins at θ = 0 and ends at θ = π, covering 180 degrees of rotation rather than a full turn.

Two common variants are used to define semispirals. The Archimedean semispiral is given by r = a θ with θ ∈ [0, π], producing a spiral where the radius increases linearly with angle. The logarithmic (or equiangular) semispiral has the form r = a e^(b θ) with θ ∈ [0, π], yielding a curve whose radius grows exponentially with angle. In some contexts, the term semispiral may be applied more loosely to describe any spiral segment whose angular extent is less than a full circle.

Applications and appearances. In architecture, semispirals appear in semicircular staircases and ramps that wind around a central core, providing a compact vertical circulation path. In computer graphics and design, semispiral segments are used for rendering partial spirals and decorative elements. In mathematical modeling, semispirals serve as finite, easy-to-analyze curves for calculating arc length, curvature, and related properties over a bounded angular range.

See also: spiral, Archimedean spiral, logarithmic spiral, polar coordinates, curvature.