semihelix

Semihelix is a term used in geometry to describe a class of space curves that interpolate between a straight line and a classical helix along a fixed axis. The phrase is not standard in formal mathematics, and definitions vary across authors.

A common way to define a semihelix is in cylindrical coordinates (r, θ, z) with z proportional to θ and r a smooth function that transitions from zero toward a finite radius. In general, r = f(θ), z = p θ, where f is monotone and f(0) = 0, and as θ grows, f(θ) approaches a constant value R. When f is identically zero you obtain the axis line; when f is constant R you obtain a standard helix. Intermediate choices give semihelices.

Example: r(θ) = a(1 - e^{−b θ}), z = c θ (with a>0, b>0, c>0) defines a semihelix winding around the axis while gradually increasing its radius from the axis toward a limiting radius a.

Properties and variants: The curve is smooth for appropriate parameters; curvature and torsion vary with θ; asymptotically it approaches a helix of radius R. Variants arise by choosing different f(θ) or by letting z be a nonlinear function of θ.

Applications: semihelices appear in architectural ornament, jewelry design, and computer graphics to create transitional, spiraling forms. They are also used conceptually in mechanical design for cables or springs with gradually changing radius.

See also: helix, spiral, conical spiral.