screwhelix

Screwhelix is a term that appears in discussions of screw motion and helices to describe the helical trajectory produced when a point moves under a constant screw motion: a rotation about an axis combined with a translation along the same axis. It is not a standard name for a single mathematical object, but rather a descriptive way to refer to the path created by synchronized rotation and translation in three-dimensional space.

In a coordinate system with the axis of motion taken as the z-axis, a constant screw motion has angular velocity ω and linear velocity v along the axis. A point with initial coordinates (x0, y0, z0) traces the world path given by x = x0 cos(ωt) − y0 sin(ωt), y = x0 sin(ωt) + y0 cos(ωt), z = z0 + v t. If the initial radius r = sqrt(x0^2 + y0^2) is fixed, the trajectory is a circular helix with pitch p = 2π(v/ω) per revolution. If r = 0, the path is a straight line along the axis; if v = 0, the path reduces to a circle in a plane perpendicular to the axis.

The term “screwhelix” is not widely standardized and may be used informally to distinguish helices arising from screw motions from other helices. It is sometimes encountered in discussions of screw theory, rigid-body kinematics, robotics, and computer graphics, where the interplay of rotation and translation is central.

See also: helix, screw motion, screw theory, twist, wrench, rigid-body kinematics.