PathXZ

PathXZ is a term used in geometry, computer graphics, robotics, and manufacturing to denote a path constrained to the XZ plane of a three-dimensional coordinate system. In this context, the Y coordinate is held constant, so a pathXZ can be represented as r(t) = (x(t), y0, z(t)). When y0 = 0, the path lies entirely in the horizontal XZ plane.

Mathematically, a pathXZ is a planar curve in the XZ plane. It can be described parametrically by x = f(t), z = g(t), or discretely as a polyline with vertex coordinates (x_i, z_i) and a fixed y value. It is common to lift a pathXZ to three dimensions by assigning a height function y = y0 or by restricting motion to a fixed altitude.

Applications include CNC machining, where G18 denotes the XZ plane and pathXZ describes toolpaths that move in X and Z while maintaining constant Y, typically used for depth-control operations. In computer graphics and simulation, pathXZ models ground contours, horizontal navigation, or autonomous robot paths constrained to a plane. In geospatial contexts, projecting 3D routes onto the XZ plane yields a pathXZ used for planning on a flat map when vertical height is ignored.

Computation and properties: The 2D length of a pathXZ is the integral of sqrt((dx/dt)^2 + (dz/dt)^2) dt. Curvature and other planar metrics follow standard 2D calculus. Representations include straight lines, circular arcs, polylines, and Bezier or spline curves defined in the x-z coordinates.

Relationship: PathXZ is related to pathXY and pathYZ as planar projections or components of more complex trajectories. It serves as a building block for planning and analysis when motion is constrained to a single vertical slice of space.

See also: XZ plane, CNC G18, Bezier curves, planar curves, polyline.