Survey Line Spiral and Anti-Survey Line Spiral: Definition, Geometric Properties, and Physical Significance

This paper defines a pair of mutually reverse three-dimensional spiral structures — the Survey Line Spiral and the Anti-Survey Line Spiral. The core characteristic of the Survey Line Spiral is that every 90 degrees of rotation, the spiral rotates around the central axis and scales as a whole by a fixed factor around a fixed scaling center, while its horizontal radius grows exponentially with angle and its vertical rising speed is proportional to the current radius. The Anti-Survey Line Spiral satisfies: the horizontal radius decays exponentially with angle, and the vertical descending speed is proportional to the current radius (in absolute value). This paper provides their precise mathematical definitions, geometric properties, and physical significance. It clarifies that, away from the closure points, the spiral growth rate b is the constant ratio of the vertical speed to the tangential speed. At the closure points, the velocity components v and s vanish, and the radius is exactly r₀, which is imposed as a boundary condition. Keywords: Survey Line Spiral, Anti-Survey Line Spiral, logarithmic spiral, 90-degree flip self-similarity, three-dimensional self-similarity, boundary condition