On the Conical Pappus Spiral in a Gravitational Field

Many mathematical curves are old, but their meaning can be renewed when they are read in a new language. The conical Pappus spiral is such a curve. Classically, it can be regarded as an Archimedean spiral lifted onto a right circular cone, or as curve generated by simultaneous uniform rotation and uniform motion along the cone. Its horizontal projection is an Archimedean spiral, while suitable planar projections are associated with Doppler spirals. The curve is described by the cone angle α, the planar Archimedean spiral angle β, and the conical spiral angle γ. The speed formulae reveal β and γ as velocity-ratio angles. The acceleration properties of this curve emphasize a simple but useful result: for uniform angular motion the vertical velocity can be non-zero and constant, while the vertical acceleration is zero, so all acceleration is contained in the horizontal Archimedean projection. The aim of this paper is to model the motion of a quantum particle on this conical Pappus spiral in a gravitational field. The central result is the velocity interpretation: tan γ = vφ/vg, where vφ is the horizontal projection of the braking speed caused by the gravitational field and vg is the speed along the cone generator. This turns an old equation into a compact physical language: alpha determines the cone (α = π/4), beta determines the braking in the planar projection, gamma determines the winding of the spiral in a gravitational field. This model leads to a new interpretation of the formulae using a simple mathematical language in compare with a more complicated mathematical language known in the general theory of relativity.

Keywords: Archimedean Spiral, Conical Pappus Spiral, Formulae in the General Theory of Relativity, Velocity and Acceleration in a Gravitational Field.