A Fixed Conic Section as the Locus of the Intersection Point of the Line OK with the Initial Plane

Abstract

Two circles of different radii, tangent at point I in the initial plane, are considered. For the same geometry and the same material point P₀ of the moving plane, a family of corresponding precessional motions with different inclination angles θ is considered. For each motion, the fixed point O of the spherical motion, the spatial centre of curvature K, and the intersection point T of the line OK with the initial plane are determined. It is shown that, as θ varies, T moves along a fixed conic section. The angle θ does not change the conic section; it only selects the line through I that intersects it at the corresponding point T. The resulting equation includes the ellipse, circle, parabola, and hyperbola. In particular, if the material point P₀ lies on the circle with diameter M₂I, then T lies on the circle with diameter IM₁. When the two circle radii are equal, the general result gives T = K₀, so the previously established collinearity of O, K₀, and K follows as an immediate special case of the broader relation.