In Clairaut's equation p sin α = c, what does p represent?

• A. The polar radius
• B. The parallel radius
• C. The meridian radius
• D. The curvature radius

Answer: (B)

On a surface of revolution, a geodesic obeys a conserved quantity due to rotational symmetry. The product p sin α remains constant along the curve, where p is the distance from the axis of revolution to the point on the geodesic (the radius of the parallel) and α is the angle between the geodesic and the meridian. This p sin α = c relation comes from the idea that the component of motion around the axis is conserved, tied to the parallel’s radius and the path’s inclination. Therefore, p represents the parallel radius—the distance to the axis of revolution. The other radii describe different geometric notions and do not appear in this invariant.