A Viewpoint on the Quantity "Solid Angle"

Abstract

Properties of the quantity are explored under the hypothesis that is a dimensional quantity, as was proposed in a previous paper. This exploration proceeds by considering rotations (plane angles) compounded in space. It is shown that five degrees of freedom would be sufficient to define the location and direction of a line in 3-D space. For the purpose of dimensional analysis, it is shown that the relationship between Plane and Solid Angles can be expressed as {A2} = {SA}, but it is also shown that this relationship cannot be applied directly to the existing units of plane and solid angles. The relationships of Plane Angles to the generation of areas on spherical surfaces, particularly those of spherical triangles, are presented, and brought into the context of the definitions of units of solid angle, especially that of the SI unit, the Steradian. Two special conditions are presented under which Solid Angle and Plane Angle are by coincidence directly proportional, i.e. by rotating a second plane angle around the axis common to both the plane and the solid angle in question. Other existing or proposed units of Solid Angle are explored to test possible proportionality, or physical interpretation in terms of geometric figures on the sphere. No such interpretation is discovered. It is concluded that it would not be justified to move the Steradian into the Base Units of SI, and that the status of this unit within SI requires careful re-evaluation.

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