A New Approach to the Plane Angle

Abstract

In distinguishing between `category of quantities' and `dimension' the author intends to reach a new understanding of the quantity plane angle. One basis is tensor algebra. Accordingly, the common angle φ (or θ) can be separated into the quantity `arc-radius ratio x' on the one hand and the quantity `angle γ' on the other, each belonging to different categories of quantities.Another basis is the mathematical concept of `embedding'. It allows the mapping of specific `categories of quantities' into a selected `dimension'. Therefore, the category of angles can be embedded either into the `dimension(-less) ratio' or into a specific `dimension angle'.The examples in this paper show that there is no need for a change in teaching mathematics if the angle is explicitly used in defining rotary-motion quantities.The author emphasizes transparency and perspicuity. He argues for an explicit use of the angle γ in rotational mechanics. Furthermore, he suggests that there should be an internationally applicable unit name (e.g. Ri, Re or Ro) for the reference magnitude `full angle Γ'.

Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI)

“…The dimensional constant rad is to be kept explicitly in all mathematical and physical equations where it appears naturally, thus giving dimensionally consistent results automatically. Unlike some previous proposals [17][18][19][20][21][22][23][24][25][26][27] the development of dimensionally consistent equations for rotational dynamics, here, retains conventional definitions (and units) of torque, angular momentum, and moment of inertia.…”

On the status of plane and solid angles in the International System of Units (SI)

“…Many formulas of Euclidean geometry become incorrect. In particular, even the refined relations (3) and (7) for the dimensionless numerical values of the angles are incorrect. In curved space, the length of the arc l(r, ϕ) will no longer be a linear function of the radius and the numerical value of the angle.…”

“…Some authors [7,8] proposed to change the relation for the plane angle, introducing into it a dimensional coefficient. In the papers by other authors [8][9][10][11][12], it was proposed to consider the plane angle as dimensional and refer it to the base quantities, and its unit, the radian, to the base SI units. In [13][14][15][16][17][18], the difficulties of the agreement of the non-dimensional status of angles and the existing equations of mathematics and physics are discussed.…”

“…But due to the fact that the GR space is locally flat, for small values of the radius r this deviation from linearity will be small. And the smaller the value of r, the more accurate are the expressions (3) and (7). In the limit r → 0, these expressions become exact…”

“…And this neither caused any inconvenience of work, nor broke the coherence of the system of units. In the work [7], for example, a variant with eight base units is suggested, which in addition to the list of seven base units also included the radian.…”