Three Visual Angles of Three Dimensional Orthogonal Axes and Their Visualization

Abstract

Depending on our viewpoint, an angle AOB in the 3-dimensional Euclidean space changes its “visual angle” from 0 to Pi in radian. More precisely, the visual angle of AOB from the viewpoint V is defined as the dihedral angle of the two faces AOV and BOV of the tetrahedron OABV. In this paper, we will discuss the relations among three visual angles of 3-dimensional orthogonal axes. The following property is very important in our study: Proposition Let A,B,and V be three points on the unit sphere centered at O. The visual angle of AOB from the viewpoint V is equal to the angle V of the spherical triangle ABV. The main result of our study is the next theorem. Theorem Let a, b, and c be the visual angles of YZ-axes, ZX-axes, and XY-axes, respectively, i.e.,a=Y’VZ’,b=Z’VX’, and c=X’VY’ where X’=(1,0,0), Y’=(0,1,0), Z’=(0,0,1), and the viewpoint V=(x,y,z). If xyz is not equal to 0, then these visual angles are given as Tan a=-x/(yz), Tan b=-y/(xz), Tan c=-z/(xy). As a corollary, the tangent values of three visual angles are classified into the following four cases: 1. All are positive. 2. All are negative. 3. One is equal to 0, and the others are equal to infinity. 4. One is equal to infinity, and the others are indefinite. Finally, we will realize these visual angles on the plane using stereographic projection. Stereographic projection is conformal, hence we can realize an angle on the unit sphere as the angle on the plane. We will construct it along with the dynamic geometry software Cabri II Plus.