Three Visual Angles of Three Dimensional Orthogonal Axes and Their Visualization
Masahiro Mori 4asrm010@keyaki.cc.utokai.ac.jp
Yoichi Maeda maeda@keyaki.cc.utokai.ac.jp
Mathematics/Sciencs Tokai University Japan
Abstract
Depending on our viewpoint, an angle AOB in the 3dimensional 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 3dimensional 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 YZaxes, ZXaxes, and XYaxes,
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.
