16.1: Euclidean space
Recall that Euclidean space is the set \(\mathbb{R}^3\) of all triples \((x,y,z)\) of real numbers such that the distance between a pair of points \(A=(x_A,y_A,z_A)\) and \(B=(x_B,y_B,z_B)\) is defined by the following formula:
\(AB := \sqrt{(x_A-x_B)^2+(y_A-y_B)^2+(z_A-z_B)^2}.\)
The planes in the space are defined as the set of solutions of equation
\(a\cdot x+b\cdot y+c\cdot z+d=0\)
for real numbers \(a\) , \(b\) , \(c\) , and \(d\) such that at least one of the numbers \(a\) , \(b\) or \(c\) is not zero. Any plane in the Euclidean space is isometric to the Euclidean plane.
A sphere in space is the direct analog of a circle in the plane. Formally, sphere with center \(O\) and radius \(r\) is the set of points in the space that lie on the distance \(r\) from \(O\) .
Let \(A\) and \(B\) be two points on the unit sphere centered at \(O\) . The spherical distance from \(A\) to \(B\) (briefly \(AB_s\) ) is defined as \(|\measuredangle AOB|\) .
In spherical geometry, the role of lines play the great circles ; that is, the intersection of the sphere with a plane passing thru \(O\) .
Note that the great circles do not form lines in the sense of Definition 1.5.1 . Also, any two distinct great circles intersect at two antipodal points. In particular, the sphere does not satisfy the axioms of the neutral plane.