1. Chapter XII. Transformation of Space
151. Projective metric.
A 3D point (x’,y’,z’) is represented by a projection of 4D point (x,y,z,t), with
x’=x/t, y’=y/t, z’=z/t
We usually set t=1.
A quadric surface is then be represented in a homogeneous coordinate in (x,y,z,t)

2. 152. Pole and polar as to the absolute.
Theorem I. The homogenous coordinates of the point in which a line meets the plane at the infinity are proportional to the direction cosines of the line.
The equation of a line through the given finite point and having the direction cosines are

3. The infinity point (x,y,z,0) in which the line pierces the plane at infinity is given by the equations. (hint: the stereographics pole infinity)
The absolute was defined as the imaginary circle in the plane at infinity.

4. 153. Equation of motion.
Let point P be referred to a rectangular system of coordinates x,y,z,t and to a tetrahedral system Here,
is the equation of the plane at infinity t=0.
The equation connecting the two systems of coordinates are

5. Theorem I. The most general linear transformation of the form (5) that will transform the expression
are the rotations and reflections about the point (x’,y’,z’)=(0,0,0)

6. The proof will be obviously, if we use matrix representation.
It is clear that

7. 154. Classification of projective transformations.

8. The invariant points of the equation (7) are determined by the characteristic values of matrix A, namely

9. If are roots of equation (7) then the coefficient matrix is equivalent to a standard form (Jordan Normal Form) under motion(orthogonal transformations