2.3 Geometric Invariants

The group of homogeneous linear substitutions for coordinates of a point, fixed points; line-preserving, distance-preserving, angle-preserving, and orientation-preserving transformations

Consider the group of all homogeneous linear substitutions for the coordinates of a point in space. As we did earlier, we write these as

The nine coefficients a _ij are not at all independent of one another, because their relationships will depend on the conditions of invariance that we establish. For example, we may choose relationships that arise in the isometric orthogonal substitutions, as is the case for rotation about the origin, where x' ² + y' ² + z' ² = x ² + y ² + z ². (This says that the distance from any point to the origin is invariant under a rotation.) Substitution produces the following six relationships for the nine coefficients:

This leads to a solution for the so-called transposed linear substitutions:

Now we consider functions of these coordinate variables (perhaps defining lines, planes, curves or surfaces). We limit the discussion to homogeneous functions; for example, the linear forms f = Ax + By + Cz, and the quadratic forms

or forms of higher dimension, simultaneous systems of linear or higher forms, and so on. Given a set of points and some linear, quadratic, or higher form that is a function of these point coordinates, then any other function of the coordinates in the specified form that remains unchanged under a certain set of...