Write down all of the results of Universal Hyperbolic Geometry. Organize them and interpret them in terms of
Lines are parallel:
Vector is the relation between parallel lines.
Space is divided into "infinity" and "finity".
Lines are perpendicular: Pythagorean theorem, inner product
Quadratic interpolations: law of cosines A2 - 2ABcos(theta) + B2 = C2 interpolation of (A-B)2 = C2, A2 + B2 = C2, (A+B)2 = C2. Consider triangles more generally. Or consider the function: A2 + B2 - C2 which is 0 when A and B are perpendicular, and nonzero 2ABcos(theta) otherwise.
Inner product defines the equation of a line and whether points lie on it. (a1, a2, a3)(x1, x2, 1) = 0.
Oriented area, volume given by determinant.
What if we interpret the line as ax + by + cz = 0 where z=1 ?
Why is the s1s2s3 term of the third power and not the second power?