Cayley–Klein metric

Remarks

Hyperbolic case: let $Q$ be an ellipsoid and let $x$ and $y$ lie inside it. Then $a$ and $b$ are real. In an affine chart with the points in the order $a, x, y, b$ on the line, take $c = \frac{1}{2}$ and the cross-ratio $\frac{\abs{xb} \, \abs{ya}}{\abs{xa} \, \abs{yb}}$. This ratio is greater than $1$, so $d(x, y) > 0$, and $d(x, y) \to \infty$ as $y$ tends to $b$. This is the Beltrami–Klein model of hyperbolic space with curvature $-1$.

Elliptic case: if $Q$ has no real points, such as $x_0^2 + x_1^2 + x_2^2 = 0$, then $a$ and $b$ are complex conjugate, and an imaginary constant $c$ gives the real distance of elliptic geometry.

Non-example: a degenerate quadric, such as a double plane, does not fit the definition, because a line meets it in one point only. Euclidean geometry appears only as a limit of the nondegenerate cases.

Conventions for the order in the cross-ratio differ between sources; they change the sign of the logarithm, not the geometry.