Talk:Beltrami–Klein model

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Moved from the article[edit]

I have moved from the article the following text which appears to be an essay with unclear purpose and some errors. Arcfrk (talk) 02:07, 28 March 2009 (UTC)

Angles and orthogonality[edit]

Given two intersecting lines in the Cayley–Klein model, which are intersecting chords in the unit disk, we can find the angle between the lines by mapping the chords, expressed as parametric equations for a line, to parametric functions in the Poincaré disk model, finding unit tangent vectors, and using this to determine the angle.

We may also compute the angle between the chord whose ideal point endpoints are u and v, and the chord whose endpoints are s and t, by means of a formula. Since the ideal points are the same in the Cayley–Klein model and the Poincaré disk model, the formulas are identical for each model.

If both chords are diameters, so that v=-u and t=-s, then we are merely finding the angle between two unit vectors, and the formula for the angle \theta is

\cos(\theta) = u \cdot s.

If v=-u but not t=-s, the formula becomes, in terms of the wedge product,

\cos^2(\theta) = \frac{P^2}{QR},


P = u \cdot (s-t),
Q = u \cdot u,
R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t)

If both chords are not diameters, the general formula obtains

\cos^2(\theta) = \frac{P^2}{QR},


P = (u-v) \cdot (s-t) - (u \wedge v) \cdot (s \wedge t),
Q = (u-v) \cdot (u-v) - (u \wedge v) \cdot (u \wedge v),
R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t).

Using the Binet–Cauchy identity and the fact that these are unit vectors we may rewrite the above expressions purely in terms of the dot product, as

P = (u-v) \cdot (s-t) + (u \cdot t)(v \cdot s) - (u \cdot s)(v \cdot t),
Q = (1 - u \cdot v)^2,
R = (1 - s \cdot t)^2.

Determining angles is greatly simplified when the question is to determine or construct right angles in the hyperbolic plane. A line in the Poincaré disk model corresponds to a circle orthogonal to the unit disk boundary, with the corresponding Cayley–Klein model line being the chord between the two points where this intersects the boundary. The tangents to the intersection at the two endpoints intersect in a point called the pole of the chord. Any line drawn through the pole, which is the center of the Poincaré model circle, will intersect the Poincaré model circle orthogonally, and hence the line segments intersect the chord in the Cayley–Klein model, which corresponds to the circle, as perpendicular lines.

Restating this, a chord B intersecting a given chord A of the Cayley–Klein model, which when extended to a line passes through the pole of the chord A, is perpendicular to A. This fact can be used to give an easy proof of the ultraparallel theorem.