PROJECTIVE GEOMETRY COURSE

§ 7: *PM as a base for non-euclidean geometries*

Euclid included in his set of axioms the socalled fifth postulate: "Given a point *P* that's not lying on a (straight) line *l*, there's exactly one line through *P* that doesn't
intersect *l*".

In an unrefined projective geometry this postulate isn't always satisfied. However, we only call a geometry *non-euclidean* if the concepts perpendicular and distance are defined in it in a sensible
way, whilst the fifth postulate isn't satisfied.

Example 1: the *elliptic plane*. Start with the sphere model of the real projective plane (see *O6*). We call the pairs of antipodal points *elliptic points*, and the great circles
on the sphere with radius 1 *elliptic lines*.

For elliptic points *P* and *Q*, define *elliptic distance* d(*P*,*Q*) as the length of the smallest circular arc between a representative of *P* and a
representative of *Q* on the great circle containing both pair of points. In a formula:

d(*P*,*Q*) = arccos( (p_{1}q_{1}+p_{2}q_{2}+p_{3}q_{3})/(√(p_{1}^{2}+p_{2}^{2}+p_{3}^{2})
√(q_{1}^{2}+q_{2}^{2}+q_{3}^{2}))).

So we have d(*P*,*Q*) ∈ [0,π/2].

Furthermore, we define for elliptic lines *l* and *m*: *l* is perpendicular to *m* if and only if the plane of *l* in ℜ^{3} is perpendicular to the plane of *m* in
ℜ^{3}, so if l_{1}m_{1}+l_{2}m_{2}+l_{3}m_{3}=0. See *O27*.

Example 2: the *hyperbolic plane*. Consider in ℜ^{3} the cone *k* := {x_{1}^{2}+x_{2}^{2} = x_{3}^{2}} and its
interior {x_{1}^{2}+x_{2}^{2} < x_{3}^{2}}.

We define:

i) A *hyperbolic point* is a straight line though the origin which, except for its intersecting point with *k* (the origin), lies entirely within *k*.

ii) If a plane through *O* intersects the cone in two straight lines, we call the part of the plane within *k* a *hyperbolic line*.

If we consider the intersection of the hyperbolic points and lines with the plane x_{3}=1, we get the *circle model* of the hyperbolic plane:

Now define for hyperbolic points *P*:λ(p_{1},p_{2},p_{3}) and *Q*:λ(q_{1},q_{2},q_{3}) the *hyperbolic
distance* by

d(*P*,*Q*) = arccosh( |p_{3}q_{3}-p_{1}q_{1}-p_{2}q_{2}|/(√(p_{3}^{2}-p_{}^{2}-p_{2}^{2})
√(q_{3}^{2}-q_{1}^{2}-q_{2}^{2}))).

See *O29*.

Define for hyperbolic lines *l* and *m* *hyperbolic orthogonality* as follows: *l* is perpendicular to *m* if and only if
l_{3}m_{3}-l_{1}m_{1}-l_{2}m_{2}=0.

*O27* A plane through *O* in ℜ^{3} together with its normal define an elliptic line *l _{1}* and an elliptic point

a) Each elliptic line through

b)

c)

d) When

*O28* Given an hyperbolic point *P* that doesn't lie on a hyperbolic line *l*, check that there are an infinite number of hyperbolic lines through *P* that don't intersect *l*.

*O29* Prove that hyperbolic distance is welldefined and symmetrical, and that we have: d(*P*,*Q*) ≥ 0, and [d(*P*,*Q*)=0 ⇔ *P*=*Q*].

*O30* Show that the hyperbolic plane is infinitely large if we measure it in the hyperbolic way.

*O31* Consider the circle model of the hyperbolic plane.

Check by calculation that in the following picture, where *l* is the polar line of *P*, each line "through *P*"
is hyperbolically perpendicular to *l*. (Hint: first show that the equation of *l* in homogeneous coordinates is
xx_{P} + yy_{P} = zz_{P}.)