The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

36, 37, 38] PROPER AND IMPROPER POINTS

67

We shall denote an ordinary point-a proper point-by the
usual capital letter, e.g. A. An improper point-a point at
infinity-by the Greek capital letter, e.g. 2; and a point
belonging to the other class of improper points-an ideal point
-by a Greek capital letter with a suffix, to denote the line to
which the ideal point corresponds, e.g. Pc.
Thus any two lines in the hyperbolic plane determine a
pencil.
(i) If the lines intersect in an ordinary point A, the pencil
is the set of lines through the point A in the plane.
(ii) If the lines are parallel and intersect in the improper
point Q2, the pencil is the set of lines in the plane parallel to
the given lines in the same sense.
(iii) If the two lines are perpendicular to the line c, and
thus intersect in the ideal point which we shall denote by Pc,
the pencil is the set of lines all perpendicular to the line c.
~38. We now enumerate all the cases in which two points
in the Hyperbolic Plane fix a straight line and the corresponding constructions:
(1) Two ordinary points A and B. The construction of the
line joining any two such points is included in the assumptions
of our geometry.
(2) An ordinary point [A] and a point at infinity [12]. The
line A2 is constructed by drawing the parallel through A to
the line which contains Q2, in the direction corresponding to Q2.
This construction is given below in ~~ 41-3.
(3) An ordinary point [A] and an ideal point [Pr]. This line
is constructed by drawing the perpendicular from A to the
representative line c of the ideal point.
(4) Two points at infinity [l2, 2']. The line 122' is the
common parallel to the two given lines on which 2, 2' lie.
These lines are not parallel to each other or 2 and 12' would
be the same point. The construction of this line is given below
in ~ 44.
(5) An ideal point [rc] and a point at infinity [2] not lying
on the representative line c of the ideal point. The line 1Pc2
is the line which is parallel to the direction given by 2 and
perpendicular to the representative line c of the ideal point.
The construction of this line is given below in ~ 45.