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

48, 49]

THE EQUIDISTANT-CURVE

83

Thus Q corresponds to P, and as before there can only be
one point on the second line corresponding to a given point
on the first.
2. If the lines (i), (ii), and (iii) are all perpendicular to the
same straight line, then if the point Q on (ii) corresponds to the
point P on (i), and the point R on (iii) to the point Q on (ii),
the points P and R correspond.
(i) (ii) (iii)
P R
M N S
FIG. 58.
Let the common perpendicular meet the lines in M, N, and S.
Then PM =QN and QN=RS.
Therefore PM =RS, and P and R correspond.
3. The locus of corresponding points upon a pencil of lines
whose vertex is an ideal point is called an Equidistant-Curve,
from the fact that the points upon the locus are all at the same
distance from the line to which all the lines of the pencil are
perpendicular. This line is called the base-line of the curve.
On the Euclidean Plane the Equidistant-Curve is a straight
line. On the Hyperbolic Plane the locus is concave to the
common perpendicular.
This follows at once from the properties of Saccheri's Quadrilateral (cf. ~ 29). Indeed Saccheri used this curve in his supposed refutation of the Hypothesis of the Acute Angle.
We have thus been led to three curves in this Non-Euclidean
Plane Geometry, which may all be regarded as " circles."
(a) The locus of corresponding points upon a pencil of lines,
whose vertex is an ordinary point, is an ordinary circle, with
the vertex as centre and the segment from the vertex to one of
the points as radius.
(b) The locus of the corresponding points upon a pencil of
lines, whose vertex is an improper point-a point at infinity