because many trig functions use the angle as the argument, while hyperbolic func-
tions generally do not).
But these equations are not exactly the same. In particular, when one has a prod-
uct of two sines it is replaced by minus a product of the two sinh’s. For example, for
the trig term sin^2 , the hyperbolic identity uses sinh^2 (this is called Osborn’s rule).
This is not always straightforward because the minus sign is often hidden.
OTHER GEOMETRIES
What is projective geometry?
Projective geometry is a branch of geometry that deals with the properties of geomet-
ric objects under projection; it was formerly called “higher” or “descriptive” geometry.
Projection includes the transformation of points and lines in one plane onto another
plane. This is done by connecting the corresponding points on the planes using paral-
lel lines—similar to shining a light on a person’s profile and producing a silhouette on
a nearby wall. Each point on the wall is a “projection” of the person’s head.
What is non-Euclidean geometry?
Non-Euclidean geometry is a branch of geometry that deals with Euclid’s fifth postu-
late—the “parallel postulate” that only one line is parallel to a given line through a
given external point. This postulate is replaced by one of two alternative postulates.
The results of these two alternative types of non-Euclidean geometry are similar to
those in Euclidean geometry, except for those propositions involving explicit or
implicit parallel lines. (For more information about non-Euclidean geometry, see
“History of Mathematics.”)
What are hyperbolicand elliptical geometry?
Hyperbolic and elliptical geometry are the non-Euclidean alternative geometries men-
tioned above. The first alternative is to allow two parallels through any particular exter-
nal point—or hyperbolic geometry. This studies two rays extending out in either direc-
tion from a point P,and not meeting a line L;thus, the rays are considered to be parallel
to L. This also helps prove the theorem that the sum of the angles of a triangle is less
than 180 degrees. It is called hyperbolic because a line in the hyperbolic plane has two
points at infinity; this is similar to drawing a hyperbola that has two asymptotes.
The second alternative, called elliptical geometry, has no parallels to a given line L
through an external point P. In addition, the sum of a triangle’s angles is greater than
180 degrees. Sometimes called Riemann’s geometry (who developed the idea even fur-
ther; see below), it is called elliptic in general because a line in the plane of this geom-
etry has no point at infinity (where parallels may intersect it), which is similar to an 205