Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

348

ANALYTIC GEOMETRY

(18) a=p cos 4, b= -psin E, a'=~psin 4, b'= ~pcos b,
and hence,
(19) b' = a, a' =a b.
Conversely, every transformation (1), for which (19) is true
for one set of signs, can be written in one of the forms (17)
and hence is isogonal. For, if a, b, a', b' are given, satisfying
(19) for one set of signs, values of p, cos 4), and sin l can be
found, so that equations (18) hold for the same set of signs,
These values are, namely,
a -b
p= /a2 + b2, cos = sin = /
Va2 b2' 4)Va2+b2
The following theorem summaries our results.
THEOREM 2. Thie transformation (1) is isogonal when and
only when either
b' = a and a'= - b or b' =- a and a'= b.
If it is isogonal, it can be written in one of the forms (17).
Homogeneous and Non-Homogeneous Transformations. A
polynomial in x and y is homogeneous, if its terms are all of
the same degree in x and y.* Thus, the left-hand sides of formulas (6) are homogeneous polynomials of the first degree. Accordingly, a transformation of the form (6) is called a homogeneous affine transformation; and, in distinction, a transformation of the form (1), where c and c' are not both zero, a nonhomogeneous affine transformation.
Since (1) can be reduced to (6) by means of a translation, we
have the theorem.
THEOREM 3. A non-homogeneous affine transformation is the
product of a translation and the corresponding homogeneous
transformation.
* Only those terms with non-vanishing coefficients need be considered,
since a term whose coefficient vanishes has the value of 0, and 0 is not defined as having a degree. For example, if, in the polynomial x2 + 2 xy
+ ax, a has the value 0, the polynomial is homogeneous of the second
degree.