Chapter 4: Geometry

4.1.4.1 Relations between Cartesian and polar coordinates
Consider a system of polar coordinates and a system of Cartesian coordinates with
the same origin. Assume that the initial ray of the polar coordinate system coincides
with the positive Ü-axis, and that the ray ¼ Æ coincides with the positive Ý-axis.
Then the polar coordinates ´Öµ with Ö¼ and the Cartesian coordinates ´Ü ݵ of
the same point are related as follows (Ü and Ý are assumed positive when using the
ØÒ
½ formula):
´
Ü Ö Ó×
Ý Ö ×Ò
Ô
Ö Ü¾ · Ý ¾
Ý ØÒ ½ Ü
×Ò
Ó×
Ý
Ô
ܾ · Ý ¾
Ü
Ô
ܾ · Ý ¾
4.1.5 HOMOGENEOUS COORDINATES IN THE PLANE
A triple of real numbers ´Ü Ý Øµ, with Ø ¼, is a set of homogeneous coordinates
for the point È with Cartesian coordinates ´ÜØ Ýص. Thus the same point has many
sets of homogeneous coordinates: ´Ü Ý Øµ and ´Ü ¼ Ý ¼ Ø ¼ µ represent the same
point if and only if there is some real number « such that Ü ¼ «Ü Ý ¼ «Ý Þ ¼ «Þ.
When we think of the same triple of numbers as the Cartesian coordinates of
a point in three-dimensional space (page 345), we write it as ´Ü Ý Øµ instead of
´Ü Ý Øµ. The connection between the point in space with Cartesian coordinates
´Ü Ý Øµ and the point in the plane with homogeneous coordinates ´Ü Ý Øµ becomes
apparent when we consider the plane Ø ½ in space, with Cartesian coordinates
given by the rst two coordinates Ü Ý (Figure 4.4). The point ´Ü Ý Øµ in space can
be connected to the origin by a line Ä that intersects the plane Ø ½in the point with
Cartesian coordinates ´ÜØ Ýص or homogeneous coordinates ´Ü Ý Øµ.
Homogeneous coordinates are useful for several reasons. One of the most important
is that they allow one to unify all symmetries of the plane (as well as other
transformations) under a single umbrella. All of these transformations can be regarded
as linear maps in the space of triples ´Ü Ý Øµ, and so can be expressed in
terms of matrix multiplications (see page 306).
If we consider triples ´Ü Ý Øµ such that at least one of Ü Ý Ø is non-zero,
we can name not only the points in the plane but also points “at in nity”. Thus,
´Ü Ý ¼µrepresents the point at in nity in the direction of the ray emanating from
the origin going through the point ´Ü ݵ.
4.1.6 OBLIQUE COORDINATES IN THE PLANE
The following generalization of Cartesian coordinates is sometimes useful. Consider
two axes (graduated lines), intersecting at the origin but not necessarily perpendicularly.
Let the angle between them be . In this system of oblique coordinates, a
point È is given by two real numbers indicating the positions of the projections from
the point to each axis, in the direction of the other axis (see Figure 4.5). The rst
axis (Ü-axis) is generally drawn horizontally. The case ¼ Æ yields a Cartesian
coordinate system.
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