** Next:** 2 Plane Symmetries or Isometries
**Up:** 1 Coordinate Systems in the Plane
** Previous:** 1.4 Homogeneous 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 perperdicularly. Let the angle
between them be . In this system of **oblique coordinates**,
a point *P* 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 1. The first axis (*x*-axis)
is generally drawn horizontally. The case =90° yields
a cartesian coordinate system.

**Figure 1:** In oblique coordinates, *P*=(4,3),
*Q*=(-1.3,2.5), *R*=(-1.5,-1.5), *S*=(3.5,-1), and
*T*=(4.5,0). Compare Figure 1.2.1
.

Let the two oblique coordinate systems (*x*,*y*) and
(*x'*,*y'*), with angles and *'*,
share the same origin, and suppose the
*x'*-axis makes an angle with the *x*-axis. The coordinates
(*x*,*y*) and (*x'*,*y'*) of a point in the two systems are related by

*x* = (*x'* sin(-) + *y'*
sin(-*'*-))/ sin ,

*y* = (*x'* sin + *y'* sin(*'*+))/ sin .

This formula also covers passing from a cartesian system to an oblique
system and vice versa, by taking =90° or *'*=90°.

The relation between two oblique coordinate systems that differ by a translation is the same as for cartesian systems: see (1.1.4) .

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.