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Transformation of rectangular coordinates

y

y M
ρ

φ ρ
. . . . . . . . . .
0 x x

Figure 6.3 – Polar and rectangular system

6.2. Transformation of rectangular coordinates

When solving many of tasks of analytic geometry, along with rectangular system of coordinates
other rectangular systems could be considered and coordinates of points will no change. Thus,
the task arises: knowing the coordinates of points in one system, find the coordinates of the
same point in other system of coordinates.
Let’s consider two types of transformation of rectangular coordinates: parallel shift of axes,
when position of the beginning of coordinates changes, and direction of axes remains the same,
and turn of axes of coordinates, when both axes return to the same angle, and beginning of
coordinates remains without a change.

6.2.1. Parallel shift of axes
Suppose, there is point M (x; y) in system Oxy.

y y ′

y M y  M y ′ M(x; y)

y ′
O ′ y  M x ′
b | {z }
O ′ x ′ x ′

. . . . . . . . . . . . . . . . . . . . . . O x ′ M x ′
O a x x

Figure 6.4 – Parallel shift of axes

Let’s transfer the beginning of coordinates in point O(a; b), where a and b are coordinates
of a new beginning in the previous (old) system of coordinates. New axes Ox and Oy have he
′
′
′ ′
same direction as the old axes Ox and Oy . Let’s mark coordinates of point M in system Ox y
′
′
as (x ; y ) (fig. 6.4) and get obvious equalities:
′
′
′
′
′
x = OM x = OO + O M x = OO + O M x = a + x ,
x
x
x
′
′
′
′
′
y = OM y = OO + O M y = OO + O M y = b + y .
y
y
y
Thus,
x = x + a, y = y + b or x = x − a, y = y − a. (6.3)
′
′
′
′
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