On that x-axis, there is just 1 vector
E
such that abs(
E
)=1.
The pole and the polar axis constitute the basis of the polar coordinate system.
Now, we take a point P.
On the line OP we choose an axis u.
The number t is a value of the angle from the x-axis to the u-axis.
The number r is such that
P
= r.
U
The numbers r and t define unambiguous the point P.
We say that (r,t) is a pair of polar coordinates of P.
One point P has many pairs of polar coordinates. If (r,t) is a pair of polar coordinates, (r, t + 2.k.pi) is also
a pair of polar coordinates and additionally (- r, t + (2.k+1).pi ) is a pair of polar coordinates too.
Of course, k is an integer.
Examples
Figure 1:
polar coordinates of P are (2 , 0.3) or (-2 , 3.44) or (2 , -5.98) .
..
Figure 2:
polar coordinates of P are (-1.4 , 3.6) or (1.4 , 0.46 ) or (1.4 , -5.8) .
..
Figure 3:
polar coordinates of P are (3 , 5.7) or (3 , -0.58) or (-3 , 2.56 ) .
..
The polar coordinates of the pole O are by definition (0,t) with t perfectly arbitrary.
From polar to cartesian coordinates.
A polar coordinate system is given and point P has polar coordinates (r,t). We choose a y-axis through the
pole O and perpendicular to the x-axis. So, we have cartesian axes x and y. Call (x,y) the cartesian
coordinates of P.
According to the previous definition, the cartesian coordinates of U are (cos t, sin t).