To describe a straight line r in a plane, using polar coordinates instead of the more common Cartesian coordinates, it is first necessary to establish a reference system, choosing a point to act as a pole and a polar ray with origin at the pole.
If we put the pole on the line r itself, whatever the polar ray, all the points of a half of r will have equal polar angle α, all the points of the other half will have polar angle α + π, and the polar equation of the line will simply be
r: (θ = α) V (θ = α + π) {α constant, 0≤α≤π}
This situation is analogous to that which occurs in Cartesian reference systems for the straight lines that are parallel to one of the axes.
The radial coordinate ρ can take any non-negative value.
In the more general hypothesis that the pole is outside the line r and that r form the angle γ (-π/2 ≤ γ ≤ π/2) with respect to the polar ray, the polar equation of the line can be deduced from the explicit Cartesian equation of the line, taking as pole the origin of the Cartesian plane, and as polar axis the positive direction of the x-axis.
From y = m x + q (q ≠ 0), remembering the relations between Cartesian and polar coordinates
x = ρ cos θ
y = ρ sin θ
we have
ρ sin θ = m ρ cos θ + q
Since m = tg γ, we get
Moreover
then
ρ can not be negative, then, in the expression of ρ, the denominator sin(θ-γ) must have the same sign of the numerator q. Moreover 0 ≤ θ ≤ 2π. Therefore:
if q is positive
ρ is a minimum when the denominator is a maximum, that is when sin(θ-γ)=1; then the minimum value of rho is q cosγ;
if q is negative
In this case the minimum value of ρ is -q cosγ.
In both cases the minimum value of ρ may be expressed as
The equation (8.3) can be directly used to calculate the distance of a point A(x_{A};y_{A}) from a straight line r: y = mx+q. Indeed, by translating the origin O in A, and maintaining the Cartesian axes parallel, the equation of r becomes
Let H be the orthogonal projection of A on r. We have
Examples.
Given the straight line with explicit Cartesian equation y = x + 1, we have q = 1 and γ = π/4, so its polar equation is
with
The distance of the line from the pole (ie the origin of the Cartesian plane) is given by the minimum value of ρ: .
This minimum is achieved when the denominator of the function is maximum, that is when the sine is 1
Since , we get
Given the straight line with explicit Cartesian equation y =2 x - 3, we have q = -3 and γ = arctan 2 with , then its polar equation is
with
The distance of the straight line from the pole is
The denominator is equal to 1 if
Since , we get
The following Javascript application allows the calculation of the polar equation of a line given its angular coefficient and its intercept;
it also allows the graphical representation of the line and the value of ρ given θ.
You can enter the numeric data as integers, fractions, decimals or symbols like E and P or also as values of simple functions (for example Sqrt[2], Sin[1], etc.).
The application works only if your browser allows pop-ups.