8. Polar equation of a straight line


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

Eqn001.gif

Since m = tg γ, we get

Eqn002.gif

Eqn003.gif

Moreover

Eqn023.gif

then

Eqn022.gif

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:

  1. if q is positive

    Eqn004.gif

    1. if γ<0:   Eqn005.gif
    2. if γ≥0:   Eqn006.gif

    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;

  2. if q is negative

    Eqn007.gif

    1. if γ<0:   Eqn008.gif
    2. if γ≥0:   Eqn009.gif

    In this case the minimum value of ρ is -q cosγ.

In both cases the minimum value of ρ may be expressed as

Eqn024.gif

The equation (8.3) can be directly used to calculate the distance of a point A(xA;yA) 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

Eqn025.gif

Let H be the orthogonal projection of A on r. We have

Eqn026.gif

Examples.

  1. Given the straight line with explicit Cartesian equation y = x + 1, we have q = 1 and γ = π/4, so its polar equation is

    Eqn010.gif

    with Eqn011.gif

    The distance of the line from the pole (ie the origin of the Cartesian plane) is given by the minimum value of ρ: Eqn014.gif.

    This minimum is achieved when the denominator of the function is maximum, that is when the sine is 1

    Eqn012.gif

    Since Eqn011.gif, we get Eqn013.gif

  2. Given the straight line with explicit Cartesian equation y =2 x - 3, we have q = -3 and γ = arctan 2 with Eqn015.gif, then its polar equation is

    Eqn016.gif

    with Eqn017.gif

    The distance of the straight line from the pole is Eqn018.gif

    The denominator is equal to 1 if

    Eqn019.gif

    Since Eqn017.gif, we get Eqn020.gif