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# Polar distance

Polar distance (PD) is an astronomical term associated with the celestial equatorial coordinate system Σ(α, δ) and it is an angular distance of a celestial object on its meridian measured from the celestial pole, similar as declination (dec, δ) is measured from the celestial equator:

PD = 90° ± δ .

Polar distance are expressed in degrees and cannot exceed 90° in magnitude. An object on the celestial equator has a PD of 90°.

Polar distance is not affected by the precession of the equinoxes.

If the polar distance of the Sun is equal to the observer's latitude, the shadow path of a gnomon's tip on a sundial will be a parabola; at higher latitudes it will be an ellipse and lower, a hyperbola.

In geometry a polar distance, typically denoted r is a coordinate in polar coordinate systems (r, θ) and (r, θ, h).

The distance d between two points P1 and P2 with their polar coordinates (r1, θ1) and (r2, θ2) in two dimensional circular polar coordinate system is given by the polar distance formula:

$d = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} \cos (\theta_{1} - \theta_{2})}$.

Polar distance is used also in botany in classification of pollens.

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