Distance between Two Points in Plane in Polar Coordinates/Proof 1
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Theorem
Let $A = \polar {r_1, \theta_1}$ and $B = \polar {r_2, \theta_2}$ be two points in a polar coordinate plane
The distance $d$ between $A$ and $B$ is given by:
- $d = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \map \cos {\theta_1 - \theta_2} }$
Proof
Let $A$ and $B$ be embedded as suggested in a polar coordinate plane whose pole is at $O$.
The distance $d$ is the side $AB$ of the triangle $AOB$.
We have that:
- $OA = r_1$
- $OB = r_2$
and:
- $\theta_2 - \theta_1$ is the opposite angle to $AB$.
Hence we can use the Cosine Rule:
- $AB^2 = r_1^2 + r_2^2 - 2 r_1 r_2 \map \cos {\theta_2 - \theta_1}$
From Cosine Function is Even we have that:
- $\map \cos {\theta_2 - \theta_1} = \map \cos {\theta_1 - \theta_2}$
and the result follows.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $5$. Distance between two points in polar coordinates