What's the difference in finding the distance between two polar coordinates and two rectangular coordinate?

1 Answer
Mar 11, 2015

Hello,

  • In a orthonormal basis, the distance between A(x,y) and A'(x',y') is

d = sqrt((x-x')^2 + (y-y')^2).

  • With polar coordinates, A[t, theta] and A'[r',theta'], you have to write the relations :

x = r cos theta, y = r sin theta
x' = r' cos theta', y' = r' sin theta',

So,

d = sqrt((r cos theta - r' cos theta')^2 + (r sin theta - r' sin theta')^2 )

Develop, and use the formula cos^2 x + sin^2 x = 1. So you get :

d = sqrt(r^2 - 2 rr' (cos theta cos theta' + sin theta sin theta')+ r'^2)

Finally, you know that cos theta cos theta' + sin theta sin theta' = cos(theta - theta'), therefore,

d = sqrt(r^2 + r'^2 - 2rr' cos(theta - theta')).