- #1

SamRoss

Gold Member

- 254

- 36

- Homework Statement
- Trying to get a better understanding of invariant distance. As an example, I want to find the distance between (3,0) and (0,3) (in Cartesian coordinates) but using polar coordinates.

- Relevant Equations
- ##{ds}^2={dx}^2+{dy}^2={dr}^2+r^2{d\theta}^2##

##{dx}^2+{dy}^2=3^2+3^2=18##

##{dr}^2+r^2{d\theta}^2=0^2+3^2*(\theta/2)^2\neq18##

I have a feeling that what I'm doing wrong is just plugging numbers into the polar coordinate formula instead of treating it as a curve. For example, I naively plugged in 3 for r even though I know the radius would change along the line from (3,0) to (0,3) (Cartesian coordinates). How can this be done correctly? Also, I am not looking to use the ##{r_1}^2+{r_2}^2-2{r_1}{r_2}cos(\theta_1-\theta_2)## formula which I know would solve the problem easily.

##{dr}^2+r^2{d\theta}^2=0^2+3^2*(\theta/2)^2\neq18##

I have a feeling that what I'm doing wrong is just plugging numbers into the polar coordinate formula instead of treating it as a curve. For example, I naively plugged in 3 for r even though I know the radius would change along the line from (3,0) to (0,3) (Cartesian coordinates). How can this be done correctly? Also, I am not looking to use the ##{r_1}^2+{r_2}^2-2{r_1}{r_2}cos(\theta_1-\theta_2)## formula which I know would solve the problem easily.