Say we are given a two-dimensional metric $$ds^2=f_1(x)dx^2+f_2(x)dy^2,$$ for any kind of function. How do we calculate the distance along a straight line path (not the shortest possibly) between, say, points $(0,10)$ and $(1,15)$? How do we find the equation for the trajectory?


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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Feb 17 '15 at 20:02

If by "straight line" you mean "linearly varying the coordinates from the beginning point $(x_0,y_0)$ to the endpoint $(x_1,y_1)$," then the trajectory is just that. You can parametrize it as $(x,y) = (1-\lambda) (x_0,y_0) + \lambda (x_1,y_1)$ for $\lambda$ running from $0$ to $1$. Call this path $C$. The distance is $$ \int\limits_C \sqrt{ds^2} = \int_0^1 \sqrt{f_1 \cdot \left(\frac{\mathrm{d}x}{\mathrm{d}\lambda}\right)^2 + f_2 \cdot \left(\frac{\mathrm{d}y}{\mathrm{d}\lambda}\right)^2}\ \mathrm{d}\lambda. $$


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