Question

Is this a valid way to change the arc length formula by changing the angle between the coordinate axis?

Answer 1

Just based on the law of cosines.

\[ds=\int\limits_{a}^{b}\sqrt{(\frac{ dx }{ dt })^2+(\frac{ dy }{ dt })^2-2\frac{ dx }{ dt }\frac{ dy }{ dt }\cos \theta} *dt\]

Answer 2

See how the curve is almost the length of the hypotenuse of our triangle?
So we can say:\[\large \Delta s^2 \approx \Delta x^2+\Delta y^2\]
In the limit, as our arc length approaches 0, this relationship becomes exact.\[\large ds^2=dx^2+dy^2\]

From there we get the form,\[\large ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}\; dx\]