Finding the Arc Length in Spherical Coordinates

The powerful relationship between the abstract integral calculation and its concrete geometric result. The animated point visually confirms that when the radial distance ( $r= R_0$) and the polar angle ( $\theta=\theta_0$ ) are constant, the complex arc length integral in spherical coordinates simplifies perfectly to the formula for the circumference of the parallel circle being traced. Specifically, the animation shows the point revolving around a circle whose radius is $R_{\|}= R_0 \sin \left(\theta_0\right)$, directly demonstrating that the calculated arc length, $L= 2 \pi R_0 \sin \left(\theta_0\right)$, is nothing more than the perimeter of the physical path traced on the sphere. This illustrates how the metric tensor components correctly translate pure coordinate changes ( $\frac{d \varphi}{d t}$ ) into measurable, real-world distances (ds).

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