Arc Length in Different Coordinate Systems
This post will deal with converting the arc length formulas in two and three dimensions from rectangular coordinates to polar (2D), cylindrical (3D) and spherical (3D) coordinates.
Two Dimensions
In , we define the arc length of a function over the interval to be
If we define a parametric function and , we observe that . Substituting this into the arc length formula yields
Getting a common denominator gives us
In polar coordinates, we know that and . If and , then we observe that
Therefore,
Three Dimensions
Let us now consider two additional coordinate systems in : the cylindrical and spherical coordinate system.
The cylindrical coordinate system is a 3D version of the polar coordinate system in 2D with an extra component for . We can slightly modify our arc length equation in polar to make it apply to the cylindrical coordinate system given that , . We start from this step:
From rectangular coordinates, the arc length of a parameterized function is
.
So from this, it follows that in cylindrical coordinates, we have arc length defined to be
To define arc length in the spherical coordinate system, we need to know first how to convert points from spherical to cylindrical coordinates (and then spherical to rectangular). I leave without proof that , , and . Since we know that , , and goes from cylindrical to rectangular, it follows that , , and takes points in spherical coordinates and converts them into points from the rectangular coordinate system.
Again we consider the arc length formula for a parametric curve in the rectangular coordinate system:
If we let , and , we now can find expressions for , and . I leave the calculations of these three to the reader, but I will, for the sake of space, write the values to each one.
Since , we have
Since , we have
Since , we have
When you plug this into the rectangular arc length formula and after a ton of painful simplifying (which I have left out to spare your sanity), we see that the arc length formula in spherical coordinates is:
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