SUNY Geneseo Department of Mathematics

Arc Length

Wednesday, March 1

Math 223
Spring 2023
Prof. Doug Baldwin

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Arc Length of a Curve

Based on “Arc Length for Vector Functions” and “Arc-Length Parameterization” in section 2.3.

Key Ideas

The formula for arc length: the arc length along r(t) from r(a) to r(b) is…

L equals integral from A to B of magnitude of R prime of T

An arc length parameterization is a parametric equation for a curve in terms of distance along it (from some reference point).

Questions

Where does the book’s equation for arc length of a helix come from? The core helix equation is simple: a component proportional to t, one to sine t, and one to cosine t. The book then adds parameters to control the radius of the helix and the spacing of its turns, which makes everything look complicated. But those parameters are all constants as far as the integration is concerned, and you can mentally treat them as ugly but constant coefficients in reading the derivation. By the end, everything that depends on the actual variable, t, simplifies away before evaluating the integral.

Examples, especially a helix?

Find the length of one turn of the helix r(t) = ⟨ 3t, sin t, cos t ⟩.

To get started, realize that “one turn” means one period of the sine and cosine functions, so from 0 to 2π.

Graph of vector 3 T, sine T, cosine T, showing 1 turn between T equals 0 and T equals 2 Pi

This gives you bounds on the integral in the arc length formula. To get ready to evaluate that integral, calculate the derivative of R, and the magnitude of the derivative:

R prime equals vector 3, cosine T, negative sine T; magnitude of R prime is square root of 10

Now integrate the magnitude of r′(t)…

Integral from 0 to 2 Pi of magnitude of R prime of T equals integral of root 10 equals 2 Pi times root 10

Now see if we can find an arc length parameterization for this curve.

Start by using the arc length formula to find a function that gives the distance along the curve from some starting point t = a to any point t = u:

D of U equals integral from A to U of magnitude of R prime of T

Applying this to the example above, and using a = 0 as the starting point, gives

D of U equals integral from 0 to U of root 10, which equals root 10 times U

This gives distance in terms of u (which started as a value for t, but once we make it an argument to the distance function, it becomes just another name for t). Rewrite the equation to get u (aka t) in terms of distance, and the result is a new parameter you can replace t with to get the arc length parameterization:

U is D over root 10 means R of D is vector 3 D over root 10, sine D over root 10, cosine D over root 10

Next

How tightly curved is a curve? And what direction does it curve in?

Please read “Curvature” and “The Normal and Binormal Vectors” in section 2.3 of the textbook.

A question for next time: How do you know which formula to use?

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