SUNY Geneseo Department of Mathematics

Arc Length

Tuesday, September 27

Math 223 01
Fall 2022
Prof. Doug Baldwin

Anything You Want to Talk About?

(No.)

LaTeX Workshop

PRISM is sponsoring, and Prof. Cesar Aguilar is leading, a workshop on LaTex (the pre-eminent document preparation software for math and scientific writing).

Thursday, September 29, 12:30 - 1:30, Welles 121.

Arc Length

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

Also drawing on the derivation of the general arc length formula in yesterday’s class.

Key Ideas

(See the next section for the equations/formulas mentioned here.)

The arc length formula.

Arc length functions.

Arc length parameterization gives position as a function of distance.

What’s Going On?

Can you put into words what each of the following formulas calculates/means/represents/etc?

Arc length between 2 specific points, arc length from reference to any point, position as a function of arc length

Examples

What is the arc length of yesterday’s helix r(t) = ⟨ cos (πt), sin (πt), t ⟩; 0 ≤ t ≤ 4?

Start by figuring out r′(t) and its magnitude. To find the magnitude, notice that there’s a chance to use the trigonometric identity sin²x + cos²x = 1, which dramatically simplifies the magnitude:

Finding magnitude of R prime by using identity sine squared plus cosine squared equals 1

Once you know the magnitude of r′(t), you can either find the specific arc length from t = 0 to t = 4, or you can find the general arc length function for arc lengths from r(0) and plug t = 4 (or any other value you’re interested in) into it:

Specific arc length and arc length function are both integrals of magnitude of R prime

Thanks to having the arc length function, we can also find the corresponding arc-length parameterization. The key to finding arc length parameterizations is expressing t as a function of s, then plugging that function into the original form for r.

Using arc length function to find T in terms of S, then plugging that into parametric form for curve

(After class I added bounds for s in the arc length parameterization; calculate those by plugging the original bounds, which are t values, into the formula for s in terms of t, i.e., the original arc length function.)

Another sometimes interesting measure related to curves: curvature, or how tightly bent the curve is and in what direction.

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

We’ll probably make heavy use of Mathematica to evaluate curvature and related formulas, since they’re tedious to calculate by hand.

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