SUNY Geneseo Department of Mathematics
Friday, March 3
Math 223
Spring 2023
Prof. Doug Baldwin
(No.)
Based on “Curvature” and “The Normal and Binormal Vectors” in section 2.3.
Curvature at a point is the reciprocal of the radius of a curve’s “osculating circle” at that point.
There are multiple formulas for curvature (see the book for each).
The principal unit normal and binormal vectors: the principal unit normal is a unit vector in the direction a curve is turning at a point, and is calculated as the derivative of the unit tangent divided by the magnitude of that derivative; the binormal is the cross product of the unit tangent and principal unit normal.
How do you know which formula to use for curvature?
I prefer the || T′ || / || r′ || formula, since it’s relatively simple and easy to remember. Of the remaining 3 formulas in the book, the first isn’t very useful for calculation, and the other two are special cases (curves in 3 dimensions and curves in 2 dimensions with an explicit equation).
What is the exact definition of “osculating circle”?
Informally, it’s a circle that is tangent to a curve and “fits” exactly in the way the curve bends. I’ll check on the formal definition and explain it at the start of class Monday.
Consider the curve r(t) = ⟨ √3 cos t, cos t, 2 sin t ⟩. It’s a circle of radius 2, perpendicular to the xy plane and forming a 30 degree angle with the x axis.
Find the curvature of this curve, using Mathematica to help with the tedious parts.
One new Mathematica feature is helpful here, namely variables. Think of a variable as like memory in a calculator, i.e., it’s a place to store values for later re-use. Unlike pocket calculators, Mathematica has an unlimited amount of “memory,” and you can give each piece of it (i.e., each variable) a name to remember it by. To define a variable, use name = value, where name is the variable name and value is any expression whose value you want to store in that variable. To retrieve the value of the variable, just type its name.
You can download this notebook to see how we calculated curvature, including lots of examples of variables.
How about the principal unit normal (N)?
The last few lines of the curvature notebook show how to calculate the principal unit normal from the unit tangent vector.
Problem set 6, on arc length and curvature, is ready.
Work on it next week, grade it in the week after break.
See the handout for more information.
I’ll give you a precise definition of “osculating circle.”
Complementing what we’ve done over the last few weeks with functions with multivariable or vector outputs, start looking at functions with multiple input variables.
Please read section 3.1 in the textbook.
We’ll look at graphing multivariable functions, so bring Mathematica if you want to try it out while we talk.