Tuesday, February 22, 2011

Filtering down to an answer

I am currently teaching Calculus III. On Monday, I covered quadric surfaces. For those of you who don't know, these are surfaces that are the graphs of equations that are of the second degree in x, y, and z. Aren't you glad you asked?

There are six types of such surfaces that we consider: ellipsoids, hyperboloids of one sheet, hyperboloids of two sheets, ellipsoidal cones, elliptical paraboloids, and hyperbolic paraboloids. (No, you CAN'T just have a copy of the Watchtower? Why do you ask?)

Ideally, we would like students to have the skill to know the graph of the equation just by looking at the equation. This is a skill that can be developed over time. What we do as a means learning that is to teach the students to "filter-down" to the answer by taking the traces in the xy-, yz-, and xz-planes of equations that are in standard form.

Consider the hyperboloid of one sheet that is pictured below:

Its cross-sections by the xy-, yz-, and xz-planes are an ellipse, a hyperbola, and a hyperbola, respectively. It is the only one of these surfaces that has these particular cross-sections. The geometrically-minded read should be able to convince himself of this with but little trouble. We can summarize the cross-sections of all six surfaces below:

  1. Ellipsoid: ellipse, ellipse, ellipse
  2. Hyperboloid of one sheet: ellipse, hyperbola, hyperbola
  3. Hyperboloid of two sheets: empty set, hyperbola, hyperbola
  4. Elliptical cone: single point, crossed lines, crossed lines
  5. Elliptical paraboloid: single point, parabola, parabola
  6. Hyperbolic paraboloid: crossed lines, parabola, parabola

By filtering down, I mean that every time we take a cross-section, we filter-out all but certain possibilities from the list. For example, if our first cross-section is an ellipse, we know out surface is 1 or 2. The next cross-section will determine the answer. Similarly, if a cross-section is crossed lines, then we have filter-down to 4 or 6.

This is a very parsimonious system. One can, given equations in standard form, apply it mindlessly and obtain correct answers without necessarily taking the surface into one's marrow. This is both a good thing and a bad thing.

The good is obvious. It takes a problem that is hard and transforms it to something that is a simple exercise in recognizing conic sections. (Provided you know how to recognize conic sections!) We apply a similar philosophy of filtering when we apply the first and second derivative tests in Calculus I, as we filter-out innumerable non-answers in favor of possible answers, and when we classify the Wallpaper Groups in Transformational Geometry using the presence of certain transformations to eliminate other possibilities. People like mathematics when it can take a chaotic situation and simplify it.

The bad is less obvious. The answer is so simply we might miss out on other insights. For example, one might miss that the surface exhibited above can be written as a disjoint union of lines.

I suppose the answer is to always leave room open for curiosity.

No comments: