Here's a pretty cubic curve: x³+xy²+y-x=0.

It has the rather curious polar representation r²+tan θ = 1.

It is asymptotic to the y-axis.

Okay, the math topic on my brain right this second is the question of whether or not it is reasonable to say a curve has multiple tangent lines at a point. I'll come right out and say I think not. It seems to me that for a point P on a curve C to have a tangent line with slope m, it must be the case that for all epsilon>0, there exists a delta>0 such that for any point Q on the curve C, if Q is within delta of P then the slope of the secant line through P and Q is within epsilon of m. This definition makes it possible to only have one tangent line.

Really, that has to be the accepted definition. But I have to go check.

In particular, the question comes up when you consider nice curves (polynomial curves, for instance) with self-crossings. I think it only reasonable to say the curve has no tangent line at the self-crossing (unless, of course, the crossings are all in the same "direction"). However, the calculus text I'm teaching out of appears to have no problem with saying the curve has multiple tangent lines at self-crossings. This latter conception leads, I think, to numerous problems, and will cause difficulties with the basic notion of differentiability. Or maybe I'm mixing up the two. Or maybe it's all just a matter of convention. I need to go check some reliable sources.

I wrote this applet to illustrate a couple facts about the tangent lines to a parabola.

First, for every tangent line to a parabola, there is another tangent line to which it is perpendicular.

Second, the set of intersections of these perpendicular tangents is a line, perpendicular to the parabola's line of symmetry. This line happens to be the parabola's directrix.

If you take any differentiable curve and form the set of intersections of perpendicular tangents, you have what is known as the orthoptic of the curve. So the orthoptic of a parabola is a line. It's pretty easy to see that the orthoptic of a circle is a concentric circle. Many other orthoptics are hard to determine.

The more general concept is the isoptic, which is the set of intersections of tangents which form a constant (not necessarily pi over 2) angle. Wikipedia doesn't say much about this topic, but I have a nice old copy of "A Handbook on Curves and Their Properties" by Robert Yates which gives a number of examples. Few seem easy to derive: with even a simple curve, the orthoptic is often quite messy.

A couple of examples, without derivation (from page 140 of Yates).

• The orthoptic of the deltoid is its inscribed circle.
• The orthoptic of the cardioid is a circle and a limacon.
• The orthoptic of y²=x³ is the parabola 729y²=180x-16.

I just wrote an applet with Processing to help my students visualize the equiangular spiral.

Here it is.

I wrote this applet for my Calculus I students to investigate, very preliminarily, the curvature of y=x² at the origin. Notice how the circle is in contact with two points of the parabola until the radius gets small enough, and then it just sits on the origin. Hmmm: what does that tell us?