This fall in our Mathematical Perspectives class I lectured on function graphs in the real projective plane . I was inspired by a graph found on page 43 of A Guide to Plane Algebraic Curves by Keith Kendig. Staring at Infinity Annotated is the PowerPoint presentation I used for the presentation for what it is worth. I used a bunch of mathematica CDF files with interactive content but what follows are still images with a brief sketch of the concepts.
The idea is to map a curve in the rectangular coordinate plane onto a sphere. This image illustrates how its done.
Mapping a Parabola onto a Sphere
The straight line goes through the center of the sphere and “etches” two curves onto the sphere as it traces along a parabola in a plane tangent to the sphere. For purposes of the projective plane the antipodal points on the two curves are “identified.”
Then the curve, one set of the mapped points, is projected onto a disk in the center of the sphere as this image roughly shows.
Sphere with the Identified Curve
Note the point at infinity. One can imagine the line tracing along the parabola going farther and farther out so that curve on the sphere eventually closes. This is the point at infinity. The disk model of the projective plane allows us to “see” the point at infinity and we can turn the sphere so that the point becomes visible.
Here is an image of the disk with the curve for a parabola with infinity at the edge of the disk.
Parabola on the Projective Plane – Infinity on the Edge
Here is an image with the sphere rotated so that the entire curve sits inside the projective plane. We can see how the parabola “goes to infinity.” This is what I meant by “staring at infinity.”
Parabola on the Projective Plane – Infinity Inside
Here is the image of the cubic equation, first with infinity on the edge and then with infinity inside.
Cubic Function – Infinity on the Edge
Cubic Function – Infinity Inside
This image compares how the second, fourth and sixth degree polynomials approach infinity in the projective plane.
Compare Even Degree Polynomials at Infinity
Here is the reciprocal function first with infinity on the edge and then a couple of views with infinity inside.
Reciprocal Function – Infinity on the Edge
Reciprocal Function – Infinity Inside
Here is a rational function, first with infinity on the edge, then inside.
Rational Function – Infinity on the Edge
Rational Function – Infinity Inside
Here is a second rational function. The first image in the infinity inside group is what got me started on all this – the image on page 43 of A Guide to Plane Algebraic Curves by Keith Kendig.
A Second Rational Function – Infinity on the Edge
A Second Rational Function – Infinity Inside
Although I didn’t do it in the lecture it seems natural to explore transcendental functions. Here is the exponential function.
The Exponential Function – Infinity on the Edge
The Exponential Function – Infinity Inside
So now the tangent function with an infinity number of vertical asymptotes. Pretty cool.
The Tangent Function – Infinity on the Edge
The Tangent Function – Infinity Inside
And finally the secant function with an infinite number of loops.
The Secant Function – Infinity on the Edge
The Secant Function – Infinity Inside
After a while the images become less surprising even predictable, but it was totally fun building the image generators in mathematica.
Math Thoughts 