Creativity in Mathematics

Now for the final post on inversive geometry!  I’ve been generating some fascinating images, and I’d like to share a bit about how I make them.

In order to create such images in Mathematica, you need to go beyond the geometrical definition of inversion and use coordinate geometry.  Let’s take a moment to see how to do this.

Recall that P′, the inverse point of P, is that point on a ray drawn from the origin through P such that

where denotes the distance from A to B.  Feel free to reread the previous two posts on inversive geometry for a refresher (here are links to the first post and the second post).

Now suppose that the point P has Cartesian coordinates  Points on a ray drawn from the origin through P will then have coordinates where  Thus, we just need to find the right so that the point satisfies the definition of an inverse point.

This is just a matter of a little algebra; the result is

What this means is that if you have an equation of a curve in terms of and if you substitute everywhere you see and substitute everywhere you see you’ll get an equation for the inverse curve.

Let’s illustrate with a simple example — in general, the computer will be doing all the work, so we won’t need to actually do the algebra in practice.  We’ll look at the line  From our previous work, we know that the inverse curve must be a circle going through the origin.

Making the substitution just discussed, we get the equation

which may be written (after completing the square) in the form

It is not hard to see that this is in fact a circle which passes through the point

Now we need to add one more step.  In the definition of an inverse point, we had the point being the origin with coordinates  What if were some other point, say with coordinates ?

Let’s proceed incrementally.  Beginning with a point translate to the point so that the point now acts like the origin.  Now use the previous formula to invert:

Finally, translate back:

This is now the inverse of the point about the point

So what you see in the above image is several copies of the parabola inverted about a series of equally spaced points along the line segment with endpoints and  This might seem a little arbitrary, but it takes quite a bit of experimentation to find a set of points to invert about in order to create an aesthetically pleasing image.

Of course there is another perspective on accomplishing the same task — just shift the parabolas first, invert about the origin, and then shift back.  This is geometrically equivalent (and the algebra is the same); it just depends on how you want to look at it.

Here is another image creating by inverting the parabola about points which lie on a circle.

And while we’re on the subject of inverting parabolas, let’s take a moment to discuss the cardioid example we looked at in our last conversation about inversion:

To prove that this construction of circles actually yields a cardioid, the trick is to take the inverse of a parabola about its focus.  If you do this, the tangent lines of the parabola will then invert to circles tangent to a cardioid.  I won’t go into all the details here, but I’ll outline how the proof goes using the following diagram.

Draw a line (shown in black) tangent to the blue parabola at its vertex; the inverse curves are shown in the same color, but dashed.  Note that the black circle must be tangent to the blue cardioid since the inverse black line is tangent to the inverse parabola.

The small red disk is the focus of the parabola.  Key to the proof is the property of the parabola that if you draw a line from the focus to a point on the black line and then bounce off at a right angle (the red lines), the resulting line is tangent to the parabola.  So the inverse of this line — the red dashed circle — must be tangent to the cardioid.

Since perpendicularity is preserved and the line from the focus inverts to itself (since we’re inverting about the focus), the red circle must be perpendicular to this line — meaning that the line from the focus in fact contains a diameter, and hence the center, of the red circle.  Then using properties of circles, you can show that all centers of circles formed in this way lie on a circle (shown dotted in purple) which is half the size of the black circle.  I’ll leave the details to you….

Finally, I’d like to show a few examples of using the other conic sections.  Here is an image with 80 inversions of an ellipse around centers which lie on a line segment.

And here is an example of 100 hyperbolas inverted around centers which lie on a line segment.  Since the tails of the branches of a hyperbola all go to infinity, they all meet at the same point when inverted.

So now you know how to work with geometrical inversion from an algebraic standpoint.  I hope seeing some of the fascinating images you can create will inspire you to try creating some yourself!

This week, I thought I’d go a little deeper into the subject of projective geometry (feel free to reread last week’s post for a refresher).  Why? I think this is a good opportunity to discuss the idea of an algebraic model of a geometry.  Certainly using Cartesian coordinates in the Euclidean plane makes many geometrical problems easier to approach.

So what would a coordinate system in projective geometry look like?  The most commonly used system is homogeneous coordinates.  Let’s see how they work.

The difficulty with adding the line at infinity is that we need infinitely many points in addition to those already in the plane.  Perhaps you might imagine that adding an “infinite” coordinate might do the trick, but there is an alternative — and algebraically simpler — approach.

First, think of the Euclidean plane as being the plane z = 1 in three-dimensional Cartesian coordinates, so that every Euclidean point (x, y) is given homogeneous coordinates (x, y, 1).  But we don’t want to make the z-coordinate special — we want all the homogeneous coordinates to have similar interpretations.  We accomplish this by giving each point infinitely many names — specifically, for any k ≠ 0, the coordinates (kx, ky, k) describe the same point as (x, y, 1).  Geometrically, you might think of this as saying that the line through  (x, y, 1) (except the origin) is, in some sense, the point (x, y, 1).

So if z ≠ 0, the point (x, y, z) in homogeneous coordinates is another name for the Euclidean point (x/z, y/z).  But why would you want to do this?

The punch line is that we can now use z = 0 to indicate that a point is on the line at infinity!  We still keep the convention that if k ≠ 0, the homogeneous coordinates (kx, ky, 0) describe the same point as (x, y, 0), as long as x and y are not both 0.

So our system of homogenous coordinates contains all points (x, y, z) such that not all three coordinates are 0.  Any point with z ≠ 0 corresponds to a Euclidean point, and any point with z = 0 corresponds to a point on the line at infinity.

Is it really worth all this trouble?  There are many interesting features of such a coordinate system — and I’d like to mention a few of them here, omitting the proofs.  There are many resources online that include all the details — one example is the classic Projective Geometry by Veblen and Young available free as an ebook.

Let’s start be looking at equations of lines in projective geometry.  In the Cartesian plane, we may represent a line in the form Ax + By + C = 0, where A, B, and C are not all zero.  In the projective plane, a line has the form form Ax + By + Cz = 0, where A, B, and C are not all zero.  Nice, isn’t it?  This form is also consistent with our system of having many names for a point:  if Ax + By + Cz = 0 and k ≠ 0, then also A(kx) + B(ky) + C(kz) = 0 as well. So no problems there.

But the really neat aspect of this representation is how you can use linear algebra in three dimensions to solve many problems in projective geometry.  This isn’t a post about linear algebra — so I’ll limit myself to just one example.  But in case you do know something about linear algebra, I think you should see how it can be used in projective geometry when homogeneous coordinate are used.

We’ll consider the problem of finding the intersection of two lines, say Ax + By + Cz = 0 and Dx + Ey + Fz = 0.  The form of these expressions should remind you of taking the dot product, so that we can rewrite these expressions as

Interpreting these coordinates and coefficients as vectors in three-dimensional space, we observe that the common point (x, y, z) is simultaneously perpendicular to both (A, B, C) and (D, E, F), since both dot products are zero.  So (x, y, z) can be found using the cross-product

Very nice!  Again, there are many such examples, but this is not the place to discuss them….

This algebraic model also suggests another geometric model for the projective plane besides adding a line at infinity to the Euclidean plane.  Begin with the surface of a sphere centered at the origin in three-dimensional Cartesian space, and observe that opposite points on the sphere have homogeneous coordinates that are different names for the same point in the projective plane.

So, in some sense, we have exactly twice as many points as we need — so we identify opposite points on this sphere, so that they are in fact the same point.  (You might recall a similar discussion about points in the post on spherical geometry.)  Thinking about it in another way, we might just consider the top hemisphere of the sphere with only half of the equator, including just one of the endpoints of the semicircle.

And while this model is fairly simple geometrically, it is important to point out that this does not mean that the projective plane lies inside three-dimensional space.  Once we have our hemisphere and semicircle, we have to think about it without any of the surrounding space.  This is not easy to do, but this type of thinking is necessary all the time when studying differential geometry, a topic for another time….

One last benefit to using homogenous coordinates:  it can easily be abstracted to any number of dimensions.  Do you want a projective space?  Just add a plane at infinity to three-dimensional Euclidean space!  Coordinates are easy — all points (x, y, z, w), with not all coordinates equal to 0.  When w ≠ 0, the point (x, y, z, w) corresponds to the Euclidean point (x/w, y/w, z/w), and when w = 0, the point is on the plane at infinity.

And clearly, there would be no need to stop at three dimensions — it is just as easy to create a projective space of four or more dimensions.

Finding a workable system of coordinates for a particular geometry is not always a simple matter — but finding a system that allows problems to be solved easily is often a key step to studying any type of geometry.  I hope this gives you some insight to yet another aspect of the diverse universe of so many different Geometries….

The last type of geometry I discussed was inversive geometry, which is obtained by adding a point at infinity to the Euclidean plane.  Recall that as long as we had a consistent, useful model of this extended plane, it made perfect sense to define this new type of geometry.

Today, we’re going to add a line at infinity — creating what is called a projective geometry.  There are in fact many different types of projective geometries, but let’s just try to understand one at a time….

You might remember one important property of ω, the point at infinity in inversive geometry:  it was on every unbounded curve, and in particular, on every line. We need to be a bit more specific with our projective geometry, in the following sense.  Every line will intersect the line at infinity — denoted by λ — but not every line will intersect at the same point.

Consider all lines with some given slope m.  We then add one point to λ which lies on these lines, but no other lines.  In other words, each point on λ corresponds to an infinite family of parallel Euclidean lines — since now, with the addition of λ, there are no parallel lines in the projective plane.  The point at which two parallel (in the Euclidean sense) lines intersect is determined by their slope.

How does this differ from inversive geometry?  Well, in inversive geometry, if two lines had different slopes, they intersected in two points:  the usual finite point of intersection you learned about in algebra class, as well as ω.  But in projective geometry, two lines with different slopes intersect in only one point, since they intersect the line at infinity in two different points.

What this means is that any two distinct lines always intersect in exactly one point.  See the difference?  Parallel lines (in the Euclidean sense) intersect in a point on λ, and nonparallel (in the Euclidean sense) lines intersect in the same point they did in Euclidean geometry.

Now let’s look at the dual question (recall the discussion of duality, an important concept in spherical geometry):  what about a line between two points?  Two finite points generate a line, as usual.  If one point is finite and one lies on λ, the line generated is that line through the finite point with the slope corresponding to the point on λ.  And if both points are infinite, then the line through them is just λ.

Thus we have the following duality:  two distinct lines determine a unique point, and two distinct points determine a unique line.  Again, duality is an extremely powerful concept in geometry, so the fact that points and lines are dual concepts really is a legitimate justification for thinking about projective geometry.

Projective plane geometry is a broad subject — but some of my favorite objects to look at in projective geometry are the conic sections:  ellipses, parabolas, and hyperbolas.  This is because from the point of view of projective geometry, they are, in a sense which we’ll look at right now, all the same.

This sounds odd at first reading, but follow along.  The first question we need to consider is what points on λ lie on unbounded curves.  For lines, it’s easy — it’s just the point on λ corresponding to the slope of the line.

Now what about a parabola?

You can see from the image that as points on the parabola move further away from its vertex, the tangents have slopes of ever-increasing magnitude.  Determining the precise slopes is an easy exercise in calculus — but what is important is that the tangent lines approach, in slope, the axis of the parabola.  In this case the axis is vertical, but of course the parabola may be rotated.

What this means is that the parabola intersects the line at infinity in the point corresponding to the slope of its axis.  Again, the tangent lines on either side approach this slope, but from two directions — from above and from below.

Now hold this thought while we look at a hyperbola.

Let’s see how we travel along the hyperbola.  As we move to the upper right toward the open red circle, we are moving closer to the red dashed asymptote — meaning we approach the point on λ corresponding to the slope of the red asymptote.  But here is where it becomes interesting:  as we cross the line at infinity, we come back along the asymptote toward the filled red circle on the other branch of the hyperbola!

Then we continue moving along the left branch, getting continually closer to the blue asymptote, approaching that point on λ corresponding to the slope of the blue asymptote as we pass through the blue filled circle.  Then we cross the line at infinity again, and jump to the right branch of the hyperbola toward the open blue circle.

Can you see what this means?  In projective geometry, a hyperbola does not have two separate branches.  Adding the line at infinity allows us a means to jump between the two branches.

Let’s look at these scenarios from a slightly different perspective.  Suppose that somehow, we can actually draw the line at infinity.  It’s the blue dashed line in the figure below.  Easy.

Can you see where we’re going with this?  The left circle represents an ellipse, since it is a bounded curve and cannot intersect λ.  The middle circle represents a parabola — it’s a curve which intersects the line at infinity at exactly one point, and so is actually tangent to λ.  The right circle represents a hyperbola, since it intersects the line at infinity at two points.  It may look strange from this perspective, but this right circle really does represent the hyperbola illustrated above.

To summarize:  if a conic does not intersect the line at infinity, we call it a Euclidean ellipse, if it is tangent to λ, we call it a Euclidean parabola, and if it intersects λ in two points, we call it a Euclidean hyperbola.

Because you see, in projective geometry, there is no distinguished line.  Every line is just like any other.  So, in some sense, these is just one conic section in projective geometry!

What we have looked at today is what would be called an embedding of the Euclidean plane in the projective plane.  We saw how we could just add a line at infinity with certain properties, and our plane would behave “projectively.”  But from the perspective of projective geometry, you’ve just got conic sections and lines, with possibly 0, 1, or 2 intersection points as illustrated above.  A little mind-blowing, but absolutely true.

Of course we’ve only been able to scratch the surface of this really amazing geometrical world today.  Hopefully you will be inspired to learn a little bit more about it….