#6 – Pseudoconics and Conic Sections
Paul N -
Hello, and welcome to week 6 of my Senior Project! If you recall, last week, I mentioned that I began learning about Non-Euclidean geometry to see if there was some way to connect it to pseudoconics. So far, I have had two ideas on how to explore this connection: my first idea was that it might be possible to apply some sort of Non-Euclidean transformation to a double cone so that a plane passing through the double cone would form a pseudoconic at the intersection. I explored this idea throughout the week this week, and while it doesn’t seem to bear any connection to Non-Euclidean geometry, I still made some interesting findings that I will be sharing in this post. The second idea that I had involved using geodesics to find the 3-dimensional surface upon which a certain “straight line” (geodesic) would produce a certain “parabola” (pseudoparabola), and I think this idea has much more potential to yield a connection between pseudoconics and Non-Euclidean geometry, but I have yet to explore this idea any further.
As for my first idea, recall from my previous post that the standard conic sections can be described by the intersection of a plane with a double-cone in 3-dimensional space. If we want to produce pseudoconics using a similar method, we will either have to change the double-cone or the plane (or, theoretically, both). For the sake of simplicity, we will keep the plane and only change the double-cone. The standard equation of a double-cone centered at (h, k, l) is the following (forgive the unusual formatting, but I am currently unable to upload images):
(z – l)2 – (x – h)2 – (y – k)2 = 0
If we want to find the (x,y) coordinates of all intersection points of the double-cone and the plane ax + b (in other words, if we want to find the equation of the intersection curve), we can plug in z = ax + b into the above equation:
(ax + b – l)2 – (x – h)2 – (y – k)2 = 0
Recall that, if we have a directrix curve y = d(x), the pseudoparabola formed by that curve can be described as two parametric equations, x(t) and y(t). Thus, if we have a directrix curve y = d(x), and if we then know x(t) and y(t), we can plug x(t) and y(t) into the above equation to find values of a, b, l, h, and k which would create the pseudoparabola produced by that directrix at the intersection of the plane ax + b and the double-cone centered at (h, k, l). This gives us a very long differential equation which only has a few solutions (only a few directrices can be inputted which satisfy the differential equation), but we can expand the possible number of directrices by generalizing the equation a little bit further:
(ax + b – l)β – (x – h)γ – (y – k)α = 0
Now, we have begun to manipulate the double-cone to be a different type of surface depending on our values for beta, gamma, and alpha. This allows for a wider range of possible directrix curves, as well as some interesting conic surfaces and planes which produce their respective pseudoparabolas, which I am unable to show here due to the fact that I am unable to upload images, but, as usual, I will link the Desmos graph associated with this topic at the end of the post. Of course, we could generalize this equation even further, but this generalization works well enough for most cases.
As I mentioned earlier, this ends up giving us a differential equation which allows us to find the equations of the conic surface and of the plane whose intersection creates the pseudoparabola resulting from a certain directrix curve. However, is it possible to find, using this equation, the directrix curve which produces a pseudoparabola which satisfies a certain form of this equation? Consider the following “simple” equation:
y = 1
Simple enough, right? This is the equation resulting from the above differential equation when beta and gamma are zero and k is equal to one. The equation for y(t), the y-component of a pseudoparabola, centered at (0,0), is as follows:
y(t) = ((d(t))2 – t2 – 2td(t)d'(t))/(2(d(t) – td'(t))
If we want to find the equation of a directrix curve which produces the pseudoparabola satisfying the equation y = 1, we would have to solve this differential equation. As of yet, I have not found an easy way to solve this differential equation, and this is only the simplest possible case! As a result, if we want to find any sort of general way to visualize the directrix curves which produce certain pseudoparabolas satisfying specific forms of the differential equation, we will have to resort to approximations and slope fields. We can easily solve the equation y(t) = 1 if we want to isolate d'(t):
d'(t) = (t2 + 2d(t) – (d(t))2)/(2t(1-d(t)))
While this does not yield an easy way to solve the differential equation, it does give us an easy way to find the slope field for the directrix curve which satisfies the differential equation. Furthermore, we can approximate a graph for the directrix curve by using Euler’s method to find a few of the points lying on or near the curve, then using a polynomial to approximate the curve going through each of these points. However, one limitation of this is that we can only approximate a small portion of the graph of the directrix curve at any time. If the spacing between iterations of Euler’s method is too large, the derivative becomes inaccurate, and even if the spacing is small, it will also produce an inaccurate curve when the derivative is very large near the point on which the solution curve is centered. Besides this limitation, however, this generally functions as a good way of approximating the slope field and the possible directrix curves which produce certain forms of the differential equation. Next week, I will begin exploring how geodesics might be able to better connect Non-Euclidean geometry with pseudoconics, but for now, I will link the graphs used in this post. I hope to see you next week!
Conic Surfaces: https://www.desmos.com/3d/imlxc9df9m
Slope Fields and Directrix Approximation: https://www.desmos.com/calculator/zffhbabsjl