Prof. Evans,
I wondered about the same thing: For epsilon=1 there is obviously a “singularity” where r -> infinity, so what distinguishes a parabola from a hyperbola, both having singularities in the equation, r = alpha/(i+epsilon*cos(x*theta))?
Attached is a combined plot for epsilon < 1, epsilon = 1, and three values of epsilon > 1 (alpha=1 and x=1 for all plots).
One important thing I did was to plot a range of theta for a large number of periods (to make sure passage thru any singularities and to show any pattern that may emerge — my plotting software has a setting to suppress divide-by-zero errors and keep going).
What I found was one difference between a parabola and a hyperbola is that the parabola has a single point per period which is singular and the value of the radius is ALWAYS POSITIVE. The singularity of the parabola can be considered an arbitrarily large POSITIVE value as well because arbitrarily large values of the radius on either side of infinity is positive.
However, plotting a multi-period hyperbola produces what at first appears to be two STRAIGHT LINES between the “other side of the universe” and r=0!
In reality, for a hyperbola, there are TWO SINGULARITIES per period, each having an associated theta value. Between these two singularity points is a range of theta which produces the ‘conventional hyperbola’ having positive radius value, but there is a range of theta producing negative radius value. Also each singularity of the hyperbola is different that of the parabola as the hyperbolic singularities are approached on one side from the positive direction but from the other side from the negative direction. This Is the reason why the ‘non-conventional’ trajectory of a hyperbola has two straight lines in polar coordinates: In Cartesian coordinates there are both (+) and (-) ‘way out there’ directions, while in polar coordinates ‘way out there’ is always positive r. If one allows negative values in the polar coordinate system, there is only one way for it to go – ‘deeper’ into the r=0 area, so a polar plot of a +/- singularity point is a straight line between r=0 and r = +infinity, but notice that polar infinities HAVE A DEFINITE VALUE OF THETA!!
I’ve made a second plot which allows negative r and one can see this. There is a surprise – each hyperbola has a loop in the negative radius area! Also, plotting for both negative and positive radius values shows with more emphasis how a hyperbola curves away in the positive radius area. The parabola appears to maintain almost straight lines which gradually spread apart as it approaches (+)infinity.
Ray Delaforce
