Deriving The Equation Of The Partial Parabola Tangential To A Circle In Polar Coordinates

In the original story of this series, we investigate the properties of a square spiral through the natural numbered points of the cartesian plane. In particular, we investigated the curve to which the average of all visited points converged as the length of the spiral approached infinity. We discovered that the curve was in fact an array of 4 partial parabolas arrayed about the sides of a 1/2 unit square centered on (1/4,1/4) and that the adjacent partial parabolas had the same derivatives as their adjacent parabolas at the point of adjacency and also to the unit circle that passes through the corners of the same square.

In the next story, we looked more closely at the properties of parabolas which are tangential to a circle, radius r, at circle radii elevated at ±𝛼 from the x-axis.

We showed that these parabolas are described by equations of the form:

with parameters, a and k, determined as follows:

where 𝛼 is the angle between the radius from the center of the circle to the point of tangential intersection with the parabola and r is the radius of the circle.

We stated, without proof, that the same partial parabola can also be represented with an equation in polar coordinates:

where rp is the distance from the center of the circle, to the parabola, at an elevation of 𝜃 above (or below) the x-axis, 𝛼 is the angle of the radius which intersects the circle at the point of tangential intersection with the parabola and r (or r_c in the image above) is the radius of the circle. For the rest of the article r will refer to the radius of the circle unless specified otherwise.

This article presents the derivation of the equation above and also investigates the properties of the ratio:

Derivation Of Equation in Polar Coordinates

The point (x,y) can be expressed in terms of rp and 𝜃

Because (x,y) also lies on the parabola, parameterized as above we have:

Rearranging we have:

The equation has a trivial solution where 𝜃 = 0, namely:

Solving the quadratic equation for rp, we find:

Next, we make the following substitutions:

which yields:

Multiplying top and bottom by:

which yields:

Factoring out cos²𝛼 and applying trigonometric identities, yields:

For the range of interest:

then the following are all true:

The following limits:

It is clear that the additive arm of the solution equation diverges as 𝜃→0 since the numerator approaches a finite value, 2 cos 𝛼, but the denominator approaches 0 indicating that the ratio increases arbitrarily as 𝜃→0

We know, however, that at 𝜃=0:

so we expect that the other solution must converge to this value. In other words, we expect that the following is true, although we cannot yet demonstrate this:

The truth of this statement is at least plausible since both the numerator and denominator approach the same limit (0).

So, we will put aside the divergent arm of the solution and concentrate on simplifying the remaining arm.

Multiplying numerator and denominator by:

yields:

Simplifying the numerator yields:

Factoring out (cos²𝜃 -1) yields:

Applying trigonometric identities and canceling common terms yields:

which is the equation we intended to derive.

Also, it is now readily apparent that the limit 𝜃→0 is k as expected:

Conclusion And Future Work

By expressing the coordinates of the points of parabola in polar coordinates and substituting those expressions into an equation of the parabola in the cartesian plane, we derived a quadratic equation for rp in terms of rc, 𝛼, and 𝜃. Solving that equation and discarding the divergent solutions yielded an expression for rp in terms of rc, 𝛼, and 𝜃 for the domain 0 ≤ 𝜃 ≤ 𝛼 < π/2

The curve can be extended beyond the domain -𝛼 ≤ 𝜃 ≤ 𝛼 with a somewhat clumsy expression involving use for a floor operator:

Next steps might be to find a way a more natural way to express the symmetry that does not rely on use othe f the floor operator.