Need formula for solving ellipse radius with equal spacing points

[sharplens]

I'm trying to build an oval bicycle wheel and need a formula to calculate the radii so that I can use the results to calculate the length for each spoke. See diagram for details. Hope someone can help.

View attachment 8645

 

[MarkFL]

Jello, and welcome to MHB! (Wave)

I think what I would do is use a parametric representation of the given ellipse:

\(\displaystyle x(t)=r_{10}\cos(t)\)

\(\displaystyle y(t)=r_{1}\sin(t)\)

And then:

\(\displaystyle r_n=\sqrt{\left(r_{10}\cos(t_n)\right)^2+\left(r_{1}\sin(t_n)\right)^2}\)

where:

\(\displaystyle t_n=\frac{\pi}{18}n\) where \(\displaystyle n\in\{1,2,3,4,5,6,7,8,9\}\)

I am assuming the angular difference between the spokes is uniform. Are you instead wanting the distance along the ellipse between spokes to be uniform?

 

[I like Serena]

Hi sharplens, welcome to MHB! ;)

Do you want equi-parametrized, which is what Mark gave the formulas for?
View attachment 8646

Or equi-angular?
View attachment 8648

Or equi-distant?
View attachment 8647

Admittedly, Mark's formulas are the simplest and most straight forward.

 

[sharplens]

I want equal distance along the arc. That's equivalent to the spoke holes on an oval bicycle rim. Thanks for the nice colorful graph. :)

 

[I like Serena]

I want equal distance along the arc. That's equivalent to the spoke holes on an oval bicycle rim. Thanks for the nice colorful graph. :)

The method is a bit involved, so please bear with me.
Or jump to the end for the results. ;)Let $a$ be the length of the longest spoke $r_{10}$.
Let $b$ be the length of the shortest spoke $r_{1}$.

First we need to find the arc length of the 90 degree angle, which is:
$$\text{arclength} = \int_0^{\pi/2} s'(t)\,dt = \int_0^{\pi/2} \sqrt{a^2\sin^2 t + b^2\cos^2 t}\,dt$$

Then we need to divide it into 9 equal parts $\Delta s$ for the 10 spokes.
$$\Delta s = \frac 19 \text{arclength}$$

Next is to find the parameters $t_i$ ($i=1,...,10$) where the spokes are.
For that we need to solve:
$$t'(s) = \frac{1}{\sqrt{a^2\sin^2 t + b^2\cos^2 t}}$$
and find the values for $t$ where $s = (i-1)\Delta s$.

And finally we can calculate the lengths of the spokes $r_i$:
$$\text{length }r_i = \sqrt{a^2\cos^2 t_i + b^2\sin^2 t_i}$$I'm afraid we can only evaluate these formulas numerically.
If I do that for $a=r_{10}=3$ and $b=r_{1}=1$, I find an arc length of $3.3412$.
And the spoke lengths are:
$$r_i = 3.0000,\ 2.8572,\ 2.5909,\ 2.2981,\ 2.0020,\ 1.7146,\ 1.4479,\ 1.2202,\ 1.0595,\ 1.0000$$

 

[I like Serena]

I found a couple of interesting articles about the Elliptic integrals of the first and second kind, and how we can deal with arc length along the ellipse here.
It means we can simplify the formulas a bit, and look up the results with online tools that are readily available.

$$\text{arclength quarter ellipse} = b\cdot E(1-\frac{a^2}{b^2})$$
where $E$ here is the so called complete elliptic integral of the second kind with parameter $m=k^2$.
The free online Wolfram|Alpha can calculate this for us with [M]b * EllipticE[1 - a^2 / b^2][/M].
Applied to the example you can see here on W|A that the result is indeed $3.3412$.To find the $t_i$ we need to solve:
$$\text{arclength}(t) = b\cdot E(t \mid 1-\frac{a^2}{b^2}) = s$$
where this time $E$ is the incomplete elliptic integral of the second kind, and $s$ is the arc length up to the spoke.

Wolfram|Alpha can do this for us with [M]FindRoot[b * EllipticE[t, 1 - a^2 / b^2] == s][/M].
Applied to spoke 9 of the example, you can see here on W|A that the result is $t_9=0.3293$.Finally, the spoke length is:
$$r_i = \sqrt{a^2\cos^2 t_i + b^2\sin^2 t_i}$$
Applied to the same spoke W|A shows us that this is indeed $r_9=2.8572$.