# Need formula for solving ellipse radius with equal spacing points

• MHB
• sharplens
In summary, the conversation discusses the calculation of radii for an oval bicycle wheel. A parametric representation of the given ellipse is suggested, along with equations to calculate the length of each spoke. It is noted that the method is involved and must be evaluated numerically. Simplified formulas and online tools are suggested to aid in the calculation process.
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.

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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?

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.

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• elliptic_spokes_equiparam.png
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• elliptic_spokes_equiangle.png
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• elliptic_spokes_equidist.png
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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. :)

sharplens said:
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 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 $r_9=2.8572$.

## 1. What is an ellipse?

An ellipse is a geometric shape that resembles a flattened circle. It is defined as a set of points in a plane, where the sum of the distances from any point on the ellipse to two fixed points (called the foci) is constant.

## 2. How is the radius of an ellipse calculated?

The radius of an ellipse is not a fixed value, as it varies depending on the direction in which it is measured. The two most commonly used methods for calculating the radius of an ellipse are the major axis method, which measures the longest distance across the ellipse, and the minor axis method, which measures the shortest distance across the ellipse.

## 3. What is the formula for solving the radius of an ellipse with equal spacing points?

The formula for solving the radius of an ellipse with equal spacing points is: r = sqrt[(a^2 - b^2) / 2], where a is the semi-major axis and b is the semi-minor axis of the ellipse.

## 4. Why is it important to calculate the radius of an ellipse with equal spacing points?

Calculating the radius of an ellipse with equal spacing points allows for the creation of a more accurate and symmetrical ellipse, which is important in many scientific and mathematical applications. It also allows for easier and more precise measurements to be taken on the ellipse.

## 5. Are there any other methods for calculating the radius of an ellipse?

Yes, there are other methods for calculating the radius of an ellipse, such as the focal distance method and the parametric equations method. These methods may be more suitable for certain situations or applications, but the formula for solving the radius with equal spacing points is generally the most commonly used method.

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