Knowing an elliptical segment centered at major semiaxis: determine $a$ and $b$ semiaxes

Question

Given a known elliptical segment which is centered in the major semi axis $a$. We know the arc lenght ($L$), the chord length ($c$) and the height ($h$). We know that chord $c$ and minor semiaxis $b$ are parallels, chord $c$ is defined by the length between $PQ$ and of course $h$ is in the semiaxis $a$.

angle theta θ and coordinates of P and Q are unknown

¿Can we disover the semiaxis a and b?

This is needed for an archeology project where we want to know if the architecture and scultures of some round decoratives figures and parts of buildings were elliptical and the relation between a and b semiaxis.

For numerical example, one of the structures is: Arc lentgh $L = 17.10 \,{\rm cm}$ chord c =12,35cm and height h =5,25cm

Numerical approach would be appreciated too.

---

My solution derivates from two others older greats answers about chords in the site.

WORKING IN A THREE EQUATION SYSTEM

I´ve just seen this answer:

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

where the chord c would be: $c=2b\sqrt{1-(a-h/a)^2}$

Now trying this marvellous answer:

The chord length of an ellipse

where, in the last equation of the answer, by simmetry of angles : $cos(θp)=cos(-θp)=$ and $sen(θp)=-sin(-θp)$ we find that the lenght of the chord c is

$c=\sqrt{b^2*(2sin(θ))^2}$ where θ is the angle at point P

so $c=2bsin(θ)$

So we have the system now where:

$b=\frac{c}{2\sqrt{1-(a-h/a)^2}}$ and

$b=c/(2sin(θ))$ and

it´s needed a third equation for solving it

Answer 1

Only the second of the last three equations that expresses $\pmb{b}$ is wrong, because it cannot be expressed only in terms of $\pmb{c}$ and $\pmb{\theta}$. The following equation shows why your equation is wrong. $$\dfrac{c}{2\sin\left(\theta\right)}=\sqrt{a^2-e^2h\left(2a-h\right)}, \quad\text{where}\quad e=\dfrac{\sqrt{a^2-b^2}}{a}$$

Therefore, you have only two valid equations and you need to find another equation that uses the given arc length $\pmb{L}$. Since no closed form equation exists for expressing $\pmb{L}$, you are forced to resort to use numerical methods which definitely needs elliptic integral of the second kind.

Or, you can carry out measurements to obtain a set of values $\pmb{c}$ and $\pmb{h}$ for another chord, which is parallel to the chord you have shown in your sketch. If you do that, then we have five points needed to define a conic.

By the way, we have resolved the given example and obtained the following equation of an ellipse, which has $\pmb{c}\approx 12.349999$ cm, $\pmb{h}=5.25$ cm, and $\pmb{L}\approx 17.10000069$ cm. $$\dfrac{x^2}{10.327485^2}+\dfrac{y^2}{7.091228^2}=1$$

This ellipse has the following prperties. $$\pmb{a}\approx 10.327485\space\text{cm}\qquad\pmb{b}\approx 7.091228\space\text{cm}\qquad\pmb{e}\approx 0.727000912$$ $$\text{Perimeter of the Ellipse} \space \approx 55.1957656\space \text{cm}\qquad\qquad\qquad\quad\space$$ $$\text{Area of the Ellipse} \space = 230.073126\space \text{cm}^2\qquad\qquad\qquad\qquad\qquad$$

The ellipse shown in the diagram is drawn to scale. If this is not the ellipse you were expecting, let us know. In the meantime, if you can carry out measurements to obtain a set of values $\pmb{c}$ and $\pmb{h}$ for another chord, which is parallel to the chord you have shown in your sketch, we will be probably able to find an ellipse which is fitting your data better than this.

The method used has lot of guess work, which are outside the mathematical realm. That is the reason why we are reluctant to give it to you.

Answer 2

My solution derivates from two others older greats answers about chords in the site.

WORKING IN A THREE EQUATION SYSTEM

I´ve just seen this answer:

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

where the chord c would be: $c=2b\sqrt{1-(a-h/a)^2}$

Now trying this marvellous answer:

The chord length of an ellipse

where, in the last equation of the answer, by simmetry of angles : $cos(θp)=cos(-θp)=$ and $sen(θp)=-sin(-θp)$ we find that the lenght of the chord c is

$c=\sqrt{b^2*(2sin(θ))^2}$ where θ is the angle at point P

so $c=2bsin(θ)$

So we have the system now where:

$b=\frac{c}{2\sqrt{1-(a-h/a)^2}}$ and

$b=c/(2sin(θ))$ and

it´s needed a third equation for solving it