Intereting Posts

Finding the points of a circle by using one set of coordinates and an angle
Math Wizardry – Formula for selecting the best spell
Presentation of $D_{2n}$
Is there any geometric explanation of relationship between Integral and derivative?
Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational.
Topological spaces in which every proper closed subset is compact
Proving that for infinite $\kappa$, $|^\lambda|=\kappa^\lambda$
Quick way to find the number of the group homomorphisms $\phi:{\bf Z}_3\to{\bf Z}_6$?
Is the tangent bundle the DISJOINT union of tangent spaces?
Find a basis of $V$ containing $v$ and $w$
Prove that 3 is a primitive root of $7^k$ for all $k \ge 1$
Does the series $\sum\limits_{n=2}^\infty(-1)^n\ln\left(1+\frac{\sin n}{\ln n}\right)$ converge?
Logic, set theory, independence proofs, etc
Find shortest distance between lines in 3D
Is there any non-abelian group with the property $AB=BA$?

I want to determine the length of an arc from the ellipse in the picture below:

How can I determine the length of $d$?

- Two Circles and Tangents from Their Centers Problem
- Calculating radius of circles which are a product of Circle Intersections using Polygons
- How can I find the points at which two circles intersect?
- How to calculate the two tangent points to a circle with radius R from two lines given by three points
- Find the circle circumscribing a triangle related to a parabola
- Why can a Venn diagram for 4+ sets not be constructed using circles?

- Calculate average angle after crossing 360 degrees
- How to find the vertex of a 'non-standard' parabola? $ 9x^2-24xy+16y^2-20x-15y-60=0 $
- Finding the points of a circle by using one set of coordinates and an angle
- Convergence and closed form of this infinite series?
- Is $x^2-y^2=1$ Merely $\frac 1x$ Rotated -$45^\circ$?
- Prove that $AH^2+BC^2=4AO^2$
- Definition of an ellipsoid based on its focal points
- Find normal vector of circle in 3D space given circle size and a single perspective
- Line intersects conic at exactly one point implies the line is tangent to conic
- Circle revolutions rolling around another circle

Let $a=3.05,\ b=2.23.$ Then a parametric equation for the ellipse is $x=a\cos t,\ y=b \sin t.$ When $t=0$ the point is at $(a,0)=(3.05,0)$, the starting point of the arc on the ellipse whose length you seek. Now it’s important to realize that the parameter $t$ is *not* the central angle, so you need to get the value of $t$ which corresponds to the top end of your arc. At that end you have $y/x=\tan 50$ (degrees). And in terms of $t$ you have $y/x=(b/a)\tan t$. Solving for $t$ then gives

$$t=t_1=\arctan \left( \frac{a}{b}\tan 50 \right).$$

[note I’d suggest using radians here, replacing the $50$ by $5\pi/18.$]

For the arclength use the general formula of integrating $\sqrt{x’^2+y’^2}$ for $t$ in the desired range. In your case $x’=-a \sin t,\ y’=b \cos t$, so that you are integrating

$$\sqrt{a^2 \sin^2t+b^2 \cos^2t}$$

with respect to $t$ from $0$ to the above $t_1$. There not being a simple closed form for the antiderivative (it’s an “elliptic integral), the simplest approach now would be to do the integral numerically. This seems the more appropriate in your problem as you only know $a,b$ to two decimals, apparently.

*** When I did this numerically on maple I got about $2.531419$ for the arclength.

You can compute this as

$$d=b\,E\bigl(\tan^{-1}(a/b\,\tan(\theta))\,\big|\,1-(a/b)^2\bigr)$$

using the incomplete elliptic integral of the second kind $E(\varphi\,|\,m)$. In Mathematica-Syntax (and suitable for Wolfram Alpha) this can be written as

```
2.23*EllipticE[ArcTan[3.05/2.23*Tan[50°]],1-(3.05/2.23)^2]
```

I adapted this from this post which investigates the converse problem (given arc length, find angle) but along the way treats this direction of the problem as well. As noted there, this angle conversion will only work for the first and last quadrant. Otherwise, either adjust the angle or look at that post for an alternative formula to use in its place.

With a few more digits of precision, the answer is returned as $2.5314195265536624417$ which essentially matches both the other answers here. Of course, printing that many digits in the answer is very bad style if the input is only given to two decimals. It does show that the numerical integration by Jyrki is a bit less precise than what coffeemath did, but even he should theoretically have rounded in the other direction.

Giving a Mathematica calculation. Same result as coffeemath (+1)

```
In[1]:= ArcTan[3.05*Tan[5Pi/18]/2.23]
Out[1]= 1.02051
In[2]:= x=3.05 Cos[t];
In[3]:= y=2.23 Sin[t];
In[4]:= NIntegrate[Sqrt[D[x,t]^2+D[y,t]^2],{t,0,1.02051}]
Out[4]= 2.53143
```

- Are there an equal number of positive and negative numbers?
- Geodesics on a polyhedron
- Spectral Measures: Scale Embeddings
- Sylow 7-subgroups in a group of order 168
- Necessary condition for positive-semidefiniteness — is it sufficient?
- Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
- Differentiation under the integral sign – line integral?
- Are there infinitely many nonnegative integers not of the following four forms?
- Eigenvalues and eigenvectors in physics?
- The longest word in Weyl group and positive roots.
- Integral of a Gaussian process
- Sniper probability question
- Simple proof involving eigenvectors and eigenvalues
- Can different tetrations have the same value?
- RSA: Fast factorization of N if d and e are known