How to calculate length of Clothoid segment? I want to calculate the length of a clothoid segment from the following available information.


*

*initial radius of clothoid segment 

*final radius of clothoid segment

*angle (i am not really sure which angle is this, and its not
documented anywhere)


As a test case: I need to find length of a clothoid(left) that starts at $(1000, 0)$ and ends at approximately $(3911.5, 943.3)$. The arguments are: $initialRadius=10000$, $endRadius=2500$, $angle=45(deg)$.
Previously I have worked on a similar problem where initial radius, final radius, and length are given. So I want to get the length so I can solve it the same way.
I am working on a map conversion problem. The format does not specify what are the details of this angle parameter.
Please help. I have been stuck at this for 2 days now.
 A: Hint:
For a unit clothoid, of parametric equations
$$x=\int_0^t\cos t^2\,dt,\\y=\int_0^t\sin t^2\,dt,$$
the direction of the curve is given by
$$\tan\theta=\dfrac{\dot y}{\dot x}=\tan t^2,$$ and this is independent of scale.
A: The basic equation for a clothoid is  $R·L = A^2$ where R is radious, L is the length from the point where R=infinite and A is a constant (a scale factor). You can also write $L=\frac{A^2}{R}$
The length along the spiral (not a segment) between two points is $L= L_2 - L_1$
The local X-axis is the line tangent at R=inf, (and then $L=0$).
The angle $\phi$ from the X-axis to the tangent at a point $r=R_i$ is $\phi=\frac{L_i}{2R_i}=\frac{L_i^2}{2A^2}=\frac{A^2}{2R_i^2}$
So, $L_2-L1= \frac{A^2}{R_2} - \frac{A^2}{R_1} = A^2(\frac{1}{R_2}-\frac{1}{R_1})$
To get A you can use the angle at one point. For example, for the second point $(r=R_2$ and $\phi=\phi_2)$ you use $A^2= \phi_2·2·R_2^2$
Note that if the first point is R=inf, $L=0$ then the required length is
$L=A^2/R_2 = \phi_2·2·R_2$
