I need to write a program that will calculate coordinates along the designed rails of a proposed subway. The designed polyline representing the rail is composed of lines, arcs (circular curves) and transition spirals which are clothoids (Euler spirals). My problem concerns the clothoids. The simplest case is as follows: A tangent line to a clothoid to an arc to another clothoid to another tangent line. See the first image below. standard clothoid transition spiral
After searching the internet I found a solution for the standard case. It is possible to calculate the variables y and x which are the distance along the tangent and the perpendicular distance to a point on the spiral, defined by the length along the spiral. Using y and x it is easy to calculate the coordinate of the point on the spiral in relation to the tangent line. The given or known values for the solutions are extracted from the Autocad design drawing. They include the coordinates of the tangent points, start and end of arcs, start and end of clothoids, radiuses of arcs and lengths of clothoids. The formula used to calculate y and x are shown above. The calculation is an expansion of Fresnel integrals. A clothoid is used as a transition spiral because the radius changes linearly along the length of the spiral from infinity at the tangent point to the radius of the arc segment at the end of the spiral. My problem is with the more complicated cases the first of which is as follows: Tangent line to Spiral to Curve to Spiral to CURVE to Spiral to Tangent line. The difference between this and the first case is, how do I solve a spiral that is between two arcs? See the spiral labeled S2 below: a spiral between 2 arcs
Thank you John
September 15, 2019 Hello The only way I will be able to understand the solution is by seeing a numerical solution. Here is an example spiral between 2 circular curves. If someone has the time to do this I would be extremely grateful. I have the data in other formats but I can't post the files here. Thanks John
You have the special case where the radius of curvature of the clothoid is $0$ at one endpoint. To obtain the general case with two non-zero radii of curvature, $R_1$ and $R_2$, calculate two sets of coordinates along a clothoid, with the radius of curvature $0$ at one endpoint, $R_1$ at the other endpoint in one calculation and $R_2$ in the other calculation. Then, if you want to, you can translate the coordinate system such that the point with radius of curvature $R_1$ is at the origin and rotate it such that the tangent at that point points along the $X$ axis. The angle of rotation is the total angle subtended by the clothoid up to that point, as given in your first image (with $R=R_1$).