I need to calculate coordinates along a polyline made up of lines, arcs (of circles) and transition spirals (clothoids). The general case is line-clothoid-arc-clothoid-line (see attachment A1). The coordinates of the start and end points of each entity are known, the radius R of the arcs and the length of the clothoids S are also given. I have found that it is possible to calculate the value of x, the distance along the prolongation of the tangent line, and y, the perpendicular distance to a point on the clothoid for any given distance along the clothoid s. From here it is easy to calculate the coordinate of the point on the spiral using the tangent as a baseline. There is another case where the clothoid is between 2 arcs. I have not been able to find a solution for this case. See attachments A2, A3, and A4.