Equation of Curve Using Arc Length and Two Points

[Kurani]

This is an engineering problem that involves geometry so I thought this would be the best forum to post in. I am attempting to interpolate elevations of roadway lines offset from a baseline (center line of a roadway) between two points. The coordinate system used is a combination of stations and offsets. In other words, the baseline is considered straight despite any curvature and the other lines that run alongside it are defined by perpendicular lines drawn from the baseline to the other lines. The offset is the distance in the x direction the point on the line is from the baseline. The station is the distance along the baseline. Unfortunately, the lines next to a baseline are often not parallel to the baseline (they sometimes curve) and in order to use a linear interpolation of elevations between two points, I first need to know the offset of points on the curve which requires an equation for the curve. I am given arc length and the offset and station of two points (which also means I know the chord length) on a line next to the baseline. Can anyone direct me to a method that uses arc length and two points to find the equation of an arc so that I can find the offset between any point on the curve and the baseline? Any help is appreciated.

 

[Unrest]

Are you in 2D? If not then you can't know the orientation of the curve. Even in 2D you can't know if it's concave 'in' or concave 'out'. But if you did know that, and you assume a circular arc, you could use:

arc length = r theta (arc length of a segment of a circle)
chord length = 2r²(1-cos theta ) (cosine rule)

Then solve them simultaneously to find r, the radius of curvature, and theta, the angle subtended by the two end points at a distance r from each.

 

[Franksaei]

Baseline and offsets are used better in the field. So, if you have a base line and offset, specially the offsets are in x direction, then your baseline is the y direction. Assuming it's a simple curve, you can just use the offset and base as coordinates. Hence, you can find the cord and half theta, which is the inclination of the cord from the baseline. Then you can find any part of a simple circular curve such as: T, Bc, Ec, etc. Remember the Theta you'l find is a tangential angle and must be doubled in order to find the inclination angle (Theta). Otherwise, you are going into a lot of trouble solving from the following equations:
Arc length, l=RxTheta Theta is in Radians
Chord length, C=2xRxsin(Theta/2)

 

[Kurani]

Those equations are fine. I need to write a program that calculates the arc and I am given the radius, not just the arc length, so the equations become a lot simpler. Thanks for the help.

 

[blue_raver22]

I understand the importance of finding accurate equations to represent curves in engineering problems. In this case, it seems like you are trying to find the equation of an arc using arc length and two points. This can be achieved by using the arc length formula, which relates the length of an arc to the radius of the circle it belongs to.

To find the equation of the arc, you will need to know the radius of the circle and the coordinates of the center of the circle. This can be calculated using the two known points and their offset from the baseline. Once you have the radius and center coordinates, you can use the standard equation for a circle (x^2 + y^2 = r^2) to find the equation of the arc.

Another method that can be used is the parametric equation for a circle, which involves using the angle of the arc and the radius to calculate the coordinates of any point on the arc. This can be useful if you need to find the offset at a specific point on the curve.

In both cases, it is important to keep in mind that the baseline is considered straight despite any curvature, and the perpendicular lines drawn from the baseline to the other lines will intersect at the center of the circle. This will help you accurately determine the radius and center coordinates needed for the equations.

I hope this information helps you in finding the equation of the curve and interpolating elevations between two points. It is always important to carefully consider the geometry and coordinate systems used in engineering problems to find the most accurate solutions.