Just to close out my particular contributions to this topic, I'd like to develop the proper equations for circular arc curves on a grade. We start with the three equations we developed to start the Rigid-Body form, x’, y’, and z’, except we can drop the “prime” embellishment because we won’t need it.
x = -R (1 - cos ϴ)
y = 0
z = R sin ϴ
We’re going to modify the second equation for y to reflect a constant grade. So far, it represents the projection of the curve in the x-z plane.
First, we need to develop some notation to describe the grade. As you know, a grade is defined as a ratio, rise / run. Let’s call that fraction G. The rise at arbitrary point P is just y. The run at P is just R ϴ meters, but that’s only true when ϴ is expressed as radians. As well, the total run is R ϴ
T when ϴ
T is expressed as radians. Of course, R and ϴ
T are what MSTS uses to characterize circular arc curves.
Let’s take a couple of examples illustrating different ways of specifying grades. First up is the MSTS way, using the grade angle in degrees. Let’s consider 3°. G = rise / run = tan 3° = 0.0524.
Another way is as a percentage, say 3%. G = 0.03. It’s that simple!
Now let’s develop the function that generates the change in elevation, y, as a function of parameter ϴ. We know that at the end of the curve y
T = G R ϴ
T. Note that factor G, the grade parameter, scales R ϴ
T, the length of the curve in meters (in the horizontal plane).
But what we want is a linear increase in y as we progress around the curve. We can achieve that by weighting the arc length progressed with respect to the total arc length –
Note that the two factors (R ϴ
T) cancel out, leaving:
y = G R ϴ
As simple as that!
Plugging this function into our set of three equations:
x = -R (1 - cos ϴ)
y = G R ϴ
z = R sin ϴ
Note that these three equations are a lot simpler than the rigid-body equations, perhaps even elegant.
Let’s see if they yield correct results. Here’s a plot of the same fashion as before. I plotted points and lines so that you could see the plot interval –
Helix.gif (4.09K)
Number of downloads: 3
Note that the slope of the points is the same throughout, and, overall, the curve achieves its full specified elevation. And there is no twist, unless superelevation is superimposed.
Q.E.D.