Dividing the SPC grid distance between two points by the average scale factor of the line that joins them gives the ellipsoid distance between the points. James Stem tells us to calculate the average scale factor of a line by applying Simpson's Rule to the scale factors of the end points and the midpoint of the line. But, how accurate is this approximation?
To get an idea, I used distances between Primary Airport Control Stations in Wilmington and Winston-Salem, NC. Here are the figures:
Over a distance of more than 296 km, the recommended SPC approximation is 1 mm different from the NGS FORWARD solution. Using the simple mean of the two endpoint scale factors is worse than using unadjusted grid distance, and adding two terms to the recommended Simpson's Rule approximation does not improve it.
Oscar Adams et al were a smart bunch!
:good:
Ok, I'll show my ignorance.
Can you expose the excel formula for the 1/6 Simpsons rule?
Here are the formulas for both Simpson's Rule computations. These give the average scale factor, so dividing Grid Distance by either of these results gives the ellipsoidal distance.
The scale factors for the intermediate points were calculated in Corpscon 6.
Simpson's Rule creates a quadratic approximation to a curve. Scale factors have elements of both quadratics and trig functions, so the rule fits the application very well.
At the risk of ...
belaboring the point, here are the results of some further work with the line from Wilmington to Winston-Salem. First, plot the scale factors for the end points and the three calculated intermediate points:
(0, 1.00001924)
(0.25, 0.99991252)
(0.50, 0.99987290)
(0.75, 0.99989956)
(1.00, 0.99999170)
Here's the graph, a fine 2nd degree figure:
Then calculate a second-degree regression equation from those points; note the R-square:
Finally, graph the equation with the plotted points:
Note that the simple average of the end points ignores the majority of the scale factors along the line. Simpson's Rule, on the other hand, weights the intermediate points more heavily than the end points.
While this is a very long line by State Plane standards, and an extreme case by any measure, it does illustrate the potential dangers of using simple averages of scale factors to determine the scale factor of a line. Since combined factors include scale factors and elevation factors, the same holds true for averages of combined factors.
All of the figures are screen shots from a (blush) TI-84 Plus. I'm sorry about that, but TI is the standard in the NC public schools, and I got this one for free at a seminar this past summer.
At the risk of ...
forgive me, but what is R^2 in this context?
Visualizing the way that the conic projection surface cuts through the ellipsoid, I can see how this makes perfect sense. If the points are generally East to West orientation in a Lambert or North to South in a Transverse Mercator, the differences may not be this substantial. However, if the points are, for example near the intersection of the conic and the ellipsoid (Southern and Northern Standard parallels in a Lambert system), there is a lot of bulge in the ellipsoid between them that will not be accounted for in the average scale factors, just as your parabolic curve suggests.
Really good discussion, Teach. Thank you.
At the risk of ...
It's the coefficient of determination, the percent of variation in data that is explained by a regression equation. Technically, it's the ratio of the variance of predicted values from a regression equation to the variance of the original data.
Because the predicted values are generated by the regression equation, they will have a smaller variance than the original data. If the equation is a perfect fit to the data, the predicted values will be the same as the original data and this ratio will be 1. If there is a poor fit, the ratio will be near 0.
For linear regressions, the square root of this number is the correlation coefficient.
In this case, R^2 = 0.99998, which means that 99.998% of the variance in the original scale factors is explained by the regression equation. That is, scale factors along a grid line are quadratic, or very, very nearly so.
There's actually a bit more going on mathematically in the Simpson's Rule application than meets the eye. If anyone is interested in a bit of calculus, we can explore that.
By the way, your publications and postings are a great source of learning for me, as I'm sure they are for many others. Please don't stop sharing.
At the risk of ...
thank you. like you, I am a student, though not nearly as well educated.
I made it through Calculus II in college, bombed Calculus III, but stayed in the class because I enjoyed mathematics and never made it to differential equations (at the time it was considered the pinnacle of mathematics, but I don't know).
Didn't Simpson's rule also work for approximating the length of a curve (or something along those lines)?
I had a great mathematics education in high school from some amazing math teachers. I think of them often and fondly as I use those skills routinely to this day. I didn't learn about determining the order of a polynomial until I helped my son with his math homework a few years ago (somehow we never covered that in any of my classes). I had no idea that subtracting the coordinates and subtracting the results until the results were equal (or nearly so), that the order of the curve could be figured. Like I said, I'm still a student, and what little I know about polynomial curve fitting is pretty rudimentary - I don't follow the matrix algebra very well.
But with that limited knowledge, I could see what you are doing there and the sample you selected is perfect because the parabolic curve is easily seen.
And thank you as well. James Stem, on pages 18 and 19, tells us in an intuitive way that scale factors do not result from linear functions. In figure 2.5, which illustrates azimuths, he shows that a "straight" geodetic line on the ellipsoid projects as a curved line on the state plane surface. That's the source of a need for an arc-to-tangent correction.
The scale factor is the vehicle that converts a geodetic line to a grid line, so, if the projected line is to curve, it must be the scale factor that makes it do so. Later, on page 50 in his discussion of the scale factor of a line, he gives us the Simpson's Rule formula.
Scale factors are not linear, but elevation factors are. So, if I wanted to calculate a combined factor for the Wilmington to Winston-Salem line, I would use Simpson's Rule to compute the scale factor, the average of the end point elevation factors to compute the elevation factor, and multiply the two to get the combined factor. Dr. Ghilani discussed this process in detail in the September 2013 issue of Professional Surveyor.
It's when we average combined factors instead of working with the pieces that we open ourselves up to errors.
At the risk of ...
Getting through Calculus II is a pretty strong credential. A few years ago, Calc II was the most-failed math course at UNC-Chapel Hill. I haven't kept up with recent statistics, but more than one of my successful AP Calculus students struggled with that course in college.
Simpson's Rule is an approximate integration formula, so it can be used to calculate the length of a curve as well as the area under a curve. There's another formula involving integrals that is used to calculate the average value of a function over a given interval.
Substituting Simpson's Rule for the integral expression in the average value formula produces the formula in James Stem's manual. Because the numerical overhead factors present in both formulas cancel out upon substitution, what remains is a very simple expression.
I just work backward from published results. I can't imagine the intellects of the guys who developed those results going forward.
