Just a quick note about scale factors, elevation factors and combined factors.
It's probably common knowledge that multiplying a ground distance by an Average Combined Factor produces a grid distance:
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ Grid = Ground * Average Combined Factor
Using a little algebra, dividing both sides of this equation by Ground, gives us:
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ Average Combined Factor = Grid/Ground
If we know a Grid distance between two points and a Ground Distance between those same two points, then dividing the Grid Distance by the Ground Distance produces the Average Combined Factor.
Dividing both sides of the first equation by Grid and rearranging terms gives us:
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ1/(Average Combined Factor) = Ground/Grid
Concerning Scale Factors, the analogous equations are:
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿGrid = Ellipsoidal * Average Scale Factor
and
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿAverage Scale Factor = Grid/Ellipsoidal
We can also get from a Ground Distance to a Grid Distance using two steps:
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ Ellipsoidal = Ground * Average Elevation Factor
and
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ Grid = Ellipsoidal * Average Scale Factor
Noting that this could be done in a single equation:
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ Grid = Ground * Average Elevation Factor *Average Scale Factor
leads to pre-computing the Average Combined Factor as:
?ÿ ?ÿ Average Combined Factor = Average Elevation Factor * Average Scale Factor
The Scale Factor at a point is actually the ratio of two infinitesimal distances, one on the Grid and the other on the Ellipsoid. The relationship works for State Plane computations because both the Transverse Mercator projection and the Lambert Conformal projections are conformal. The point figures can be used to compute the average values needed for distance reductions.
Conformal projections work on the characteristics of similar figures, thus preserving proportional distances and exact angles over infinitesimal distances.
Lots of big words.?ÿ Now my head hurts.?ÿ Need more Diet Mountain Dew. ???? ???? ?????ÿ
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BTW, as a young lad I was reading a big book that was not typical reading for kids my age when I first encountered the word infinitesimal.?ÿ I had no a clue what it meant, so I looked it up in a dictionary.?ÿ The reason it stands out in my memory is that, in the story, a woman had just given birth to a baby boy.?ÿ Another lady present looked at the baby and marveled at his infinitesimal (potty mouth).?ÿ She was referring to his man parts being so teeny.?ÿ I was shocked!
Now HC, you know all about them differentials and infinitesimals. They're the bedrock of engineering.
And you also know that once you use them to derive the formulas, you don't have to go back to them ever again. There's no telling how much of that stuff it took to get those GNSS satellites up there and those lat/lon numbers appearing in those receivers.
But we do need to know what's been derived and how to make the best use of it. Compared to GNSS, conceptually, these factors are nothing.
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@holy-cow i wish you would have been my math teacher in school. ?ÿNice job. And ya made it relatable. Now for those big words I will have to let you have that. To much for this boy. Your write up reminded me of a math professor that was tutoring me in my first college math class. He quickly realized all the numbers and letters were scaring the heck out of and he asked me to come by his house and do some work. Labor chores. So I did and we built a fence around a small area for his wifes flower garden. I learned the Pythagorean right triangle that way. On paper I would freak and he had me doing it without thinking. Taught me percent slopes and other surveying math the same way. He was not only a math teacher but a LS and PE. I am thankful to him to this day. Because I was about to go in another direction. It still took me years and I still struggle with math formulas but I don??t give up i just keep reading and trying to figure out how to apply it and relate it then the dim light bulb flickers a bit.
Maybe 20 years ago I had as a client a young man who had worked for me for quite some time prior to his survey.?ÿ He was to buy an exact number of acres on which he would place a doublewide.?ÿ We were to use the north section line as one constraint.?ÿ The other was a straight, but oblique, former railroad right of way.?ÿ I told him this was an algebra problem that he could solve.?ÿ The west line was to be perpendicular to the north section line and the south line was to be parallel with the north section line.?ÿ A wedge shape.?ÿ Reminded him that one acre was 43,560 square feet.?ÿ Told him to tell me where the corners needed to be.?ÿ The only data I provided to him was the relative angle between the north section line and the railroad line.
He had excelled at math in school and used that to acquire his degree in Economics.?ÿ His brother was a Junior High math teacher.?ÿ They both chewed on this problem for a bit before finding the solution.?ÿ We are all handed problems to solve in our math classes.?ÿ Recognizing a real world problem and turning it into a solvable equation is quite a different circumstance.?ÿ This is why educators need to learn how to flip that switch in the brains of their students.?ÿ Memorizing equations is a minor step ins true learning.
Good story and very true overall. It's harder to go from the map to the ground or from the ground to the map than it seems.
Math is abstract, problems are concrete, and fitting the two together is the field of applied mathematics. But much of our mathematics came about to solve problems that already existed. Probability and the statistics it spawned has roots in games of chance. Calculus has old roots in the area of circles and "newer" roots in finding the volumes of wine casks and how much of earth's gravity reaches to the moon.
I used to tell my students that the problem of communicating from a camp site to home existed in the time of the caveman, but it was only recently solved with cell phones.
