I struggled with mapping a square grid over an area and then turning those points into Lat/Lon. Came close, but not exact. Gave up and asked Dr. Math:
(you can use the exact equations to determine error bounds in feet)
***** DO NOT REPLY VIA EMAIL *****
Replies will not be received.
See other options at the end of this message.
**********************************
Hi, Dennis.
As Dennis wrote to Dr. Math
On 01/14/2016 at 12:00:36 (Eastern Time),
>[Question]
>Hi, I'm trying to depth survey a small lake. The intent is to create
>a square grid for the survey points. I've read the archives
>concerning latitude and longitude problems. I now have the problem
>solved to my satisfaction, but remain intrigued. I'm looking for an
>analytic answer. Grid points are 50 feet apart, so a flat projection
>is easily supported. I'd like to be able to calculate how many
>degrees(a very small number) per foot there are in both the latitude
>and longitude directions.
>
>[Difficulty]
>Instead of a "distance between" problem, I'm looking at the limiting
>case of a single latitude, longitude point. At that point, how many
>degrees per foot are there in either direction?
>
>[Thoughts]
>For 41 degrees latitude, I get 2.75E-6 degrees per foot N-S
>direction.
>
>For -75 degrees longitude, I get 3.64E-6 degrees per foot E-W
>direction.
>
>These are approximations, would really appreciate the analytic
>approach to this for more precision and general interest.
ANSWER:
The mathematical basis for the calculations you want to do is found here:
Transformation between (x,y) and (longitude, latitude)
http://mathforum.org/library/drmath/view/51833.html
There you'll find the formulas:
y = R*(b2-b1)*pi/180
x = R*(a2-a1)*(pi/180)*cos(b1)
where x is the east-west separation, and y the north-south separation,
of two nearby points; b1 and b2 are the latitudes of the points; a1
and a2 are the longitudes of the points; and R is the radius of the
earth in the desired units (feet in your case).
You want to know the ratio of the separation (x and y) in feet to the
difference in angle (b2-a1 and a2-a1) in degrees. From the formulas
above we get:
North-south: (b2-b1)/y = 180/(pi*R)
East-west: (a2-a1)/y = 180/(pi*R*cos(b1))
Using R = 3956 miles = 20 887 680 feet, we have
North-south: 2.743*10^-6 degrees/foot
East-west: 2.743*10^-6/cos(latitude) degrees/foot
For your example, at latitude 41 degrees, cos(41) = 0.7547096, so the
east-west factor is 3.635*10^-6 degrees/foot. The north-south factor
is always 2.743*10^-6 degrees/foot. These match your calculations. I
don't know whether all four digits in these numbers are valid, as the
earth is not a perfect sphere, so the radius I used is an average number.
I'm curious how you arrived at your numbers without an "analytic
approach".
- Doctor Rick, The Math Forum
<
http://mathforum.org/dr.math/>