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Unable to shift coordinates in python

I have problems with shifting coordinates by 100 meters horizontaly/verticaly in Python. I found that If your displacements aren't too great (less than a few kilometers) and you're not right at the poles, use the quick and dirty estimate that 111,111 meters (111.111 km) in the y direction is 1 degree (of latitude) and 111,111 * cos(latitude) meters in the x direction is 1 degree (of longitude). here: https://gis.stackexchange.com/questions/2951/algorithm-for-offsetting-a-latitude-longitude-by-some-amount-of-meters
I wrote two functions in Python. This function works, that is it computes the shift in latitude:
def vertikalne(shift):
return shift/111_111
Check that distance between (49.550586, 18.859254) and (49.550586 + vertikalne(100), 18.859254) is really 100 meters.
But this function does not work and I don`t know what is the problem:
import math
def horizontalne(latitude, shift):
return (shift/111_111) * math.cos(latitude*math.pi/180) # UPDATE: I just converted to radians
Distance between (49.550586, 18.859254) and (49.550586 , 18.859254 + horizontalne(49.550586 , 100)) is 49 which is nonsense.
Can you help me please?

Correct solutions is:
import math
def horizontalne(latitude, shift):
return shift/(111_111 * math.cos(math.radians(latitude)))

Change math.cos(latitude) to math.cos(math.pi/180*latitude) as the math.cos() expects angles in radians and not in degrees.

math.cos() takes radians as input not degrees. You need to convert.
def horizontalne(latitude, shift):
return (shift/111_111) * math.cos(2*math.pi()*latitude/360)
something like this.
And if you want to calculate distances you need to apply Pythagoras' theorem. Maybe even take the average of the latitudes for the cos.
--
Mine is wrong. #vojtam is correct. (parentheses...)
It is rather unclear what the function is supposed to do. Maybe better to give your params more meaningful names. e.g. shift_distance, shift_long, shift_lat etc.
I think it is supposed to do the following:
The functions both take a distance as input and as output it gives the fraction of 1 degree of the arc of the intended circle around the earth. (for vertikalne it is the meridian, for horizontalne it is the parallel at the given latitude. (https://www.britannica.com/science/latitude )
111_111 meters is indeed more or less the distance over the earth of 1 degree. (at least along a meridian, the equator is slightly larger....)
So your evaluation should be different:
A = (49.550586, 18.859254) # lat,long
B = (49.550586, 18.859254 + horizontalne(49.550586 , 100)) # B is 100m east of A along the parallel at lat 49.550586
(B[0] - A[0], (B[1] - A[1]) )
gives:
Out[42]: (0.0, 0.0005838993787818936)
Which means: moving 100 meters horizontally along the earth's parallel at latitude 49.55... means changing the longitude angle for 0.0005838993787818936 of a degree. Makes sense to me.

Dividing City by Longitude and Latitude into Matrix

How do you divide a city such as San Francisco into equally sized blocks according to longitude and latitude coordinates? The aim of this is when I receive a coordinate of a location in the city I want to assign it automatically into one of these blocks.

Here's a simple use of the publicly available python library module Geohash (https://github.com/vinsci/geohash).
A python hashmap (dictionary) is used to map Geohash strings to lists of lat/lng points which have been added in the area ("block") represented by the Geohash. A query is performed by computing a position's Geohash and accessing the hashmap for that entry to retrieve all of the entries for that "block".
### Add position to Geohash
### Each hash mapping contains a list of points in the "block"
###
def addToGeohash(m, latitude, longitude):
p = (latitude, longitude)
ph = encode(p[0],p[1],5)
if ph not in m:
m[ph] = []
m[ph].append(p)
return
### Get positions in same "block" or empty list if none found
###
def getFromGeohash(m, latitude, longitude):
p = (latitude, longitude)
ph = encode(p[0],p[1],5)
print ("query hash: " + ph)
if ph in m:
return m[ph]
return []
### Test
m = dict()
# Add 2 points in general area (1st near vista del mar)
addToGeohash(m, 37.779914,-122.509431)
addToGeohash(m, 37.780546,-122.366189)
# Query a point 769 meters from vista del mar point
n = getFromGeohash(m, 37.779642,-122.502993)
print ("Length of hashmap: "+str(len(m)))
print ("Contents of map : "+str(m))
print ("Query result : ")
print (n)
The default precision is 12 characters (5 was used in example) and will affect the dictionary mapping efficiency or "block" size.
Note that using a lat/lng based approach is a non-linear and so over vast areas or closer to poles would not be "equal sized blocks". However,
over the area of San Fransisco and with sufficient precision, this non-linearity is reduced significantly.
Output
query hash: 9q8yu
Length of hashmap: 2
Contents of map : {'9q8yu': [(37.779914, -122.509431)], '9q8yz': [(37.780546, -122.366189)]}
Query result :
[(37.779914, -122.509431)]
To get a sense of the block size for various precisions use this link and enter for example 37.779914,-122.509431 and a precision of 5. Experiment with precision.
Here are the approximate box sizes for precisions 5-8:
5 ≤ 4.89km × 4.89km
6 ≤ 1.22km × 0.61km
7 ≤ 153m × 153m
8 ≤ 38.2m × 19.1m
An interesting feature of the Geohash which works mostly (and always with the SanFran area) is you can easily find the 8 adjacent neighbors by manipulating the last character. So, with minimal effort you could use a higher precision (e.g. 8) and reduce it to a size between 7 and 8.

the naive approach is to find a big enough rectangle around the whole city (area you want to cover) and by the number of blocks desired you can deduce to how many parts divide the rectangular edges, it should be a fairly basic math
given a point you can assign it to it's block in a very fast way (just check it's lan and long see where it falls in the grid)

I've written different versions of this algorithm over the years. Spent a lot of time mulling over the problem. There are two issues involved:
Mapping 2-dimensional coordinates into a 1-dimensional value. (Fundamentally, in order to do this optimally, you must know the bounds of both dimensions)
Cutting the plane covering a sphere roughly into "squares". (The squares are different sizes as you get closer to the poles. Also, they're never actually squares, but the earth is so huge this usually doesn't matter)
This is some code that I did that suited my purpose. A few things to consider:
This creates blocks of a set size that you designate. The python Geohash library does this cool thing where the hash can be truncated to produce hashes of larger sizes. This algorithm does not do that, but on the flip side you can specify your desired block size (roughly).
This actually creates two-dimensional coordinates that I treat as a 1-dimensional string. I do this because its good enough for me and I can easily manipulate the string data to get the adjacent blocks in a way that is clear and makes sense. For example the hash: "-23,407" is 23 blocks west and 407 blocks north of the center point. So if you want to move one block to the east, you just add 1 to -23: "-22,407".
The center point that I used here is the middle of Washington DC. You could use center point 0,0, or the middle of San Francisco or whatever. But do not use center points near the poles: -90,-180. because when the algorithm goes to calculate the longitude offset in kilometers, it will calculate the distance between (-90, your-longitude) and (-90, -180). These points are at the south pole (I think the south pole?) and the distance will be infinitesimally small because at the poles all of these blocks are extremely small.
This is the main algorithm to hash the points:
# Define center point
CENTER_LAT = 38.893
CENTER_LNG = -77.084
def geohash(lat, lng, BLOCK_SIZE_KM=.05):
# Get Latitude Offset
lat_distance = haversine_km(
(CENTER_LAT, CENTER_LNG),
(lat, CENTER_LNG)
)
if lat < CENTER_LAT:
lat_distance = lat_distance*-1
lat_offset = int(lat_distance/BLOCK_SIZE_KM)
# Get Longitude offset
lng_distance = haversine_km(
(CENTER_LAT, CENTER_LNG),
(CENTER_LAT, lng)
)
if lng < CENTER_LNG:
lng_distance = lng_distance*-1
lng_offset = int(lng_distance/BLOCK_SIZE_KM)
block_str = '%s,%s' % (lat_offset, lng_offset)
return block_str
I included these helper functions for calculating the distance between two coordinates:
def haversine_km(origin, destination):
return haversine(origin, destination, 6371)
def haversine(origin, destination, radius):
lat1, lon1 = origin
lat2, lon2 = destination
lat1 = float(lat1)
lon1 = float(lon1)
lat2 = float(lat2)
lon2 = float(lon2)
dlat = math.radians(lat2-lat1)
dlon = math.radians(lon2-lon1)
a = math.sin(dlat/2) * math.sin(dlat/2) + math.cos(math.radians(lat1)) \
* math.cos(math.radians(lat2)) * math.sin(dlon/2) * math.sin(dlon/2)
c = 2 * math.atan2(math.sqrt(a), math.sqrt(1-a))
d = radius * c
return d

Distance between point and arc in 3D

I want to compute the distance between an arc and a point in a 3D space. All I found is the distance between a circle and a point link (which is either wrong, or where I made a mistake, as I get wrong values):
P = np.array([1,0,1])
center = np.array([0,0,0])
radius = 1
n2 = np.array([0,0,1])
Delta = P-center
dist_tmp = np.sqrt( (n2*Delta)**2 + (np.abs(np.cross(n2, Delta))-radius)**2 )
dist = np.linalg.norm(dist_tmp)
I have a circle in the x-y-plane with origin at x-y-z = 0 and radius = 1. The point of interest is in distance 1 above the circle. The result of the distance from the code is 1.73.. and not 1.
What is the right equation for point-circle distance?
How can I extend it to point-arc distance?

You have several errors in your code. Here is the answer to your first question.
First, you try to implement the dot product of n2 and Delta as n2*Delta, but that is not what the multiplication of 2 np arrays does. Use np.dot() instead. Next, you try to take the "absolute value" (magnitude) of a vector with np.abs, but that latter is for real and complex numbers only. One way to get the vector magnitude is np.linalg.norm(). Changing those gives you the proper answer, and you don't need the calculation you used for variable dist. So use
Delta = P-center
dist = np.sqrt(np.dot(n2, Delta)**2 + (np.linalg.norm(np.cross(n2, Delta))- radius)**2)
That gives the proper answer for dist, 1.0.

Map point to closest point on fibonacci lattice

I use the following code to generate the fibonacci lattice, see page 4 for the unit sphere. I think the code is working correctly. Next, I have a list of points (specified by latitude and longitude in radians, just as the generated fibonacci lattice points). For each of the points I want to find the index of the closest point on the fibonacci lattice. I.e. I have latitude and longitude and want to get i. How would I do this?
I specifically don't want to iterate over all the points from the lattice and find the one with minimal distance, as in practice I generate much more than just 50 points and I don't want the runtime to be O(n*m) if O(m) is possible.
FWIW, when talking about distance, I mean haversine distance.
#!/usr/bin/env python2
import math
import sys
n = 50
phi = (math.sqrt(5.0) + 1.0) / 2.0
phi_inv = phi - 1.0
ga = 2.0 * phi_inv * math.pi
for i in xrange(-n, n + 1):
longitude = ga * i
longitude = (longitude % phi) - phi if longitude < 0 else longitude % phi
latitude = math.asin(2.0 * float(i) / (2.0 * n + 1.0))
print("{}-th point: ".format(i + n + 1))
print("\tLongitude is {}".format(longitude))
print("\tLatitude is {}".format(latitude))
// Given latitude and longitude of point A, determine index i of point which is closest to A
// ???

What you are probably looking for is a spatial index: https://en.wikipedia.org/wiki/Spatial_database#Spatial_index. Since you only care about nearest neighbor search, you might want to use something relatively simple like http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.spatial.KDTree.html.
Note that spatial indexes usually consider points on a plane rather than a sphere. To adapt it to your situation, you'll probably want to split up the sphere into several regions that can be approximated by rectangles. You can then find several of the nearest neighbors according to the rectangular approximation and compute their actual haversine distances to identify the true nearest neighbor.

It's somewhat easier to use spherical coordinates here.
Your spherical coordinates are given by lat = arcsin(2 * i / (2 * N + 1)), and lon = 2 * PI * i / the golden ratio.
Reversing this is not a dead end - it's a great way to determine latitude. The issue with the reverse approach is only that it fails to represent longitude.
sin(lat) = 2 * i / (2 * N + 1)
i = (2 * N + 1) * sin(lat) / 2
This i is an exact representation of the index of a point matching the latitude of your input point. The next step is your choice - brute force, or choosing a different spiral.
The Fibonacci spiral is great at covering a sphere, but one of its properties is that it does not preserve locality between consecutive indices. Thus, if you want to find the closest points, you have to search a wide range - it is difficult to even estimate bounds for this search. Brute force is expensive. However, this is already a significant improvement over the original problem of checking every point - if you like, you can threshhold your results and bound your search in any way you like and get approximately accurate results. If you want to accomplish this in a more deterministic way, though, you'll have to dig deeper.
My solution to this problem looks a bit like this (and apologies, this is written in C# not Python)
// Take a stored index on a spiral on a sphere and convert it to a normal vector
public Vector3 UI2N(uint i)
{
double h = -1 + 2 * (i/n);
double phi = math.acos(h);
double theta = sqnpi*phi;
return new Vector3((float)(math.cos(theta) * math.sin(phi)), (float)math.cos(phi), (float)(math.sin(theta) * math.sin(phi)));
}
// Take a normalized vector and return the closest matching index on a spiral on a sphere
public uint N2UI(Vector3 v)
{
double iT = sqnpi * math.acos(v.y); // theta calculated to match latitude
double vT = math.atan2(v.z, v.x); // theta calculated to match longitude
double iTR = (iT - vT + math.PI_DBL)%(twoPi); // Remainder from iTR, preserving the coarse number of turns
double nT = iT - iTR + math.PI_DBL; // new theta, containing info from both
return (uint)math.round(n * (math.cos(nT / sqnpi) + 1) / 2);
}
Where n is the spiral's resolution, and sqnpi is sqrt(n * PI).
This is not the most efficient possible implementation, nor is it particularly clear. However, it is a middle ground where I can attempt to explain it.
The spiral I am using is one I found here:
https://web.archive.org/web/20121103201321/http://groups.google.com/group/sci.math/browse_thread/thread/983105fb1ced42c/e803d9e3e9ba3d23#e803d9e3e9ba3d23%22%22
(Orion's spiral is basically the one I'm using here)
From this I can reverse the function to get both a coarse and a fine measure of Theta (distance along the spiral), and combine them to find the best-fitting index. The way this works is that iT is cumulative, but vT is periodic. vT is a more correct measure of the longitude, but iT is a more correct measure of latitude.
I strongly encourage that anyone reading this try things other than what I'm doing with my code, as I know that it can be improved from here - that's what I did, and I would do well to do more. Using doubles is absolutely necessary here with the current implementation - otherwise too much information would be lost, particularly with the trig functions and the conversion to uint.

python: elegant way of finding the GPS coordinates of a circle around a certain GPS location

I have a set of GPS coordinates in decimal notation, and I'm looking for a way to find the coordinates in a circle with variable radius around each location.
Here is an example of what I need. It is a circle with 1km radius around the coordinate 47,11.
What I need is the algorithm for finding the coordinates of the circle, so I can use it in my kml file using a polygon. Ideally for python.

see also Adding distance to a GPS coordinate for simple relations between lat/lon and short-range distances.
this works:
import math
# inputs
radius = 1000.0 # m - the following code is an approximation that stays reasonably accurate for distances < 100km
centerLat = 30.0 # latitude of circle center, decimal degrees
centerLon = -100.0 # Longitude of circle center, decimal degrees
# parameters
N = 10 # number of discrete sample points to be generated along the circle
# generate points
circlePoints = []
for k in xrange(N):
# compute
angle = math.pi*2*k/N
dx = radius*math.cos(angle)
dy = radius*math.sin(angle)
point = {}
point['lat']=centerLat + (180/math.pi)*(dy/6378137)
point['lon']=centerLon + (180/math.pi)*(dx/6378137)/math.cos(centerLat*math.pi/180)
# add to list
circlePoints.append(point)
print circlePoints

Use the formula for "Destination point given distance and bearing from start point" here:
http://www.movable-type.co.uk/scripts/latlong.html
with your centre point as start point, your radius as distance, and loop over a number of bearings from 0 degrees to 360 degrees. That will give you the points on a circle, and will work at the poles because it uses great circles everywhere.

It is a simple trigonometry problem.
Set your coordinate system XOY at your circle centre. Start from y = 0 and find your x value with x = r. Then just rotate your radius around origin by angle a (in radians). You can find the coordinates of your next point on the circle with Xi = r * cos(a), Yi = r * sin(a). Repeat the last 2 * Pi / a times.
That's all.
UPDATE
Taking the comment of #poolie into account, the problem can be solved in the following way (assuming the Earth being the right sphere). Consider a cross section of the Earth with its largest diameter D through our point (call it L). The diameter of 1 km length of our circle then becomes a chord (call it AB) of the Earth cross section circle. So, the length of the arc AB becomes (AB) = D * Theta, where Theta = 2 * sin(|AB| / 2). Further, it is easy to find all other dimensions.