Calculating area volume from longitude, latitude and altitude

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Hello! I'm trying to calculate the volume of airspace(3d polygon) from its longitude, latitude and altitude.
The polygon's vertices are given with longitude and latitude (ex. [128 129 129 128 128], [37 37 38 38 37])
and its altitude ranging from 1000~5000 feet.
How can I get the volume of this 3d polygon?

Accepted Answer

darova on 11 Sep 2021
  • Convert spherical coordinates into cartesian using sph2cart
  • Use alphaShape to build an object and calculate volume
hye wook Kim
hye wook Kim on 14 Dec 2021
Thank you for your help! @Walter Roberson.
As you said, I'm trying to solve this problem using the triplet integral (integral3 in matlab).
and I got 3 questions as follows;
Q1. How to calculate the volume in mile unit? (ex. 1000 nm3)
Q2. Does R need to be calculated in the loop for every latitude?
Q3. how to construct formula for integral3 to apply your recommand; 'Integrate altitude + R over altitude and latitude in radians, and multiply by radians spanned by longitude'.
This is the code that I wrote and seems to show wrong answer...
%% Coordinates(Lat,Lon,Alt) of Target Airspace
Latitude = [128 129 129 128 128]; % Degree
Longitude = [37 37 38 38 37]; % Degree
Bottom_Altitude = distdim(1000,'feet','nauticalmiles'); % Convert FT to NM
Top_Altitude = distdim(5000,'feet','nauticalmiles'); % Convert FT to NM
%% Get Centroid coordinates of Target Airspace
polyin = polyshape(Latitude, Longitude);
[centroid_latitutde, centroid_longitude] = centroid(polyin);
%% Get Radius of the Earth in target airspace
r1 = 3963.191; % Earth's radius (nm) at sea level at the EQUATOR.
r2 = 3949.903; % Earth's radius (nm) at sea level at the POLES.
B = centroid_latitutde; % Latitude of target airspace's centroid.
R = sqrt(((r1^2*cos(B))^2+(r2^2*sin(B))^2)/((r1*cos(B))^2+(r2*sin(B))^2)); % Earth's radius (nm) at sea level from Target centroid
%% Get Spherical Coordinates from Lat,Lon,Alt
min_rho = Bottom_Altitude + R; max_rho = Top_Altitude + R; % Height (nm) from the Earth's center.
min_phi = deg2rad(min(Latitude)); max_phi = deg2rad(max(Latitude)); % Latitude
min_theta = deg2rad(min(Longitude)); max_theta = deg2rad(max(Longitude)); % Longitude
%% Apply Triplet Integral to get the volume
vol_fun = @(rho,phi,theta) rho.*phi.*theta;
vol_Boundary = integral3(vol_fun,min_rho,max_rho,min_phi,max_phi,min_theta,max_theta);

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