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%% Octave code
format long
% First, some utility functions
% Convert degrees to radians
deg2rad = @(x) x * pi / 180;
% Convert radians to degrees
rad2deg = @(x) x * 180 / pi;
% Compute the angle between two vectors
vector_angle = @(v,w) acos(dot(v,w) / (norm(v) * norm(w)));
% The inputs for our calculation
% Aspect ratio of a default minetest window (e.g. 800/600)
aspect_ratio = 4/3;
% Default (vertical) field of view in radians
fov_y = deg2rad(72);
% The horizontal field of view
fov_x = aspect_ratio * fov_y;
% Now, the calculation
% Assume we are looking in +Z direction (the direction has no effect on the end result)
center_of_screen = [0, 0, 1];
corner_of_screen = [tan(fov_x/2), tan(fov_y/2), 1];
angle_to_corner = vector_angle(center_of_screen, corner_of_screen);
camera_fov = max(fov_x, fov_y);
required_factor = angle_to_corner / (camera_fov / 2)
%
% Output: required_factor = 1.10421165985322
%
% Changing the inputs (e.g. aspect_ratio = 1, fov_y = deg2rad(0.1))
% results in required_factor approaching sqrt(2) as fov_y gets close to zero.
%
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