% octave/MATLAB program to solve problem
% ellipse parameters
A = 165.0 / 2.0; % [m]
B = 135.0 / 2.0; % [m]
L0 = 34.6; % [m]
% approximation terms
N = 2048; % [1]
% compute angles
theta = linspace(0.0, 2.0 * pi, N);
% compute ellipse points
r = [A * cos(theta); B * sin(theta)];
% compute delta vectors
dr = r(:, 2 : end) - r(:, 1 : end - 1);
% compute delta vector lengths and sum up
L = sqrt(sum(dr .* dr));
L = cumsum(L);
% estimate angle of specified arc length
thetai = find(L > L0, 1);
theta0 = L(thetai);
% compute x
x = A - r(1, thetai);
% display solution
disp(x);
the result is x = 10.1122, which seems reasonable
note, this program estimates the differential lengths by subtracting adjacent points, not with the analytic expression in
>>12143840, but both are equivalent in the limit
