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\nOptimal Path Around Quicksand - Math Puzzle (HARD)\n

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\nWhat is the optimal path to travel from start to finish in the least time? Your speed is = the distance you are from the pit. Here I show one optimal path! #SoME1Timestamps:0:00 - Introduction & Question0:52 - Total Time: Cartesian...\nWhat is the optimal path to travel from start to finish in the least time? Your speed is = the distance you are from the pit.

Here I show one optimal path! #SoME1

Timestamps:
0:00 - Introduction & Question
0:52 - Total Time: Cartesian Coordinates
1:23 - Total Time: Polar Coordinates
3:07 - Euler Lagrange / Beltrami Identity
3:41 - Solve for r
5:15 - Graph r
6:05 - MATLAB optimization confirmation

MATLAB code below:
clc; clear all; close all;

%% Calculations

%create empty history file
history= struct('x',[],'fval',[]); % create the structure with empty fields
history(1)= []; % create the empty structure array with the specified fields
save('history.mat');

%enter in start and end positions
start_radii = 1;
end_radii = 3;

%enter number of points
n = 30;
d_theta = pi/(n-1);

objective = @(x) sum(sqrt( diff(x).^2 + x(1:end-1).^2 .* d_theta^2 ) ./ x(1:end-1));
x0 = [start_radii 0.2*ones(1,n-2) end_radii];
A = [];
b = [];
Aeq = [];
beq = [];
lb = [start_radii zeros(1,n-2) end_radii];
ub = [start_radii Inf(1,n-2) end_radii];
nonlcon = [];
options = optimoptions(@fmincon,'OutputFcn',@outfun2);

[x,fval,~,output] = fmincon(objective,x0,A,b,Aeq,beq,lb,ub,nonlcon,options);
rho = x;
theta = linspace(0,pi,n);
[x,y] = pol2cart(theta,rho);

set(gcf, 'Position', [100, 100, 1600, 900], 'color', 'w')
plot(x,y,'-o') %plot of optimal path


%% Video

load('history.mat');
[m, num_iterations] = size(history);
figure()
set(gcf,'renderer','painters')
for i = 1:num_iterations
rho = history(i).x; %each iteration of rho
theta = linspace(0,pi,n);
[x,y] = pol2cart(theta,rho);

set(gcf, 'Position', [100, 100, 1600, 900], 'color', 'w')
plot(x,y,'-o')
grid on
xlabel('x')
ylabel('y')
% title('Beam Animation')
xlim([-end_radii-1 start_radii+1])
ylim([0 2*max(start_radii,end_radii)]);

movieVector(i) = getframe(gcf);
end

myWriter = VideoWriter('Math_puzzle_optimal_path_video');
myWriter.FrameRate = 5;
open(myWriter);
writeVideo(myWriter, movieVector)
close(myWriter)

%% Functions

%For movie: saves iterates in separate history file
function stop = outfun2(x,optimValues,state)
stop = false;
load('history.mat','history');
history(end+1)= struct('x',x,'fval',optimValues.fval);
save('history.mat');
end




Chapters








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Introduction & Question
0:00










Total Time: Cartesian Coordinates
0:52










Total Time: Polar Coordinates
1:23










Euler Lagrange / Beltrami Identity
3:07










Solve for r
3:41










Graph r
5:15\n
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6:35
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Higher Ed
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\nVirtually Passed\n\n
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Date
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2021
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English
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mathematics, math puzzle, brachistochrone, optimization, pit center, starting position, finished position, hand side, square root, optimal path, path, curve, theta, expression, constant, find, start, finish, minimize, 1 meter, solve, equal, integrating, case, written, plug, call, dash, integrate, minimized, long, satisfies, result
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