Cone containing set of points

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Ouassim Benhamouche 2024-4-8

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https://ww2.mathworks.cn/matlabcentral/answers/2104301-cone-containing-set-of-points

评论： Ouassim Benhamouche 2024-4-8

I want to enforce the constraint : " the cone with apex P and opening angle of theta must coontains point1 and point2 " P is the decision variable i need to find the cone that contains those 2 points, is there some suggestion on how to write this constraint ?

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Manikanta Aditya 2024-4-8

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Hi,

The constraint you’re trying to enforce can be expressed mathematically using the dot product and the definition of a cone.

Check this example to get more better understanding:

% Define the points and the apex of the cone
P = [Px, Py, Pz];  % the apex of the cone
point1 = [x1, y1, z1];
point2 = [x2, y2, z2];
% Calculate the vectors from P to the points
v1 = point1 - P;
v2 = point2 - P;
% Calculate the cosine of the angle between the vectors
cos_angle = dot(v1, v2) / (norm(v1) * norm(v2));
% Check if the points are within the cone
if cos_angle > cosd(theta)
    disp('The points are within the cone.')
else
    disp('The points are not within the cone.')
end

Thanks.

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Answer 1

Hassaan 2024-4-8

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2104301-cone-containing-set-of-points#answer_1438196

在 MATLAB Online 中打开

% Define the opening angle theta (in radians), and two points
theta = pi / 6; % Example value for theta
point1 = [1, 2, 3]; % Example value for point1
point2 = [4, 5, 6]; % Example value for point2
% Initial guess for P (the apex of the cone)
P0 = [0, 0, 0]; % Adjust as necessary
% Optimization options
options = optimoptions('fmincon', 'Display', 'iter', 'Algorithm', 'sqp');
% Define the objective function as a handle to an anonymous function
objectiveFunction = @(P) 0; % Dummy objective function
% Define the nonlinear constraint as an anonymous function
coneConstraint = @(P) deal(...
    [cos(pi/6) - dot(point1 - P, (point1 + point2)/2 - P) / (norm(point1 - P) * norm((point1 + point2)/2 - P)), ...
     cos(pi/6) - dot(point2 - P, (point1 + point2)/2 - P) / (norm(point2 - P) * norm((point1 + point2)/2 - P))], ...
    []);
% Running the optimization
[Popt, fval, exitflag, output] = fmincon(objectiveFunction, P0, [], [], [], [], [], [], coneConstraint, options);
 Iter  Func-count            Fval   Feasibility   Step Length       Norm of   First-order  
                                                                       step    optimality
    0           4    0.000000e+00     0.000e+00     1.000e+00     0.000e+00     0.000e+00  

Initial point is a local minimum that satisfies the constraints.

Optimization completed because at the initial point, the objective function is non-decreasing 
in feasible directions to within the value of the optimality tolerance, and 
constraints are satisfied to within the value of the constraint tolerance.
if exitflag > 0
    fprintf('Optimization succeeded. Optimal apex of the cone is:\n');
    disp(Popt);
else
    fprintf('Optimization did not converge to a solution.\n');
end
Optimization succeeded. Optimal apex of the cone is:
     0     0     0

-----------------------------------------------------------------------------------------------------------------------------------------------------

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