Merge pull request #1 from tirimatangi/add-contents

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matlab/demo2D_2.m

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+%% This test demonstrates intersections of 2 ellipses
+
+% Coordinate of the shared focal point
+sharedFocalPoint = [-10; -20];
+
+% Coordinates of the other focal points of 2 ellipses.
+numEllipses = 2;
+otherFocalPoints = zeros(2, numEllipses);
+
+%% (Case 1: The major axes are collinear and upright)
+otherFocalPoints(:,1) = [0; 20] + sharedFocalPoint;
+otherFocalPoints(:,2) = [0; 10] + sharedFocalPoint;
+
+% Select point p at which the ellipses intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones. 
+% The model error is not exacly the actual distance between 
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate ellipse coordinates and plot the coordinates and the intersection points.
+plotEllipses(sharedFocalPoint, otherFocalPoints, diameters, intersections, "2 ellipses, Case 1");
+
+disp("Case 1: intersection coordinates as columns: ");
+disp(intersections);
+
+%% (Case 2: The major axes are collinear and tilted)
+otherFocalPoints(:,1) = [-10; 20] + sharedFocalPoint;
+otherFocalPoints(:,2) = [-5; 10] + sharedFocalPoint;
+
+% Select point p at which the ellipses intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones. 
+% The model error is not exacly the actual distance between 
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate ellipse coordinates and plot the coordinates and the intersection points.
+plotEllipses(sharedFocalPoint, otherFocalPoints, diameters, intersections, "2 ellipses, Case 2");
+
+disp("Case 2: intersection coordinates as columns:");
+disp(intersections);
+
+%% (Case 3: General case with 2 ellipses)
+otherFocalPoints(:,1) = [10; 4] + sharedFocalPoint;
+otherFocalPoints(:,2) = [0; 10] + sharedFocalPoint;
+
+% Select point p at which the ellipses intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones. 
+% The model error is not exacly the actual distance between 
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate ellipse coordinates and plot the coordinates and the intersection points.
+plotEllipses(sharedFocalPoint, otherFocalPoints, diameters, intersections, "2 ellipses, Case 3");
+
+disp("Case 3: intersection coordinates as columns:");
+disp(intersections);

matlab/demo2D_3.m

@@ -0,0 +1,112 @@
+%% This test demonstrates intersections of 3 ellipses
+
+% Coordinate of the shared focal point
+sharedFocalPoint = [-10; -20];
+
+% Coordinates of the other focal points of 2 ellipses.
+numEllipses = 3;
+otherFocalPoints = zeros(2, numEllipses);
+
+%% (Case 1: The major axes are collinear and upright)
+otherFocalPoints(:,1) = [0; 20] + sharedFocalPoint;
+otherFocalPoints(:,2) = [0; 10] + sharedFocalPoint;
+otherFocalPoints(:,3) = [0; -16] + sharedFocalPoint;
+
+% Select point p at which the ellipses intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones.
+% The model error is not exacly the actual distance between
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate ellipse coordinates and plot the coordinates and the intersection points.
+plotEllipses(sharedFocalPoint, otherFocalPoints, diameters, intersections, "3 ellipses, Case 1");
+
+disp("Case 1: intersection coordinates as columns: ");
+disp(intersections);
+
+%% (Case 2: The major axes are collinear and tilted)
+otherFocalPoints(:,1) = [-10; 20] + sharedFocalPoint;
+otherFocalPoints(:,2) = [-5; 10] + sharedFocalPoint;
+otherFocalPoints(:,3) = [-8; 16] + sharedFocalPoint;
+
+% Select point p at which the ellipses intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones.
+% The model error is not exacly the actual distance between
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate ellipse coordinates and plot the coordinates and the intersection points.
+plotEllipses(sharedFocalPoint, otherFocalPoints, diameters, intersections, "3 ellipses, Case 2");
+
+disp("Case 2: intersection coordinates as columns:");
+disp(intersections);
+
+%% (Case 3: General case with 3 ellipses)
+otherFocalPoints(:,1) = [10; 4] + sharedFocalPoint;
+otherFocalPoints(:,2) = [0; 10] + sharedFocalPoint;
+otherFocalPoints(:,3) = [10; 0] + sharedFocalPoint;
+
+% Select point p at which the ellipses intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones.
+% The model error is not exacly the actual distance between
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate ellipse coordinates and plot the coordinates and the intersection points.
+plotEllipses(sharedFocalPoint, otherFocalPoints, diameters, intersections, "3 ellipses, Case 3");
+
+disp("Case 3: (One intersection only) Intersection coordinates as columns:");
+disp(intersections);
+
+%% (Case 4: Non-solvable case with 3 ellipses)
+% This is an example of the solver giving a wrong answer and returning a high
+% model error when the requirement that the ellipses intersect at one point does not hold.
+otherFocalPoints(:,1) = [10; 4] + sharedFocalPoint;
+otherFocalPoints(:,2) = [0; 10] + sharedFocalPoint;
+otherFocalPoints(:,3) = [10; 0] + sharedFocalPoint;
+
+% Select point p at which the ellipses intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% The 3rd ellipse does not intersect at the same point where the 1st and 2nd ellipses intersect
+pp = p + [1; 1];
+diameters(1) = threePointDistance(otherFocalPoints(:,1), pp, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones.
+% The model error is not exacly the actual distance between
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+% Calculate ellipse coordinates and plot the coordinates and the intersection points.
+plotEllipses(sharedFocalPoint, otherFocalPoints, diameters, intersections, ...
+             ["3 ellipses intersecting at different points, rel.model error=" num2str(relativeModelError)]);
+
+disp(["Case 4: intersection with model error " num2str(relativeModelError)]);
+disp(intersections);

matlab/demo3D_3.m

@@ -0,0 +1,107 @@
+%% This test demonstrates intersections of 3 spheroids
+
+% Coordinate of the shared focal point
+sharedFocalPoint = [-10; -20; 30];
+
+% Coordinates of the other focal points of 2 spheroids.
+numSpheroids = 3;
+otherFocalPoints = zeros(3, numSpheroids);
+
+%% (Case 1: The major axes are on the same yz-plane where x is constant)
+otherFocalPoints(:,1) = [0; 20; 1] + sharedFocalPoint;
+otherFocalPoints(:,2) = [0; 10; -2] + sharedFocalPoint;
+otherFocalPoints(:,3) = [0; -16; 3] + sharedFocalPoint;
+
+% Select point p at which the spheroids intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3; 11] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones. 
+% The model error is not exacly the actual distance between 
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate spheroid coordinates and plot the coordinates and the intersection points.
+plotSpheroids(sharedFocalPoint, otherFocalPoints, diameters, intersections, "3 spheroids, Case 1");
+
+disp("Case 1: intersection coordinates as columns: ");
+disp(intersections);
+
+%% (Case 2: The major axes are on the same tilted plane (x and y coordinates are linearly dependent))
+otherFocalPoints(:,1) = [10; 20; 1] + sharedFocalPoint;
+otherFocalPoints(:,2) = [5; 10; -2] + sharedFocalPoint;
+otherFocalPoints(:,3) = [-8; -16; 3] + sharedFocalPoint;
+
+% Select point p at which the spheroids intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3; 11] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones. 
+% The model error is not exacly the actual distance between 
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate spheroid coordinates and plot the coordinates and the intersection points.
+plotSpheroids(sharedFocalPoint, otherFocalPoints, diameters, intersections, "3 spheroids, Case 2");
+
+disp("Case 2: intersection coordinates as columns:");
+disp(intersections);
+
+%% (Case 3: z coordinates of the major axes are linearly dependent on x- and y-coordinates.)
+otherFocalPoints(:,1) =[10; 1; 10+1] + sharedFocalPoint;
+otherFocalPoints(:,2) = [5; -2; 5-2] + sharedFocalPoint;
+otherFocalPoints(:,3) = [-8; 3; -8+3] + sharedFocalPoint;
+
+% Select point p at which the spheroids intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3; 11] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones. 
+% The model error is not exacly the actual distance between 
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate spheroid coordinates and plot the coordinates and the intersection points.
+plotSpheroids(sharedFocalPoint, otherFocalPoints, diameters, intersections, "3 spheroids, Case 3");
+
+disp("Case 3: intersection coordinates as columns:");
+disp(intersections);
+%% (Case 4: General case with 3 spheroids)
+otherFocalPoints(:,1) = [-2; -5; 1] + sharedFocalPoint;
+otherFocalPoints(:,2) = [0; 10; 2] + sharedFocalPoint;
+otherFocalPoints(:,3) = [10; 0; 3] + sharedFocalPoint;
+
+% Select point p at which the spheroids intersect and determine the diameters accordingly.
+% There will be 2 points of intersection, one of which will be p.
+p = [4; 3; 11] + sharedFocalPoint;
+diameters = threePointDistance(otherFocalPoints, p, sharedFocalPoint)';
+
+% Solve intersections and an estimate on the model error.
+% The bigger the error, the further away the estimated intersections are from the real ones. 
+% The model error is not exacly the actual distance between 
+% the estimate and the correct intersection so it should be taken with a pinch of salt.
+[intersections, relativeModelError] = solveEllipseIntersections(sharedFocalPoint, otherFocalPoints, diameters);
+
+assert(min(vecnorm(intersections - p)) < 1e-6, "Intersection point is not correct")
+assert(max(relativeModelError) < 1e-6, "Model error in the solver is too large")
+
+% Calculate spheroid coordinates and plot the coordinates and the intersection points.
+plotSpheroids(sharedFocalPoint, otherFocalPoints, diameters, intersections, "3 spheroids, Case 4");
+
+disp("Case 4: intersection coordinates as columns:");
+disp(intersections);