How to solve explicit equation of ellipse
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Hello,
I have the following equation:
(A*x^2)+(B*x*y)+(C*x)+(D*y^2)+(E*y)+F==0
Where all of the coefficients are already known and I am trying to find all values of x and y that satify the equation for the rotated ellipse. I cannot use fimplicit because it is not accurate enough, I need on the order of machine precision. I also tried to use solve(), but it only gives me two solutions for x and y. Do you guys know what other options I could try to solve this?
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James Tursa
2020-10-20
编辑:James Tursa
2020-10-20
What do you mean you want "all values"? For an ellipse this is an infinite set. Do you mean you want the ellipse reoriented along principle axes? And you want to find the rotation? Or ...? Can you post some example sets of coefficients?
回答(2 个)
Matt J
2020-10-20
If you convert the ellipse to its polar representation
that will give you an explicit formula with which to generate sample points.
Bruno Luong
2020-10-20
编辑:Bruno Luong
2020-10-20
% Random coefficients for test:
A = 0.5+rand;
D = 0.5+rand;
B = rand;
C = rand;
E = rand;
F = -rand;
H = [A, B/2;
B/2, D];
g = 0.5*[C; E];
c = F;
% FEX https://www.mathworks.com/matlabcentral/fileexchange/27711-euclidian-projection-on-ellipsoid-and-conic
[radii, U, x0] = EllAlg2Geo(H, g, c);
% points on ellipse, parametric
Ea = U*diag(radii);
theta = linspace(0,2*pi,181);
ellipse = x0 + Ea*[cos(theta); sin(theta)];
% implicit function on grid
[minxy, maxxy] = bounds(ellipse,2);
x = linspace(minxy(1),maxxy(1));
y = linspace(minxy(2),maxxy(2));
[X,Y] = meshgrid(x,y);
XY = [X(:) Y(:)]';
Z = reshape(sum(XY.*(H*XY + g),1) + c, size(X)); % == (A*x^2)+(B*x*y)+(C*x)+(D*y^2)+(E*y)+F
Z = reshape(Z, size(X));
figure
imagesc(x,y,Z);
hold on
% implicit curve
% contour(x,y,Z,[0 0],'ro'); % gives also points on ellipse
plot(ellipse(1,:),ellipse(2,:),'w');
axis equal;
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