How to find the axes of a rotated ellipseI have a set of 17 points which I know are on an ellipse. I have the x,y co-ordinates of each point; the y-axis is vertical and the x-axis is horizontal. I want to prove these points are on an ellipse, but the ellipse is rotated clockwise by approximately 14 degrees (determined visually - I want to calculate the exact value of the rotation). I need to find the exact position of the major and minor axes (x' and y') and I do not know the values of the semi-major axis (a) or the semi-minor axis (b). Is this possible?I have tried to find the x,y co-ordinates of the points furthest from (and nearest to) the centre of the ellipse, but I've only managed this through very many tedious iterations and it isn't exact -- is there a better way? Thank you.

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How to find the axes of a rotated ellipseI have a set of 17 points which I know are on an ellipse. I have the x,y co-ordinates of each point; the y-axis is vertical and the x-axis is horizontal. I want to prove these points are on an ellipse, but the ellipse is rotated clockwise by approximately 14 degrees (determined visually - I want to calculate the exact value of the rotation). I need to find the exact position of the major and minor axes (x&#039; and y&#039;) and I do not know the values of the semi-major axis (a) or the semi-minor axis (b). Is this possible?I have tried to find the x,y co-ordinates of the points furthest from (and nearest to) the centre of the ellipse, but I&#039;ve only managed this through very many tedious iterations and it isn&#039;t exact -- is there a better way? Thank you.

Pedro Taylor · Accepted Answer

An ellipse on the xy plane is uniquely determined by five independent parameters: B,C,D,E,F such that x2+Bxy+Cy2+Dx+Ey+F=0 with the restriction that B2−4C&amp;lt;0. So, you need to solve for 5 unknown variables in a linear equation, which requires at least 5 pairs (x,y) such that the corresponding 5 vectors (xy,y2,x,y,1)  are linearly independent.From your observations, choose any 5 (x,y)-pairs such that the (xy,y2,x,y,1) vectors are linearly independent. Then input the values of x2,xy,y2,x,y∈x2+Bxy+Cy2+Dx+Ey+F=0 to form 5 linear equations in B,C,D,E,F, and solve this system of linear equations to obtain the values of  B,C,D,E,F.So, the parameter A is set to 1 above. This amounts to dividing each of the other parameters by A. Since A is the coefficient of the term x2 in the equation of the ellipse, it cannot be zero.The rotation θ, the possible linear shift of the origin to (x0,y0), the major and lengths of the major axis 2a and the minor axis 2b can be computed from these 5 parameters B,C,D,E,F (with setting A=1).

How to find the axes of a rotated ellipse I have a set of 17 points which I know are on an ellip

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