Finding the axis and orientation of an ellipse



Answered question


Finding the axis and orientation of an ellipse with matrices
So I have this ellipse equation:
I'm asked to get the lengh of the semi-major and semi-minor axis, and it's orientation.

Answer & Explanation



Beginner2022-04-06Added 10 answers

Given 5x2+10y212xy=14
When this is translated into matrix-vector form, we define the position vector as p=xy, then we write the equation as pTAp+bTp+c=0
Where A is a symmetric 2×2 matrix, b is a 2×1 vector, and c is a scalar.
The first term is pTAp=x2A11+y2A22+2xyA12
Comparing with the given equation, one finds that
A=5 -6-6 10
The second term is bTp=b1x+b2y
Comparing with the given equation, we find that b is the zero vector.
Finally, taking 14 to the left hand side, we deduce that c=14
Therefore, the equation in matrix-vector form is
The next thing you need to do is diagonlize matrix A, i.e. find a rotation matrix R and a diagonal matrix D such that A=RDRT
There is a standard way to do this diagonalization that you should memorize
1. Calculate θ=12tan-12A12A11-A22
2. Calculate the rotation matrix R=cosθ -sinθsinθ cosθ
3. Calculate the diagonal matrix D=D11 00 D22 where
Following the above steps, we find that θ=12tan1125
Therefore θ (and (2θ) are in the first quadrant. Using the trigonometric identities
Since tan(2θ)=125
Hence cosθ=12(1813)=313 and sinθ=12(813)=213
Thus for the second step, we have the rotation matrix as
R=1133 -22 3
For the third step, we have for diagonal matrix
With all these calculations, we can now write the equation of the ellipse as
To put this in the standard form divide by (c)=(14)=14, then pTRERTp=1
where the matrix E=(D14) is equal to
E=D/14=114 00 1
Now define the vector q=RTp so that p=Rq, then it follows that qTEq=1
Hence q12E11+q22E22=q1214+q221=1
Thus the coordinate vector of q lies on an ellipse (in standard orientation) with semi-major axis =14, and semi-minor axis =1=1. Our ellipse which is in terms of the vector p is just a rotation of q-ellipse by the angle θ because p=Rq, and R is a rotation matrix by angle θ (counter clockwise).

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