1
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0answers
24 views

Axis aligned and non aligned ellipses and semi definite programming

Let's say I have a equation $$X_1^T \Omega X_1 =1 $$ $X_1$ is a $2\times 1$ matrix. $\Omega$ is a $2\times 2$ matrix. This defines an ellipse. $\Omega$ is a positive, semi definite, symmetric ...
1
vote
1answer
44 views

Properties of the positive definite Hessian matrix of a convex function

I'm reading about nonlinear programming and I'm having trouble understanding the cool properties that a positive definite Hessian matrix $Q$ of $n$-dimensional function $f: \mathbf{R}^n\rightarrow ...
1
vote
0answers
21 views

find an ellipsoid given its intersection with axes and knowing the lengths of its principal axes

My question is about ellipsoids. I have an ellipsoid in 3D centered at zero so it has an equation: $x^T U \Sigma^2 U^T x = 1$ I know the lengths of it's principal axes (therefore the $\Sigma$ ...
1
vote
2answers
77 views

How to find the determinant of this matrix

I'd like to find the determinant of following matrix $$ \begin{pmatrix} {x_1}^2 & x_1y_1 & {y_1}^2 & x_1 & y_1 \\ {x_2}^2 & x_2y_2 & ...
0
votes
1answer
56 views

Type of this Conic section

I want to determine, to which type the following Conic sections belong to: $$ \begin{align} \textrm{(i)}&\quad-8x^2+12xy-6x+8y^2-18y+8=0\\ \textrm{(ii)}&\quad5x^2-8xy+2x+5y^2+2y+1=0 ...
1
vote
1answer
51 views

Given a skewed ellipse, how to determine its axis lengths?

I am mentoring a student who is working on a library to import Adobe Pagemaker documents into LibreOffice. Pagemaker represents ellipses as a bounding box (of the original, untransformed ellipse) and ...
2
votes
1answer
94 views

Rotation of conics sections using linear algebra

When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ where $B \not= 0$ and it is not a degenerate conic, then you can use $\Delta = B^2 -4AC $ to see what type of conic it is, and then ...
1
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0answers
20 views

Number of ellipses to uniquely define a co-centered circumscribing ellipse

I have a bit of a tricky problem that has come up in my engineering research, but I haven't quite got the brains to figure it out, though I've gotten pretty far. Suppose that there is an unknown ...
1
vote
1answer
29 views

Moving between different ellipse representations

I have a representation of an ellipse that is the affine transform of the unit ball, $\|Ax + b\| <= 1$. My question is, how can I change this ellipse representation? I would like to have it in ...
1
vote
2answers
62 views

Recognize conics from the standard equation

Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if ...
2
votes
1answer
75 views

Coordinate System Rotation and Cross Term

If I have a conic equation $$ 5x^2 - 4xy + 8y^2 = 36 $$ and $ \left[\begin{array}{cc} 5 & -2\\ -2 & 8 \end{array}\right] $ in matrix form, whose eigenvalues are 4 and 9, how would I rotate ...
0
votes
1answer
64 views

Conic matrix and diagonalization

If I have the conic $C$: $$ 5x^2 - 4xy + 8y^2 = 36 $$ How would I express it as a matrix of the form: $$ \begin{bmatrix}x&y\end{bmatrix} \begin{bmatrix}a&b/2\\b/2&c\end{bmatrix} ...
2
votes
0answers
54 views

An Easier way to solve simple equations of this type

Im currently working with ellipses and I've been given two points on a ellipse whose major axis is along the x-axis, $(4,3)$ and $(-1,4)$. The question asks me to find the length of the major and ...
0
votes
1answer
78 views

Solution to a quadratic form

I'm trying to find a closed form solution of the following quadratic form for $x$. $x^{T}Dx = c$ where $c$ is just a constant placeholder for some terms on the other side. I know that, because $D$ ...
0
votes
1answer
169 views

how to simplify a general plane conic section's equation by linear algebra?

When encountering a general plane conic section a11x^2+a12xy+a22y^2+b1x+b2y+c=0, i can write it in matrix form as a quadratic form of the vector [x,y,1]. by what then? what should be done to reach the ...
3
votes
2answers
1k views

Finding Intersection of an ellipse with another ellipse when both are rotated

Equation of first ellipse=> $$\dfrac {((x-xFirstEllipseCenterPoint)\cdot \cos(A)+(y-yFirstEllipseCenterPoint)\cdot \sin(A))^2}{(a_1^2)}+\dfrac{((x-xFirstEllipseCenterPoint)\cdot ...
0
votes
0answers
46 views

Representing an imperfect ellipse in 2 linear variables

I have several shapes which are roughly elliptical. I know the major and minor axes and the true circumference, so I store them like this: $$a={\text{axis}}_{\text{major}}\\ ...
2
votes
1answer
367 views

Find enpoints of major axis of an arbitrary ellipse using its general equation

I have a general equation of an ellipse in the form of $Ax^2+Bxy+Cy^2+Dx+Ey+f=0$. How do I find the equation of (or even endpoints would work) major axis of an ellipse. I am aware of following ...
1
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1answer
281 views

Solving a Conic Matrix given these Equations

Given a conic $\Gamma$ that has the equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, $\Gamma$ can be represented by the symmetric matrix $$\mathbf{C} = \begin{bmatrix} A & B/2 & D/2\\ B/2 & ...
1
vote
1answer
386 views

Properties of ellipses and hyperbola related to matrix operation

$5x^2+8xy+5y^2=\mathbf{x}^TA\mathbf{x}= > (S^T\mathbf{x})^TD(S^T\mathbf{x})=1\left(\frac{x-y}{\sqrt{2}}\right)^2+9\left(\frac{x+y}{\sqrt{2}}\right)^2$ Thus, the equation is that of an ellipse, ...
1
vote
1answer
450 views

Finding the major and minor axes of an $n$-dimensional ellipse

Here are two n-dimensional vectors: $V_1$ and $V_2$ $V_1 (v_1,v_2, \dots ,v_n)$ $V_2 (v_1,v_2, \dots ,v_n)$ $V_1 \cos(\theta) + V_2 \sin(\theta)$ is an ellipse in the $n$-D space. (Its center is ...
1
vote
1answer
190 views

Nearest point on an ellipse in n-dimensional space.

Here are two n-dimensional vectors: $V_1$ and $V_2$ $V_1 (v_1,v_2, \dots ,v_n)$ $V_2 (v_1,v_2, \dots ,v_n)$ It seems that $V_1*cos(\theta) + V_2*sin(\theta)$ is an ellipse in the n-D space. (Its ...
3
votes
1answer
170 views

Introduction to parabolae with linear algebra

Linear algebra helps to introduce ellipses and hyberbolas. For example an ellipse can be seen as a transformed circle by a linear application. There is also this theorem for the curve ...
2
votes
2answers
183 views

Finding minimum of the modulus of a 2 variable function

How to find the minimum value of $$|f(x,y)|$$ where $f(x,y)$ is a 2nd degree function in x and y with no 'xy' term. $$f(x,y)=ax^2+by^2+cx+dy+e$$ How is the process different from finding the minimum ...
3
votes
1answer
332 views

Finding minimum of a two variable 2nd degree function under a certain constraint?

How to find the the minimum non-negative value of a function: $$f(x,y)=ax^2+by^2+cx+dy+e$$ s.t. $x$ lies in $[0, A]$ and $y$ lies in $[A, \infty),$ where $A$ is a known constant. or simply $0\leq ...
1
vote
0answers
71 views

Some question concerning curve of second order

Let $$F(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f,$$ $$\phi(x,y)=ax^2+2bxy+cy^2,$$ $x,y \in \mathbb{R}$. Assume that for some $x_0, y_0 \in \mathbb{R}$ and for some $\alpha, \beta \in \mathbb{R}$ such that ...
0
votes
1answer
273 views

Irreducible conic implies that the underlying matrix is invertible

I guess that it is true that a conic (2nd degree homogeneous equation in complex variables) is irreducible (i.e can't be factorized over polynomials) if and only if the underlying matrix of ...