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### 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 ...
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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 $f: \mathbf{R}^n\rightarrow ... 0answers 21 views ### find an ellipsoid given its intersection with axes and knowing the lengths of its principal axes My question is about ellipsoids. I have an ellipsoid in 3D centered at zero so it has an equation:$x^T U \Sigma^2 U^T x = 1$I know the lengths of it's principal axes (therefore the$\Sigma... 2answers 77 views ### How to find the determinant of this matrix I'd like to find the determinant of following matrix $$\begin{pmatrix} {x_1}^2 & x_1y_1 & {y_1}^2 & x_1 & y_1 \\ {x_2}^2 & x_2y_2 & ... 1answer 56 views ### Type of this Conic section I want to determine, to which type the following Conic sections belong to:$$ \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 ... 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 ... 1answer 94 views ### Rotation of conics sections using linear algebra When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ whereB \not= 0$and it is not a degenerate conic, then you can use$\Delta = B^2 -4AC $to see what type of conic it is, and then ... 0answers 20 views ### Number of ellipses to uniquely define a co-centered circumscribing ellipse I have a bit of a tricky problem that has come up in my engineering research, but I haven't quite got the brains to figure it out, though I've gotten pretty far. Suppose that there is an unknown ... 1answer 29 views ### Moving between different ellipse representations I have a representation of an ellipse that is the affine transform of the unit ball,$\|Ax + b\| <= 1$. My question is, how can I change this ellipse representation? I would like to have it in ... 2answers 62 views ### Recognize conics from the standard equation Suppose$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 ... 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] $in matrix form, whose eigenvalues are 4 and 9, how would I rotate ... 1answer 64 views ### Conic matrix and diagonalization If I have the conic$C$: $$5x^2 - 4xy + 8y^2 = 36$$ How would I express it as a matrix of the form: $$\begin{bmatrix}x&y\end{bmatrix} \begin{bmatrix}a&b/2\\b/2&c\end{bmatrix} ... 0answers 54 views ### An Easier way to solve simple equations of this type Im currently working with ellipses and I've been given two points on a ellipse whose major axis is along the x-axis, (4,3) and (-1,4). The question asks me to find the length of the major and ... 1answer 78 views ### Solution to a quadratic form I'm trying to find a closed form solution of the following quadratic form for x. x^{T}Dx = c where c is just a constant placeholder for some terms on the other side. I know that, because D ... 1answer 169 views ### how to simplify a general plane conic section's equation by linear algebra? When encountering a general plane conic section a11x^2+a12xy+a22y^2+b1x+b2y+c=0, i can write it in matrix form as a quadratic form of the vector [x,y,1]. by what then? what should be done to reach the ... 2answers 1k views ### Finding Intersection of an ellipse with another ellipse when both are rotated Equation of first ellipse=>$$\dfrac {((x-xFirstEllipseCenterPoint)\cdot \cos(A)+(y-yFirstEllipseCenterPoint)\cdot \sin(A))^2}{(a_1^2)}+\dfrac{((x-xFirstEllipseCenterPoint)\cdot ... 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: $$a={\text{axis}}_{\text{major}}\\ ... 1answer 367 views ### Find enpoints of major axis of an arbitrary ellipse using its general equation I have a general equation of an ellipse in the form of Ax^2+Bxy+Cy^2+Dx+Ey+f=0. How do I find the equation of (or even endpoints would work) major axis of an ellipse. I am aware of following ... 1answer 281 views ### Solving a Conic Matrix given these Equations Given a conic \Gamma that has the equation Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, \Gamma can be represented by the symmetric matrix$$\mathbf{C} = \begin{bmatrix} A & B/2 & D/2\\ B/2 & ... 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, ... 1answer 450 views ### Finding the major and minor axes of an$n$-dimensional ellipse Here are two n-dimensional vectors:$V_1$and$V_2V_1 (v_1,v_2, \dots ,v_n)V_2 (v_1,v_2, \dots ,v_n)V_1 \cos(\theta) + V_2 \sin(\theta)$is an ellipse in the$n$-D space. (Its center is ... 1answer 190 views ### Nearest point on an ellipse in n-dimensional space. Here are two n-dimensional vectors:$V_1$and$V_2V_1 (v_1,v_2, \dots ,v_n)V_2 (v_1,v_2, \dots ,v_n)$It seems that$V_1*cos(\theta) + V_2*sin(\theta)$is an ellipse in the n-D space. (Its ... 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 ... 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 ... 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 ...
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 ...