# Tagged Questions

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### dot product of vectors with not orthogonal basis

The dot produt (inner product in the context of Euclidean space) of two vectors $\mathbf{a}=\left [ a_{1},a_{2},...,a_{n} \right]$ and $\mathbf{b}=\left [ b_{1},b_{2},...,b_{n} \right ]$ is defined ...
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### Unique Euclidean isometry between affinely independent points

Let $u_0,\dots,u_n$ be vectors in $\mathbb{R}^n$ such that $u_1-u_0,\dots,u_n-u_0$ are linearly independent and similarly let $v_0,\dots,v_n$ be vectors in $\mathbb{R}^n$ such that ...
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### Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
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### Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
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### Axis of rotation of composition of rotations (Artin's Algebra)

Say $R_1$, $R_2$ are rotations in $\mathbb{R}^3$ with axes and angles $(v_1,\theta_1), (v_2,\theta_2)$ respectively. Since $SO_3$ is a group, we have that $R_2 \circ R_1$ is a rotation with some axis ...
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### Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
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### Is there relations between earth mover's distance and vector norms?

Say I have two vectors $a$ and $b$. Can I estimate $\mbox{EMD}(a,b)$ via some combination of things like $\|a-b\|_p$ and such?
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Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$\det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ... 0answers 31 views ### determine in what grid rhombus is a point i have a rhombus ( i.e. diamond) grid determined by these equations ... 0answers 16 views ### Updating vector components based on change of starting position I have an xyz point, and a 3D vector originating at that point. I would like to be able to shift the starting xyz point and update the 3D vector accordingly. For example: Starting at the xyz point ... 1answer 31 views ### Calculating incremental coordinate change along a 3D vector This is pretty trivial, but I've not managed to resolve it to my satisfaction and it has reached the stage where fresh eyes are definitely useful! I have an xyz point, and a 3D vector originating at ... 0answers 133 views ### Find features in a Signed Distance Field I'm currently trying to improve a meshing algorithm for signed distance fields (which are of the form f(x,y,z) = w, where x,y,z is the location of my query, w indicates the distance to the ... 1answer 42 views ### the “unit speed” anlogue of the evolute of the curve Given a curve, \gamma: \mathbb{R} \to \mathbb{R}^2 define the flow in the normal direction by \gamma(t) + \epsilon \, \mathbf{n}(t). This is different from the evolute which moves at speed ... 1answer 118 views ### Are Euclidean distances a monotone function of inner products? Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of k vectors happens to decrease? Similarly-if decrease is ... 1answer 30 views ### Prove that orthogonality in Euclidean space is geometrically perpendicularity? Is this simply true by definition (that is, taken as axioms?) How would one to prove that for ||\vec{x}||=1 and ||\vec{y}||=1, if (\vec{x},\vec{y})=0, then \vec{x}\perp\vec{y}? In other ... 0answers 17 views ### Alternative coordinates for the complex plane \mathrm{Re}[e^{-is}z]=a, \mathrm{Re}[e^{-it}z]=b  I am defining coordintes on \mathbb{C} using a "generalized" real and imaginary part. Here a,b \in \mathbb{R}. \begin{eqnarray*} \mathrm{Re}[e^{-is}z]&=&a \\ ... 3answers 68 views ### Simple proof that symmetries of regular polyhedron fix its center? Let P be some regular polyhedron in \mathbb{R}^3 (i.e. a regular n-hedron with n = 4, 6, 8, 12, or 20), centered at the origin o = (0, 0, 0), and with vertices v_1, ..., v_n all lying on ... 2answers 97 views ### Orthornomal matrices [duplicate] Is there a more direct reason for the following: If the columns of n\times n square matrix are orthonormal, then its rows are also orthonormal. The standard proof involves showing that left ... 1answer 52 views ### Unit circle - how to prevent backward rotation Let's assume we have a unit circle (0, 2\pi). Basically I have a point on this circle who is supposed to move only forward. This point is controlled by the user mouse and constantly calculate 25 ... 1answer 73 views ### Prove, that f is a linear map. U,V - Euclidean spaces f:U \rightarrow V f(0)=0  \forall _{u,v \in U}:d(f(u),f(v))=d(u,v) Prove that f is a linear map. I'm thinking about something like this: ||f(u+v)|| =d(f(u+v),0) = ... 1answer 96 views ### Is there a more conceptual proof of this fact? Equip {\mathbb R}^3 with the usual scalar product (.|.). Let A be the matrix$$ A= \left(\begin{matrix} 1 & 2 & 3 \\ -2 & 4 & 5 \\ -3 & -5 & 6 \\ \end{matrix}\right) ... 1answer 136 views ### Rotations and reflections in {\bf R}^3. By a rotation in {\bf R}^3 I mean an orthogonal linear transformation f:{\bf R}^3\to {\bf R}^3 represented by a matrix A (i.e. fx=Ax) with \det A=1. By a reflection (through S) I mean an ... 1answer 70 views ### Finding the “middle 2” of four lines This may seem like an overly abstract problem, but it's the best generalization I could make of a specific problem I'm trying to tackle. This problem works in 2-dimensional Euclidean space. A ... 1answer 189 views ### Line-preserving transformations Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines? They are not all affine transformations; consider a perspective projection p in ... 1answer 167 views ### How to find the affine transformation(s) ( if any ) that maps one quadrilateral into another. Given the Fundamental theorem of affine geometry. Let P,Q,R be any three non-collinear points in R2, and let U, V,W be any three other such points. Then there is exactly one affine transformation ... 0answers 25 views ### flat subspace : minimal characterization In the euclidean space {\mathbb R}^3, I can define a plane by three points (x_1,y_1), (x_2,y_2) and (x_3,y_3), using six reals. Of course I can give an equation ax+by+cz=d, using only 4 ... 2answers 102 views ### Distance between 3\times3 matrices (isometries) The problem is pretty long, so I'm going to describe it while writting what I have so far. First I have to asociate the 3\times3 matrices with \Bbb R^9, so we defined \phi:M_{3\times3}(\Bbb R) ... 2answers 29 views ### Find the axis of reflection I have to determine the axis of reflection of the composition of a rotation and a reflection, y show that the order of composition matters. So I multiply the matrices that represent each isometry, ... 2answers 1k views ### Composition of two reflections is a rotation I have this problem that says: Prove that in the plane, every rotation about the origin is composition of two reflections in axis on the origin. First I have to say that this is a translation, off my ... 2answers 239 views ### What is a the intuition behind a parametric equation? I have always used equations for the line (y=a + bx) in R2. Recently I came upon this thing called parametric equations. I cannot grasp the difference between them and the equations for lines that I ... 1answer 17 views ### Let Ax = b be a system of hyper-planes that form a bounded convex D. Can D be partitioned into union of adjacent simplices? Let Ax = b be linear system that forms q bounded region D. If the columns of A are independent, can D be written as a union of adjacent simplices? 1answer 84 views ### Optimal rotation to align a circle with external points I have a circle C with radius r and a set of finite points P=\left \{ p_1,p_2,\ldots,p_n \right \} are identified external to the circle C. These points may lie on the exterior or the interior ... 1answer 153 views ### Angular alignment of points on two concentric circles I have two concentric circles C_1 and C_2 with radii r_1,r_2 such that r_1< r_2and a set of finite points P=\left \{ p_1,p_2...p_n \right \} and Q=\left \{ q_1,q_2...q_n \right \} are ... 2answers 103 views ### Find eigenspaces using ruler and compasses I think this is an interesting question: In the 2-dimensional real vector space, we are given a linear transformation f. Suppose we already know the images of the standard bases, say ... 1answer 162 views ### How to reflect a vector across a rotated plane? Let's say I have vector \vec{A} which has points (-5,6,3) and (4,-2,4). I also have plane P which is defined by the three points, (4,-2,4), (5,-3,4), and (4,-3,4). My question is, how would I ... 1answer 123 views ### Why does aliasing cause loss of a degree of freedom in Euler angles? I'm reading a book on 3D game math where the author points out that when using Euler angles the same orientation can be reached by doing two different operations; say rotating a cube 90 degrees around ... 1answer 111 views ### Questions related to vectors and linear algebra 1) Given vectors u = (1, 1, -2) and v = (0, 1, 1), find the value of t such that magnitude of u + t(v) has the smallest value I have no idea how to begin. 2) Given vectors u = (1, 1, -2), ... 1answer 94 views ### A particular ILP where the existence of a relaxed solution implies the existence of an integer solution This question emerged from a discussion on my previous question Determining quickly whether this Integer Linear Program has any solution at all, which I would like to elaborate separately. I am ... 1answer 152 views ### Solve for Angles in a Matrix So, I have a linear equation, and I need to solve for some variables in said equation. However, since I don't know much about matrices, I don't know how to solve for the variables. The equation in ... 1answer 284 views ### How to Calculate the Direction of a Vector Let's say I have points a, b, and c. We also have \vec{ab} and \vec{ac}. Finally, we know neither vector's direction. (That is, the vector's angle on each axis as if the vector were ... 1answer 142 views ### Separation Theorem in Euclidean Space. I want to show the following: Let A,B \subseteq \mathbb{R}^n disjoint, nonempty, closed and convex sets. Then there exists a h \in \mathbb{R}^n, such that A and B gets separated in the ... 1answer 267 views ### How to determine oriented angle of rotation in 3-dimensional space Let V be oriented two-dimensional Euclidean space. Then we can define an oriented angle \phi between two nonzero vectors u,v\in X by formulas:  \phi=\arccos \frac{\langle u,v\rangle}{\|u\| ... 1answer 76 views ### Locus perpendicular to a plane in \mathcal{R}^4 I have solved an exercise but I'm not sure to have solved it perfectly. Could you check it? It's very important for me.. In \mathcal{R}^4 I have a plane \pi and a point P. I have to find the ... 2answers 1k views ### Solid angle between vectors in n-dimensional space There is a formula of to calculate the angle between two normalized vectors:\alpha=\arccos \frac {\vec{a} \cdot\ \vec{b}} {|\vec {a}||\vec {b}|}. The formula of 3D solid angle between three ...
Suppose $f$ is an isometric (i.e., distance preserving) function on $\mathbb{E}^2$ such that $f(0,0) = (0,0)$. Then I want to show that $f$ is necessarily linear. Now $f$ is linear iff $f$ is both ...
This is a notational question: What does $\overset{\underset{\mathrm{\Delta }}{}}{=}$ denote in $-0.5JDJ\overset{\underset{\mathrm{\Delta }}{}}{=} X^TX$ where $J=I-n^{-1}ee^T$ with $e$ being a ...