0
votes
1answer
27 views

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 ...
0
votes
0answers
26 views

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 ...
1
vote
1answer
48 views

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 ...
3
votes
2answers
45 views

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 ...
0
votes
0answers
14 views

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 ...
0
votes
1answer
10 views

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?
0
votes
2answers
52 views

How to prove this Gram determinant

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 \\ ...
0
votes
0answers
27 views

determine in what grid rhombus is a point

i have a rhombus ( i.e. diamond) grid determined by these equations ...
0
votes
0answers
12 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 ...
0
votes
1answer
27 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 ...
1
vote
0answers
126 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 ...
1
vote
1answer
40 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 ...
0
votes
1answer
98 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 ...
1
vote
1answer
27 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 ...
1
vote
0answers
16 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 \\ ...
3
votes
3answers
65 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 ...
5
votes
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 ...
1
vote
1answer
51 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 ...
5
votes
1answer
71 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) = ...
4
votes
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 $$ A= \left(\begin{matrix} 1 & 2 & 3 \\ -2 & 4 & 5 \\ -3 & -5 & 6 \\ \end{matrix}\right) $$ ...
5
votes
1answer
135 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 ...
0
votes
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 ...
1
vote
1answer
166 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 ...
0
votes
1answer
161 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 ...
1
vote
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 ...
3
votes
2answers
100 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) ...
0
votes
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, ...
2
votes
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 ...
4
votes
2answers
223 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 ...
1
vote
1answer
16 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?
4
votes
1answer
78 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 ...
4
votes
1answer
142 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 $r_1< r_2$and 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 ...
7
votes
2answers
102 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 ...
0
votes
1answer
141 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 ...
3
votes
1answer
120 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 ...
0
votes
1answer
108 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 $u = (1, 1, -2)$, ...
3
votes
1answer
91 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 ...
2
votes
1answer
140 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 ...
0
votes
1answer
258 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 ...
1
vote
1answer
140 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 ...
1
vote
1answer
249 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\| ...
1
vote
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 ...
1
vote
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 ...
7
votes
2answers
258 views

Showing that an Isometry on the Euclidean Plane fixing the origin is Linear

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 ...
0
votes
1answer
62 views

Notation-Linear Algebra/ Euclidean Geometry

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 ...
0
votes
1answer
57 views

In an N-dimensional space filled with points, systematically find the point with highest spearmans correlation to a given-point

I asked a question exactly like this a while ago, so I do not know if it is appropriate to ask pretty much the same question with a single tweak. For the record, my first question is In an ...
2
votes
1answer
152 views

In an N-dimensional space filled with points, systematically find the closest point to a specified point

This is my first post on the mathematics stack exchange site. If I am posting this question on the wrong site, I apologize and please let me know so I can delete it. I am a programmer, and one thing ...
3
votes
2answers
216 views

This classic from euclid's elements, is it accepted everywhere?

I was reading linear vector spaces. When doing some exercise to prove some statements based on the properties defined for linear vector spaces, i suddenly noticed, outside the things defined, i'm ...
0
votes
1answer
87 views

Formula rendering a triangle based off of radius and center?

Let's say I have a triangle (just a basic equilateral). What I'd like to do is render it by specifying its radius and center, and then calculate the vertices from there. What is the formula for this? ...
2
votes
3answers
776 views

Is there a uniform way to define angle bisectors using vectors?

Look at the left figure. $x_1$ and $x_2$ are two vectors with the same length (norm). Then $x_1+x_2$ is along the bisector of the angle subtended by $x_1$ and $x_2$. But look at the upper right ...