0
votes
0answers
4 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
10 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
118 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
33 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
66 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
23 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
15 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
62 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
90 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
46 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
67 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
95 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
125 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
69 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
150 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
147 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
28 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
911 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
181 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
73 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
127 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
97 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
117 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
110 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
104 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
83 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
121 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
227 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
137 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
215 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
75 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
958 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
226 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
59 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
56 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
141 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
206 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
85 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
666 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 ...
0
votes
1answer
75 views

Getting as close to a target vector as possible

Say I have P number of political parties in an election that I'm trying to rig. My boss has decided what number of percentage of the votes it is best that each party should get. N number of votes have ...
7
votes
1answer
156 views

A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
3
votes
1answer
102 views

Determinant and Measure

The determinant of the matrix of its vectors gives the measure of an $n$-dimensional parallelogram. For example, in $2$ dimensions, the area spanned by vectors $v$ and $w$ is \begin{array}{|cc|} v_1 ...
11
votes
1answer
362 views

Sub-determinants of an orthogonal matrix

Let $A$ be a matrix in the special orthogonal group, $A \in SO_n$. This means that $A$ is real, $n \times n$, $A^t A = I$ and $Det(A)=1$, that is, the column vectors of $A$ make a positively-oriented ...
1
vote
1answer
92 views

Need some help to understanding the formula

This is pinhole camera model (I don't get, is there [R t], or (R, t)) This formula is used to model the projection from a space point M to an image point m. Projection drawing Tilde over vector, ...
1
vote
0answers
74 views

Polynomial function from $S^3$ to $S^3$ and quaternions

I am searching the polynomial functions from $S^3$ to $S^3$. We say $g$ is a polynomial function from $S^3$ to $S^3$, if there exists $f_1,f_2,f_3,f_4 \in \mathbb{R}[X_1,X_2,X_3,X_4]$ and ...
1
vote
3answers
277 views

Closest Points in Euclidean Space

Let $C \subset R^n$ be the convex hull of the set of points $C' \subset R^n$ such that for every $c \in C'$, $c_i \in \{0,1\}$. Let $b \not \in C$. Is there an algorithm that will generate the ...
2
votes
1answer
195 views

What is wrong with this proof that isometries must be surjective?

Let $\phi : \mathbb{R}^2\rightarrow\mathbb{R}^2$ be an isometry. Suppose $\phi$ is not surjective, that is there exists some $v \in \mathbb{R}^2$ whose fiber $\phi^{-1}(v)$ is empty. Then by the ...