0
votes
1answer
24 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
2answers
57 views

Shortest Distance of a Point in $R^3$ to a Cone

I'm having a problem how to figure out the shortest distance of a point $\vec{p} = [x_p, y_p, z_p]$ to the surface of a cone given by: Start vertex $\vec{a} = [x_a, y_a, z_a]$. This is the center of ...
1
vote
0answers
39 views

Polyhedron Basis

Given a Polyhedron formed by linear constraints $$(a^Tx <= b)$$ and you have given a orthogonal basis. Then every point inside the Polyhedron can be expressed by a linear combination of elements ...
0
votes
2answers
53 views

2 vectors form a plane in $\mathbb{R}^3$?

I was told that 2 linearly independent vectors in $\mathbb{R}^3$ define a plane. However, if vectors themselves are defined by their "head" or coordinate points, wouldn't 2 vectors define 2 points and ...
2
votes
2answers
21 views

Basic geometry proof about tetrahedron

Suppose tetrahedron ABCD such that AB is orthogonal to CD and AC is orgthongal to BD. Show that AD is orthogonal to BC. So i made a picture of a tetrahedron in 3 space and sort of look down at it ...
1
vote
1answer
51 views

Neighbour Points in N-Dimensional Space

if you got a integer point in the n-dimensional space how many neighbor integer points does it have? 1D you have 2 2D you have 8 3D you have 26 i came to the formula $$n_i = 2*(n_{i-1}+1)+n_{i-1} ...
1
vote
0answers
19 views

Geometric meaning ov VO with O unitary matrix [closed]

What is the geometric meaning of VO where V is a n nxn Matrix and O is a unitary matrix?
0
votes
0answers
38 views

Why does distance lose meaning in high-dimensional space?

I'm working on an algorithm that clusters points in extremely high-dimensional space (thousands, if not more). However, I came across this wikipedia page: ...
0
votes
0answers
23 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 ...
2
votes
1answer
19 views

Geometric characterization of an Euclidean norm

Show that $N$ is an Euclidean norm if and only if the intersection of the unit ball with any plane is an ellipse. I'm stuck on this one. I do not see how can I connect the definition of an ...
1
vote
1answer
47 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 ...
0
votes
1answer
15 views

Finding end point of straight line given starting point and angle

I have a program which computes the angle of skew of a scanned photograph. It returns the angle of skew in degrees. I now need to draw lines across the image which follow the angle of skew. These ...
1
vote
2answers
58 views

Rotate XYZ frame in 3D space

Given a XYZ frame in 3D space at origin O(0,0,0). And given a plane equation: ...
0
votes
1answer
26 views

Find points near end point of a line

Any equation to find points near to both start and end points of lines with different slopes. See image. Need P and Q. If Endpoints are named A and B, AP and BQ should be 1 cm
0
votes
0answers
15 views

How to find iso function value points without exploring all points in 2D space

Consider a 2D graph with dim1 and dim2 represented as X and Y respectively. The range of X and Y are 1 to 100. Hence there are 10000 points in the 2D space. Each point in the space is some function of ...
1
vote
1answer
59 views

Geometric interpretation of Laplace's formula for determinants

Coming from the geometric point of view, the determinant of an $n \times n$-Matrix computes the volume of an parallelepiped spanned by the columns of the matrix. In context of this question, let the ...
2
votes
3answers
41 views

$SL(V)$ and $PSL(V)$ act $k$-transitively on the space of all $1$-dimensional subspaces.

A group $G$ acts $k$-transitive on some set $X$ if for every two $k$-tupels $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_k)$ there exists some $g \in G$ such that $$ g\cdot x_1 = y_1, \ldots, g\cdot ...
0
votes
0answers
20 views

Position vectors of sphere/circle touching central one

I am trying to understand the meaning of an expression describing the "kissing" number problem. On Wiki, it states the following: Let $x_n$ be a set of $N$ $D$-dimensional position vectors of the ...
1
vote
1answer
39 views

Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
2
votes
3answers
42 views

$\mathbb{R}^2$ rotations

Considering $\mathbb{R}^2$ for the two rotations $g_1,g_2$ with centers of rotation $x_1$ and $x_2$ by $\theta_1$ and $\theta_2$ I have to show that $g_1\circ g_2$ is a rotation iff ...
0
votes
1answer
25 views

how to find iso-cost contours on a 2d plot efficiently

Consider a 2D plot in which dimension 1 and 2 represent quantity 1 and 2 respectively ranging over 0 to 100. Each point in the space corresponding to (x,y) represent cost of choosing quantity 1 as x ...
1
vote
0answers
21 views

Regions of intersected hyperplanes

Let $\mathcal{A}$ be a linear arrangement of hyperplanes in $\mathbb{R}^n$ and $\mathcal{B} := \bigcap_{h \in \mathcal{A}} h$. Show that $ \mathcal{C} := \{ h / \mathcal{B} : h \in ...
0
votes
2answers
17 views

“Reverse” of a rotation matrix for superposition $\colon A$ on $B \to B$ on $A$

I have a (right-multiplying) 3D rotation matrix which best superposes one set of coordinates $A$ onto the system of another one $B$. Is there an operation that can be performed to obtain the rotation ...
0
votes
0answers
23 views

Reversing the weighted average of an edited 2D point

I have 2 points in 2D space, val_A[x,y] and val_B[x,y]. The values of val_A & val_B use the same starting point, val_Orig[x,y], and are generated by applying different rotations about different ...
3
votes
2answers
44 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 ...
3
votes
0answers
70 views

Pythagorean like theorem for general spaces

There are laws, for example Pythagorean theorem, for calculating distance of two points, say $d(a,b)$, using third point and knowing $d(a,c)$ and $d(c,b)$ in vector spaces. My question is that for ...
0
votes
4answers
193 views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
0
votes
0answers
11 views

Is this system solution for plane intersection possible?

I have na exercise that asks me to find the parametric equation of the line formed by the intersection of the two planes: $$\pi_1: X = (1,-2,0) + \lambda_1(1,0,-1) + \mu_1(0,0,-1) $$ $$\pi_2: X = ...
-1
votes
0answers
35 views

elementary linear algebra question: how to find the linear transformation?

I have to find the volume of an ellipsoid (x/a)^2 + (y/b)^2 + (z/c)^2 = 1, by finding a linear transformation from an ellipsoid to a sphere and then calculating the determinate. It seems like that ...
1
vote
1answer
20 views

Given $E=\mathbb{R}^3$, let $f$ be an endomorphism of $E$ defined by the matrix $A=(a_{i,j})$ on the canonic basis. Let $v,w$ be two eigenvectors.

Given $E=\mathbb{R}^3$, let $f$ be an endomorphism of $E$ defined by the matrix $A=(a_{i,j})$ on the canonic basis. Let $v,w$ be two linearly independent eigenvectors. Give a plane that is invariant ...
1
vote
1answer
18 views

Finding grid intersections with linear algebra?

Typically if you want to find where a given line intersects with a grid you just compute a division and you check if that division generates a remainder that is equals to $0$, and you do that for both ...
3
votes
3answers
160 views

How should I prove $(a+b)^3= a^3+3ab(a+b)+b^3$ — Model or figure?

In what way can I prove/verify $(a+b)^3= a^3+3ab(a+b)+b^3$ ? Should I make a 3D model, or create 2D figure? In the case of 3D model, I have made $a^3$ and $b^3$; i.e cube'a' and cube'b'. I don't know ...
0
votes
3answers
21 views

Find a point $p_1$ on the line $l$ with distance d from the point $p_2$ on the same line

I have tried to find posts that are related to the question but they end up with the terms like 'find a distance' etc. What I want is not to find the distance, I already have the distance, I want ...
1
vote
0answers
47 views

Regular pentagon vector proof

Given that $v = DC = \lambda EB$, prove that $\lambda v = CB + ED$. Whatever I try seems to end up with $CB + ED = (\frac {1}{\lambda} - 1)v$, ie: $$CB + ED = CD + DE + EB + ED = EB - DC = EB - ...
1
vote
1answer
37 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 ...
0
votes
1answer
16 views

Coordinates of octahedron's vertices and checking if a point is inside it.

Given that I have the distance between the center of an octahedron and any of its faces (regular octahedron, so all the distances are equal), how can I calculate the coordinates of its vertices, ...
0
votes
1answer
65 views

How to determine the visibility of an object from the top of a hill

We are developing software to train children how to cross the street safely. Part of the training is to teach them not to cross when they don't have enough visibility due to obstacles. In this case, ...
2
votes
1answer
33 views

How does a measurement error change the volume of a tetrahedron?

Consider that I have a tetrahedron $T$ whose the lengths of edges are $(a,b,c,d,e,f)$. I want to calculate the volume of the tetrahedron by Cayley-Menger Determinant. However, I know that, the ...
1
vote
3answers
37 views

Is this triangle question missing information?

In the $\Delta KLP$, find $a+b$: My question is that: isn't some information missing from the question? Because all I can see is is that $ \usepackage{ gensymb } \angle SKP = \angle LTS = ...
0
votes
3answers
58 views

If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$?

Have $n$ vectors in $\mathbb{R}^n$. If the $n$ vectors are linearly independent, can we conclude that their span is $\mathbb{R}^n$?
-1
votes
1answer
42 views

Show that n noncollinear points on a plane determine at least n lines

Let $P$ be a set of $n$ noncollinear points on a plane. Show that pairs of points from $P$ determine at least $n$ lines. Edit: I don't have my notebook. I know I should use a certain linear algebra ...
0
votes
1answer
14 views

Newbie with barycentric coordinates: why one is zero when on a vertex?

I'm trying to calculate if a 2D point lies inside a triangle and I solved the following system: ...
0
votes
2answers
143 views

Reflection in a plane.

What is the exact definition of a reflection through the plane $a.r=0$ for a given vector a and $r=(x,y,z)$. Of course I know what it is but I don't know what's part of its definition and what's part ...
2
votes
1answer
36 views

Question about geometry in a finite projective space

I apologize again for a dumb question! To add some context (though I think it'll largely be unnecessary): suppose $q$ is a prime, $F:= \displaystyle \mathbb{Z}/q\mathbb{Z}$ is a field. I've defined ...
0
votes
0answers
38 views

distance between 2 parallel hyperplanes (non-trivial)

I am trying to solve the problem shown below in the image. And I was looking at the solution, I didn't know how the solution managed to come up with $$x_1 = (b_1 / || a ||^2) a $$ and $$x_2 = ...
0
votes
2answers
49 views

Volume of tetrahedron

For a tetrahedron defined by vectors $OA=\mathbf{a},\;OB=\mathbf{b},\;OC=\mathbf{c}$, the volume is $\frac{1}{6} |\mathbf{a}\cdot (\mathbf{b}\times\mathbf{c})|$ I am wondering, what does the sign of ...
3
votes
2answers
63 views

Find point in 3D space based on plane and known point

I'm struggling with drawing geometry in 3D spaces via OpenGL. My current task is to find coordinates of point. Assume we have such input data: Points $a$, $b$ and $k$ define a plane. Point $c$ ...
4
votes
4answers
130 views

rank($A$)=rank($A^T$) [duplicate]

Is there an elementary explanation of why the row-rank of a matrix equals its column-rank (without using adjoint maps, resp. lots of technical computations)? What is the geometric intuition behind ...
0
votes
2answers
36 views

Projecting 3D Point to Plane

I have a plane defined by the equation $Ax + By + Cz + D = 0$. It does not pass through the origin. I have projected the origin of my global coordinate system onto the plane, so it is at $(a, b, c)$. ...
2
votes
2answers
42 views

Axis of symmetry of a binary image

I want to calculate the axis of symmetry of a binary image. In other words I have an image that has a black irregular shaped object with a white background. I want to find the best approximation of ...