0
votes
0answers
16 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
1
vote
1answer
23 views

Transformation matrix from a translated-rotated coordinate system to the general coordinate system

In Figure 1, suppose $XYZ$ (in black) as my general coordinate system and $X'Y'Z'$ (orange) as another system with parallel axes respect to $XYZ$. Consider $xyz$ (green) is my 3rd coordinate system ...
0
votes
1answer
30 views
+50

rotate secondary Vanishing points to the primary vanishing points to find new length of object

all though only the 2D data is available, the best way to think of this problem is a piece of paper pinned at one corner to a wall, but the paper is sitting at an angle to the wall, see illustration ...
0
votes
2answers
31 views

How to find if a point lies in the area covered by 2 straight lines

Given 2 lines Y1 = m1X1 + C1 Y2 = m2X2 + C2 Now given a point (X3,Y3), how will one find whether the point lies in the area enclosed by the 2 straight lines? In ...
0
votes
2answers
21 views

Rotate and translate a line so that it passes through two given points

I have 2 point and a line segment in 2d space. The line only rotates and translates using its mid point. How do I calculate the translation and rotation required for the line to be touching the 2 ...
0
votes
1answer
34 views

Comparing a vector with a directed line segment

Let $x$ and $y$ be two vectors in $\mathbb{R}^2$. The parallelogram law of vector addition says that the vector corresponding to the diagonal of the parallelogram formed by these vectors is $x+y$. For ...
0
votes
0answers
21 views

Find the following formulas for the images of the line $ax+by+C=0$ under translation and rotation.

We have lamar lines in the form of $ax+by+c=0$, where $a,$ $b,$ and $c$ are fixed reals satisfying $a^2+b^2\neq0$. We need to find the following formulas for the images of the line $ax+by+C=0$: ...
1
vote
2answers
22 views

How to define a cloud of points relative to a vector path?

I've been researching and playing with examples of particle clouds in a graphics visualization. Most use shape geometries to define a field of particles, or parameters for distributing them randomly ...
1
vote
0answers
39 views

Calculate 3D-coordinates of a cube's points from the points on the projections

I have a following optical system: 3 cams (left and top, which is orthogonal to the left, and right, which is parallel to the left and orthogonal to the top) and the 2 cubes in the 3D-space with ...
3
votes
2answers
130 views

Transforming $2D$ coordinates

Lets say from coordinate system 1, we have 3 points which consists of a triangle. The vertices are located at $(50,120) , (70,150) , (100,100)$. Now, coordinate system 2 consists also of a triangle, ...
0
votes
0answers
20 views

How to find the tangent vector to a curve at a special point

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. The line segments $AC$ and $BC$ make equal angle $\alpha$ with the horizontal plane through $C$. The angle ...
1
vote
1answer
35 views

Intersection of two lines in 3D

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. I want to find the coordiantes of the point $C=(x,y,z) $. The line segments $AC$ and $BC$ make equal angle $\alpha$ with ...
2
votes
1answer
67 views

Is there any proof for this simple observation? [duplicate]

If we consider the Euclidean space $R^3$, it is simply the space where we live. Here we can find only four point such that distance between any two points is a constant. If we consider the Euclidean ...
0
votes
1answer
31 views

Geometric interpretation of an ill-conditioned matrix

Given a non-singular matrix $A\in$ $\Bbb R^{n\times n}$ (invertible) with SVD decomposition $(U, \Sigma, V)$, how would you interpret geometrically $A$ being ill conditioned? From what I know, $A$ is ...
0
votes
2answers
47 views

Solve nonliner equations

We are trying to find intersection of hyperbolas and we ended up in five equations $$\begin{align} A_1X^2+B_1Y^2+C_1XY+D_1X+E_1Y+F1&=0\\ A_2X^2+B_2Y^2+C_2XY+D_2X+E_2Y+F2&=0\\ ...
2
votes
2answers
26 views

Line parallel to a plane and have 45 degrees between another

I need to find a direction vector for a line parallel to a plane $x+y+z = 0$ and that have $45$ degrees with the plane $x-y = 0$ So, i've assumed the vector $\vec V_r = (a,b,c)$ and since it is ...
0
votes
1answer
22 views

Translation of basis for a vector space on the specified distance

In the Euclidean space $XYZ$ is a basis $X_1Y_1Z_1$ defined that is specified by the vectors $\overrightarrow {O_1X_1}$, $\overrightarrow {O_1Y_1}$ and $\overrightarrow {O_1Z_1}$. How to calculate ...
1
vote
1answer
20 views

Weighted average of multiple points

Let's say I have a triangle whose three corners are $$(x_1,y_1),(x_2,y_2),(x_3,y_3).$$ I have a weight assigned to each one as a percentage, so the first point might be $75\%$, the second $15\%$ and ...
1
vote
1answer
23 views

How could I calculate the local size of an object given its distance and actual size?

Lets say I take a picture of a sign. I know that sign is 20ft (width), 10ft height. I'm standing 40 feet away. If I were to take a picture, how could I calculate how wide and how high the sign is in ...
2
votes
1answer
53 views

Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
4
votes
1answer
49 views

Geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts

What's the geometric meaning of a matrix decomposed into its symmetric and skew-symmetric parts? For example, a skew-symmetric matrix on its own can be interpreted as an infinitesimal rotation. As ...
2
votes
1answer
30 views

Finding equation of line with given slope

Find the distances of the point (1,2) from a straight line. The slope is given to be 5 and the line passes through the intersection point of the lines $x+2y = 5$ and $x - 3y = 7$ Obviously I could ...
1
vote
1answer
43 views

Keeping the arc length constant between points in a spiral

I'm making a visualization of points in a logarithmic spiral and want to keep the arc length between points (image particles) constant. I read that in an Archemedian spiral arc length is ...
1
vote
3answers
110 views

Strange proof of Schwarz Inequality with Pythagorean Theorem

Does anyone know what is going on in this proof of the Schwarz inequality? Most importantly: how can one assume that $c^2\leqq \|A\|^2$, or later on, that $c^2\|B\| \leqq \|A\|^2$? This would imply ...
0
votes
1answer
25 views

Logarithmic spiral appears inverted

I'm learning to code the equation for a logarithmic spiral for a graphics visualization. However, it appears to be inverted with the radius getting smaller (rather than larger) toward the outside of ...
3
votes
1answer
43 views

Do invariant lines of linear transformations contain a fixed point?

Suppose $A$ is a $2$-by-$2$ matrix, and $\mathcal{l}$ is an invariant line under $A$, so $(x,mx+c)$ is mapped to $(X,mX+c)$ for some variable $X$ linear in $x$. Then is there a point on the line ...
0
votes
0answers
19 views

Detecting coplanarity by given pairwise distances

Given a 3D point set $P$, where $|P| \gg 4$ is there a way other than using Cayley-Menger determinant to detect if a group of points are coplanar or not? In other words, what are the methods to ...
0
votes
1answer
26 views

Dimension of convex polytope

Hyperplanes, which are affine subspace of $R^d$ has a dimension of at most $d-1$. For hyperplanes satisfying the equations:- $a_{j_1}x_1 + ... +a_{j_d}x_d = b_j ;\quad j = 1,...,m $ (1), the ...
0
votes
1answer
22 views

Find a projectivity to create a graph.

I have the tetrahedron {xyzt=0} in projective space with homogeneous coordinate (x,y,z,t). I need to create a graph but the tetrahedron in affine coordinate is {xyz=0} and I can't visualize the ...
0
votes
1answer
28 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
67 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
46 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
56 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
30 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 ...
2
votes
1answer
53 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} ...
0
votes
0answers
30 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
21 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
51 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
29 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
70 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
31 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
17 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
68 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
45 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
23 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
41 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
43 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
27 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
22 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
23 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 ...