0
votes
0answers
5 views

Collinearity in geometry

Let S be the intersection of diagonals in a cyclic quadrilateral. Let p be a circumcircle of a triangle ABS and it intersects BC in M and q is a circumcircle of a triangle ADS and q intersects CD in ...
1
vote
1answer
17 views

Prove uniqueness of polar-coordinates $(R>0, \theta)$ up to angle $\theta+ 2 \pi$.

Prove uniqueness of polar-coordinates $(R>0, \theta)$ up to angle $\theta+ 2 \pi$. Suppose we have $(x,y) \in \mathbb R^2$. Then we can transform this point to polar-cordinates $(R>0, ...
1
vote
2answers
45 views

Finding a normal to an ellipsoid

Let $E$ be an ellipsoid centered at $v = (x,y,z) \in \mathbb{R}^3$ and let $T:\mathbb{R}^3 \to \mathbb{R}^3 $ be a linear transformation which transforms $E$ to a sphere $S$ with a radius of length ...
5
votes
2answers
49 views

How to characterize rotations in $\mathbb{R}^n$?

I am studying the performance of an optimizer algorithm to find the $$ \textrm{argmin}_{x\in \mathbb{R}^n} f(x) \text{ where } f : \mathbb{R}^n \rightarrow \mathbb{R} $$ I would like to test how the ...
0
votes
1answer
35 views

Transformation of 2D profile to 3D coordinates

I am sure that answer for similar questions have being given more than one thousandth times, but correct answer that suits my needs I still haven't found. Currently I am developing simple 3D app. My ...
0
votes
0answers
30 views

Decide coordinates for a vector in a triangle (Image attached)

I have the following triangle. I have to express the line $\overline{AT}$ as a linear combination of $\overline{AC}$ & $\overline{AB}$. A hint was to use the knowledge of $\overline{AT} = ...
0
votes
0answers
21 views

How to prove a parallel $(u=u_0) $it self curvature?

My name is Gita, and I had aproblem with my math. and need help, I know that a parallel $C$ in a surface of revolution in $M$ be a geodesic if and only if $f'(u_0)=0$. and $C$ is non arc lenght ...
0
votes
1answer
61 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
1
vote
1answer
26 views

Equation of hyperplane in Matlab

Given $n$ points in $n$-dimensions, using MatLab, how should we find the equation of the $(n-1)$-dimensional hyperplane passing through these $n$ points.
2
votes
1answer
27 views

Collinear points in 3dimension

Given three $3D$ points: $A,B$ and $C$, what is the procedure to check if they are collinear? In general, given $n$ points in $m$-dimension, how should one find out, if these $n$-points defines a ...
0
votes
4answers
38 views

How to determine the side on which a point lies?

Suppose we have a linear equation and a point in the plane, then how can one determine on which side of the line the point lies?
0
votes
3answers
22 views

Standard equation of a line

I'm a bit confused. I read in many places that the standard equation of a line in $R^2$ is the following: $w_1 x_1 + w_2 x_2 = d$ but I found a resource that mentions it as: $w_1 x_1 + w_2 x_2 + d ...
6
votes
0answers
158 views
+50

Set geometry and inclusion

I would like to prove that the set of the symmetric positive semi-definite matrices which is defined as $$\Delta_2= \{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad ...
15
votes
2answers
303 views
+200

Can a cube always be fitted into the projection of a cube?

If we project the unit cube, i.e. a axis parallel cube with side length 1 centered at the origin, in $\mathbb{R}^n$ onto a $k$-dimensional subspace of $\mathbb{R}^n$ which contains the origin, can we ...
1
vote
0answers
34 views

Isometry in Euclidean space

The question is to show that an isometry from $\mathbb{E}^{1} \to \mathbb{E}^{1}$ is of the form $x \to ax + b $ from first principles, and determine the values $a$ can take. From my notes I know for ...
0
votes
0answers
22 views

Finding the equation of a plane by using point-to-point distances

Assume that we have a plane $P_1$ whose equation is known. I need to find the equation of plane $P_2$. If we choose a point set $N = \{n_1, n_2, ...\}$ on $P_1$ and another point set $M = \{m_1, m_2, ...
0
votes
0answers
18 views

How to prove the distributive law of cross product by geometric definition

I'm asked to prove the distributive law of cross product by its geometric definition in an exercise, and I found the following answer in stack exchange. How to prove the distributive property of ...
0
votes
1answer
48 views

System of 4 equations:

There is a famous geometry problem named "Hardest easy problem" where we are required to find an angle(x). After drawing some lines and doing some calculations, I ended up with a system of 4 ...
2
votes
0answers
75 views

A probability problem with multivariate Gaussian distribution

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description. I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...
0
votes
2answers
29 views

Find perpendicular vectors in subspace of $V_{3}$

Find all vectors of $V_{3}$ which are perpendicular to the vector $(7,0,-7)$ and belong to the subspace $L((0,-1,4), (6,-3,0)$. As a note, this is an extra question of a long exercise, the ...
1
vote
3answers
155 views

Estimating Eigenvalues and Eigenvectors from an 'eignpicture'

I was given this question at my school but it really does not make sense to me: The unit vectors $x$ in $\mathbb{R}^2$ and their images $Ax$ under the action of a $2x2$ matrix A are drawn ...
1
vote
1answer
30 views

Dual Basis problem

I've been dealing with this but I haven't been able to understand the underlying principles of dual basis, so i don't know how to do it well. It starts like this: Have $(e_1, e_2, e_3)$ basis of the ...
1
vote
1answer
47 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
2
votes
1answer
41 views

Geometric interpretation of matrices

I'm interested in knowing some geometric interpretation of matrices. Can you suggest any lecture note or textbook or anything else about it? I've just finished an undergraduate course in linear ...
0
votes
1answer
38 views

Relationship for cosine of angle [duplicate]

If $x$ is the cosine of the angle between the vectors $a$ and $b$, $y$ is the cosine of the angle between the vectors $a$ and $p$, and $z$ is the cosine of the angle between the vectors $b$ and $p$, ...
4
votes
1answer
51 views

Get diagonal from Quadrilateral described by vectors

I have the following problem: in 2D space. I have quadrilateral which have 2 of angles = 90 degree And 2 non unit vectors h1 and h2. (Have length and direction) We, also, have point P where h2 and ...
0
votes
1answer
12 views

Making a new angle from given angles.

We know how to draw some angles.We are also able to draw the angles after performing addition and subtraction operation on known angles.Suppose there are n number of known angles and k queries, where ...
3
votes
0answers
142 views

How can higher-dimensional projection maps be described mathematically?

New question: (resulting from discussions with Sabyasachi) I am wonder how can higher-dimensional projection maps, analogous to for example the Mercator, Miller, Behrmann projections, can be ...
0
votes
1answer
38 views

$|\langle a_i, a_j\rangle|$ for $p$ points on a unit circle.

Is it true that given any $p$ points $a_1, .., a_p$ on a unit [euclidean] circle, there is always a pair $i \ne j$ such that $|\langle a_i, a_j\rangle| \ge \cos{\pi/p}$?
1
vote
0answers
26 views

Slope intercept equation where b is the point on x-axis?

In our slope intercept formula: $y = mx + b$ where we know the value of $b$ is a point on $y$-axis. Can we write this as where $b$ is a point on $x$-axis? E.g. our slope formula is: $m = (y_2 - ...
2
votes
1answer
48 views

Raising Rotation Matrix to a Power

For the general $\sin$ and $\cos$ rotation matrix, $R$, am I right to assume that for a given angle of rotation $\theta$, $R^n$ gives us the rotation matrix for the new angle $nθ$? In my question I ...
12
votes
0answers
179 views

How to find eigenvalues and eigenvectors of this matrix

Can you help to find eigenvalues and eigenvectors of the following matrix? Here is the matrix: $$C = \small \begin{pmatrix} -\sin(\theta_{2} - \theta_{M}) & \sin(\theta_{1} - \theta_{M}) & 0 ...
0
votes
2answers
47 views

Do four dimensional vectors have a cross product property? [duplicate]

we know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$ where : $ \vec A = (A_i; A_j; A_k)$ $ \vec B = (B_i; B_j; B_k)$ $ \vec C = (C_i; C_j; C_k)$ $C_i = ...
0
votes
1answer
37 views

Relationship between cosines of angles in 4 dimensions

If the cosine of the angle between the vectors $a$ and $b$ is $x$ and the cosine of the angle between the vectors $a$ and $p$ is $y$, then, if we call $z$ the cosine of the angle between the vectors ...
0
votes
0answers
14 views

Find angle-preserving transformation matrix given 2 points

I asked a similar question yesterday about finding an affine transform matrix given the same 2 points in both coordinate systems. I was told that there was only a unique solution, if the scaling was ...
0
votes
1answer
41 views

Find 2D affine transform matrix given a pair of points

I have the coordinates of two points in an initial 2d coordinate system and the corresponding coordinates in a target system. Is is possible to determine the affine transform matrix from these values? ...
2
votes
4answers
71 views

How to find $n+1$ equidistant vectors on an $n$-sphere?

Following this question, Proving the existence of a set of vectors, I'm looking for a way to find $n+1$ equidistant vectors on a Euclidean $n$-sphere. For $n=2$, you can pick the vertices of any ...
2
votes
2answers
69 views

check if point is on a plane (using Heron formula ?)

Is this true that if any of parameters a, b, c, d is equal to sum of three others then 4 points are on same plane? I am given 4 points in 3 dimensional space. Is this correct to state that all 4 ...
1
vote
1answer
54 views

How to find out if four points are on the same plane, only by using distances?

There is a method called Cayley-Menger determinant in order to find if 3 points are collinear, 4 points are coplanar etc. provided that all the pairwise distances are given. However, in 2-D, there is ...
2
votes
1answer
37 views

Volume of ellipsoid using Linear Algebra

Can someone tell me how to find the volume of an ellipsoid of dimension $\mathbb{R}^3$ by using linear algebra? I know the formula is $\frac{4}{3}\pi abc$. I am given the equation ...
0
votes
2answers
29 views

Scale a Point onto Plane

I'm trying to find the scale factor that scale a point onto plane in 3D Space. I have the following information: Point on a plane: $a = (x_1,y_1,z_1)$ Plane equation: $P\colon Ax + By + Cz +D =0$; ...
1
vote
2answers
65 views

Does congruence guarantee length conversion?

Suppose that a linear transformation $M:R^2 \rightarrow R^2$ maps a triangle $ABC$ to a congruent triangle $A'B'C'$ ($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$ are not colinear, and $A,B,C\neq O$) Is it ...
7
votes
4answers
150 views

Geometry of the Cayley Transform

I'm trying to understand the geometry of the Cayley transform. Suppose I have a $3 \times 3$ rotation matrix $R$ (i.e an orthogonal matrix with determinant equal to $1$). Let's ignore the corner case ...
0
votes
1answer
67 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 ...
2
votes
3answers
79 views

Product of reflections is a rotation, by elementary vector methods

Let $\mathbf{u}$ and $\mathbf{v}$ be two 3D unit vectors. The transform that performs reflection in the plane normal to $\mathbf{u}$ is given by $$ T_{\mathbf{u}}(\mathbf{x}) = \mathbf{x} - ...
0
votes
2answers
20 views

How do I calculate the new x y coordinate for a rectangle when centering it within a rectangle?

I need to center a rectangle inside another rectangle. I know the width and height of the parent rectangle, and I know the width and height of the child rectangle that needs to be centered. I need ...
2
votes
1answer
103 views

Equation for concentric circles?

I want an equation for concentric circles. In following image I am trying to draw concentric circles in Java but as you can see these are messed up. This is because their (x,y) coordinates (i.e. ...
0
votes
1answer
48 views

Find tangent vector to surface given a point on the surface and its normal vector (for a sphere)

I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N. I know that there are many possible tangent vectors ...
0
votes
1answer
24 views

The Geometry of a Linear Transformation

Consider a square matrix of full rank (these assumptions are made for the sake of simplicity). This matrix expresses in coordinates a Linear Mapping that sends the unit sphere to a hyperellipsoid on ...
0
votes
2answers
38 views

Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...