# Tagged Questions

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### A question on the rectangular region defined for a vector in $\mathbb{R}^N$

Let $K = (k_1,k_2,k_3,...k_N)$ be a vector in $\mathbb{R}^N$, consider the region $S_K$ consisting of all vectors $L = (l_1,l_2,l_3,...l_N)$ such that, $|l_i| \le |k_i| \forall i \in \{1,2,3,...N\}$. ...
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### what is vector $(\vec{a}\cdot \vec{b})\vec{c} + (\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$

Suppose we have three non orthogonal vectors in $R^3$ as $\vec{a}, \vec{b}, \vec{c}$. The vector of $(\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ is in the plane spanned by ...
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### What series of 'hyperpolyhedrons' do exist? Is there an effective way to derive their cross-sections by 3-d subspace?

There are two obvious series of 'hyperpolyhedrons'. 'Hyperoctahedron' with vertices $(\pm1,0...0), (0,\pm1,0,...0)...(0,...0,\pm1)$ and each vertex connected by an edge with each other vertex ...
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### Cartesian to geodetic conversion of 3D bounding box - How to calculate latitude and longitude from an axis aligned bounding box

I have a geometry with its vertices in cartesian coordinates. These cartesian coordinates are the ECEF(Earth centred earth fixed) coordinates. This geometry is actually present on an ellipsoidal model ...
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### Symmetric Parallelograms Under Linear Transfer Marticies

I am trying to show that a parallelogram which is symmetric about the origin stays symmetric about the origin under the action of a linear transfer matrix. It is a fairly trivial case to draw a ...
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### Axiomatization of angle measuring in real vector spaces

In linear algebra / analytic geometry it is common to define the angle between two vectors $u,v \in V$ of an euclidean vector space $V$ by $\angle (u,v) := \arccos \frac{(u,v)}{\|u\| \cdot \|v\|}$. Is ...
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### How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
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### 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 ...
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### I have some problems with straight lines and planes

Firstly, I need to say that English is not my first language and the problems were written in Spanish. I have never read a Math problem in English, so some words may be confusing. If they are, please ...
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### 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 ...
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### 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 ...
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### 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 ...
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### What is (fundamentally) a coordinate system ?

Consider the following construction of vectors and points. Let's start with a vector space, or more specifically a coordinate space $F^N$ over a field $F$ and of $N$ dimensions. The elements of this ...
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### Computationnal geometry: vector, basis, point and coordinate system?

I am trying to build a small geometrical library in C++, that is mathematically consistent (not so false). The goal here is to construct two concepts: vectors and points. I am not sure that the ...
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### Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system

I am trying to understand the exact relation between all these things: point vector affine space vector space basis frame coordinate system Can you explain me rigorously (in the mathematical ...
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### Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
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### Missing component of a 4D vector

I need to calculate the missing component of a 4D vector, when I know that one of the dimensions is always positive and less than or equal to the magnitude. In other words, I have four variables x, ...
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### 4D-vector calculations

For a 4D vector, how can I calculate any component as a function of the three other components and a magnitude and vice versa? I want x = f(y, z, i, m) where m is the magnitude of the vector. Will ...
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### Equation of plane without cross product

We know that vectors $(3,3,4)$ and $(-1,-1,5)$ span a plane in $\mathbb{R}^3$. Can we somehow readily infer that the plane's equation is $x_1 - x_2 = 0$? Cross-products have not yet been introduced ...
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### Converting from spherical coordinates to cartesian around arbitrary vector $N$

So if I'm given an arbitrary unit vector $N$ and another vector $V$ defined in spherical coordinates $\theta$ (polar angle between $N$ and $V$) and $\phi$ (azimuthal angle) and $r = 1$. How do I ...
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### Geometric accuracy analysis of 2d rectangular models

I have reconstructed set of rectangular objects lie on a 2D plane (for ex. ABCD). All these objects are in a one coordinate system. On the other hand, I have reference models for all of them ...
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### Number of ways to cut a square

How many ways are there to cut the unit square into two pieces? And how many ways are there if the two pieces must have equal area? Some special cases: A. If the cut is required to be a horizontal ...
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### Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
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### Coordinates relative to arbitrary 3D plane

Say that I have an arbitrary plane, $\mathcal{P}$, in $\mathbb{R}^3$ that is defined by a given vector, $\vec{v}_0$, on the plane and a normal vector, $\vec{n}$. I will be using $\vec{v}_{0}$ as a ...
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### Angle between vectors of the form $(\cos A,\cos B,\cos C)$

The question: Two vectors $S=(\cos A,\cos B,\cos C)$, $S'=(\cos A',\cos B',\cos C')$, What is the angle between them? The answer is $\cos(\theta)$ = $\cos A.\cos A'+ \cos B.\cos B'+ \cos C.\cos C'$. ...
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### What is first edge position in the Minkowski sum of two convex polygons in the plane?

I am trying to understand the informal algorithm of the Minkowski sum of two convex polygons in the plane as described here: ...
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### Simple way to parameterize two perpendicular vectors

Given are two vectors in $\mathbb{R}^3$, $\bar{u}$ and $\bar{v}$, such that they are perpendicular ($\bar{u}\cdot\bar{v}=0$) and of equal length ($|\bar{u}|=|\bar{v}|$). Is there a "nice" way to ...
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### Generating a 3d ribbon from a series of points

I am attempting to generate a 3d ribbon from a set of 3d points. The idea is to generate a realistic ribbon which follows those points. In its current state, one example looks like this: In this ...
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### projective geometry and relationship of cross-ratios

Define for pairwise different points $P_i=[v_i]$ the cross-ratio $\operatorname{CR}(P_1,P_2,P_3,P_4) = \frac{\det(v_1,v_2)}{\det(v_2,v_3)}\cdot\frac{\det(v_3,v_4)}{\det(v_4,v_1)}$ on $\mathbb{KP^1}$ ...
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