All questions with tag [math: affine-geometry]


Projective Geometry: Why is multiplication defined this way?

I am attempting to recognize this new means of increasing in projective geometry. Why is it specified similar to this? Additionally does this have anything to do with reproduction making use of a slide leader? (The image in the link reveals that $4 \cdot 4 = 16$ and also $ 4 \cdot 2 =8$. Every device is a power of 2. Slide leaders were gener...
2022-07-22 14:57:32

Finding the singularities of affine and projective varieties

I'm having trouble calculating singularities of varieties: when my professor covered it it was the last lecture of the course and a shade rushed. I'm not sure if the definition I've been given is standard, so I will quote it to be safe: the tangent space of a projective variety $X$ at a point $a$ is $T_aX = a + \mbox{ker}(\mbox{Jac}(X))$. A vari...
2022-07-21 09:12:01

Affine transformation matrixes

I could use some advise with the following problem: Lets say there is a cuboid that has two distinguished points - that is one of its vertexes ($A$) and the other one is somewhere on the surface ($B$). There is an arbitrary point $X_B$ with coordinates $(a,b,c)$ if you take B as the origin. Now this cuboid gets moved (i.e. a translation and a ro...
2022-07-20 01:00:19

$\mathbb{A}^{2}$ not isomorphic to affine space minus the origin

Why is the affine room $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the beginning?
2022-07-14 05:40:39

Domain rotations in the Mellin integral transform

Let's consider the following form of the Mellin integral transform: $$m_{pq} =\iint\limits_{D_R} \! x^p y^q f(x,y) \, dx\; dy, \, D_R={\{(x,y)\,|\,x^2 + y^2 \le R^2\}}$$ If we scale the domain of the function $f$ by a factor $\gamma$: $$f_\gamma(\gamma x, \gamma y) = f(x, y)$$ $$m_{pq}^{(\gamma)} =\iint\limits_{D_{\gamma R}} \! x^p y^q f_\gamma(...
2022-07-11 08:48:49

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a second course on representation theory, so if there is a reasonably palatable representation-theoretic proof then that's probably what's expected but on...
2022-07-10 07:53:36

Definition of an affine subspace

I read this introduction to Mechanics and also the definition it offers (following Proposition 1.1.2) for an affine subspace problems me. I cite: A part $B$ of a $\mathbb{R}$ - affine room $A$ designed on $V$ is an affine subspace if there is a subspace $U$ of $V$ with the building that $y−x \in U$ for every single $x,y \in B$ It later on ...
2022-07-10 04:59:23

What is the relation between complex numbers and transformation matrices?

I read enhancement and also reproduction with complex numbers can be stood for as translation and also turning in a 2D aircraft. I am utilizing this to walk around things on the screen. I have actually a countered number, that I increase by a number standing for an angle. I after that make use of the countered to relocate an object in a particu...
2022-07-08 20:37:13

Equation of the line in an affine plane over a polynomial field

What are some instances of this? Claim for $F_{4}$. I recognize this is a really straightforward inquiry, yet I can not locate any kind of details on it. Edit: Yes, I was considering $F_{2}[x]/(x^2+x+1)$. I was perplexed.
2022-07-08 06:44:20

finding two polynomials that their roots are a given line.

Given an area $F$ and also $A = F^3$. we specify $L$ to be the line that experiences the factors: $(8,1,-1)$, $(5,0,-1)$. My object is to locate 2 polynomials $q(X_1,X_2,X_3)$, $p(X_1,X_2,X_3)$ in $F[X_1,X_2,X_3]$ of level $\leq 1$ such that $L$ is the set of absolutely nos of $p$ and also $q$. many thanks. benny
2022-07-03 04:45:11

Definition of Affine Independence in Brondsted's Convex Polytopes?

At one factor in guide (An Introduction to Convex Polytopes, by Arne Brondsted) a definition of affine freedom is offered as adheres to, An n - family members $(x_{1},...,x_{n})$ of factors from $\mathbb{R}^d$ is claimed to be affinely independent if a straight mix $\lambda_{1} x_{1} + ... + \lambda_{n} x_{n}$ with $\lambda_{1} + ... + \lambda...
2022-07-03 02:26:13

Effect the zero vector has on the dimension of affine hulls and linear hulls

I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no full solutions (yet)! The question is as follows, For any subset $M$ of $\mathbb{R}^d$, show that $\dim(\text{aff M}) = \dim(\text{span M...
2022-07-03 02:23:21

Meaning of affine transformation

From Wikipedia, I found out that an affine makeover in between 2 vector rooms is a straight mapping adhered to by a translation. Yet in a publication Multiple view geometry in computer vision by Hartley and also Zisserman: An affine makeover (or even more merely a fondness) is a non - single straight makeover adhered to by a translation. I...
2022-07-01 19:32:42

Equiangular polygon inscribed in rectangle

In a drawing application I am writing, I would like to offer the opportunity for a user to draw an equiangular n-sided polygon inscribed in rectangular bounds drawn by their finger (this application is being written for the iPhone and iPad). My head must not be screwed on right today, because for the life of me, I cannot figure out an algorithm ...
2022-07-01 17:27:21

Is every convex-linear map an affine map?

Let is claim that a map $f: V \rightarrow W$ in between limited - dimensional actual vector rooms is convex - straight if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$. Allow is claim that a map $f: V \rightarrow W$ in between limited - dimensional actual vector rooms is affine if $f(\lambda x + (1...
2022-07-01 16:07:34

Count points and lines in $\mathbb{A}^2(\mathbb{F}_p)$

Let $p$ be a prime, then $\mathbb{F}_p$ is a finite field. $\mathbb{A}^2(\mathbb{F}_p)$ is an affine plane. Number of points in $\mathbb{A}^2(\mathbb{F}_p)$ is $p^2$. I look at a line equation $ax+by=c$ and realise that number of distinct lines equals to number of triples $(a,b,c)$, where $\gcd(a,b,c)=1,\ a,b,c \in [0,p-1]$. The question is: how...
2022-07-01 15:28:54

Further questions on barycentric coordinates

Following on from my previous post... I'm going through this PDF file describing barycentric coordinates and trying to make sure I understand everything fully as I need to implement and support these in a computer program. One page 3, area $A$ is defined as follows: $$A = \begin{vmatrix}\bf{P_1}&\bf{P_2}&\bf{P_3}\\1&1&...
2022-06-30 00:44:50

Prove that $v_0, v_1,...,v_k$ are affinely independent if and only if $v_1 - v_0,...,v_k - v_0$ are linearly independent

Definition: Let $v_0, v_1.. v_k$ be factors in $\mathbb{R}^d$. These factors are called affinely independent if there do not exist actual numbers $\alpha_0, \alpha_1...\alpha_k$ that are not all absolutely no such that $\sum_{i=0}^k \alpha_i v_i = 0$ and also $\sum_{i=0}^k \alpha_i = 0$. We require to confirm the following: The factors $v_0...
2022-06-29 02:19:33

Decomposition of a nonsquare affine matrix

I have a $2\times 3$ affine matrix $$ M = \pmatrix{a &b &c\\ d &e &f} $$ which changes a factor $(x,y)$ right into $x' = a x + by + c, y' = d x + e y + f$ Is there a means to decay such matrix right into shear, turning, translation, and also range? I recognize there is something for $4\times 4$...
2022-06-27 19:56:48

This is the most difficult question I could get without using mass point geometry

In triangular ABC, aims D and also E get on sides BC and also CA specifically, and also factors F and also G get on side abdominal muscle with G in between F and also B. BE intersects CF at factor O_1 and also BE intersects DG at factor O_2. If FG = 1, AE = AF = DB = DC = 2, and also BG = CE = 3, calculate $\tfrac{O_1O_2}{BE}$. I could not cons...
2022-06-27 17:08:42