Articles of affine geometry

Affine space $A^n$ and definition of difference.

I’m not sure if this question would be more appropriate in Physics.SE, if so let me know. I need help in understanding this quote from “Arnold – Mathematical Methods in Classical Mechanics” (This is my translation from italian): The affine $n$-dimensional space $A^n$ differs from $\mathbb R^n$ for the fact that there’s no fixed origin […]

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}) = […]

Product of two algebraic varieties is affine… are the two varieties affine?

Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then? If this is not true, could you give a counterexample?

alternative definition of Affine map

Let $f:X\longrightarrow Y$ be a function on real vector spaces (note that $X,Y$ have arbitrary dimensions). If $T(x)=f(x)-f(0)$ is linear, $f$ is called an affine map. Prove that $f$ is affine if and only if $f\left(\displaystyle\sum_{k=1}^na_kx_k\right)=\displaystyle\sum_{k=1}^n a_k f(x_k),\ \forall n\in\mathbb N,\ \forall x_1,x_2,\dots,x_n\in X,\ \forall a_k\in\mathbb R\text{ such that }\displaystyle\sum_{k=1}^na_k=1 \ .$

Polar form of (univariate) polynomials: looking for a proof

Recently I stumbled upon the following theorem — I’d like to read a comprehensible (i.e. understandable for an engineer) proof for it: Given a polynomial $F(t)$ of degree $n$, there exists a unique equivalent symmetric, multiaffine polynomial $f(u_1, u_2, \ldots, u_n)$ — $u_i$ all of degree $1$ — satisfying $f(t,t,\dots,t) = F(t)$. This $f(u_1,u_2,\ldots,u_n)$ is […]

Is perspective transform affine? If it is, why it's impossible to perspective a square by an affine transform, given by matrix and shift vector?

I’m a bit confused. I want to program a perspective transformation and thought that it is an affine one, but seemingly it is not. As an example, I want to perspective a square into a quadrilateral (as shown below), but it seems impossible to represent such a transform as a matrix multiplication+shift: 1) What I […]

Is every convex-linear map an affine map?

Let’s say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is convex-linear if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$. Let’s say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is affine if $f(\lambda x + (1-\lambda)y) = \lambda f(x) […]

fitting points into partitions of a square

A friend of mine came up with the following problem: Let $\{X_1, X_2, …, X_n\}$ be an arbitrary finite partition of the unit square $[0, 1]^2$. Let $\{P_1, P_2, …, P_m\}$ be a finite set of points in $[0, 1]^2$. Can all the points $P_\ell$ be transformed through the same affine transformation (rotation, translation, squeezing, […]

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

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

What is the difference between linear and affine function

I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated