Affine transformation

In Euclidean geometry, an affine transformation or affinity (from the Latin, affinis, "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If X is the point set of an affine space, then every affine transformation on X can be represented as the composition of a linear transformation on X and a translation of X. Unlike a purely linear transformation, an affine transformation need not preserve the origin of the affine space. Thus, every linear transformation is affine, but not every affine transformation is linear. Examples of affine transformations include translation, scaling, homothety, similarity, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. Viewing an affine space as the complement of a hyperplane at infinity of a projective space, the affine transformations are the projective transformations of that projective space that leave the hyperplane at infinity invariant, restricted to the complement of that hyperplane. A generalization of an affine transformation is an affine map (or affine homomorphism or affine mapping) between two (potentially different) affine spaces over the same field k. Let (X, V, k) and (Z, W, k) be two affine spaces with X and Z the point sets and V and W the respective associated vector spaces over the field k. A map f: X → Z is an affine map if there exists a linear map mf : V → W such that mf (x − y) = f (x) − f (y) for all x, y in X. (Wikipedia).

What are affine transformations?

Algorithm Archive: https://www.algorithm-archive.org/contents/affine_transformations/affine_transformations.html Github sponsors (Patreon for code): https://github.com/sponsors/leios Patreon: https://www.patreon.com/leiosos Twitch: https://www.twitch.tv/leioslabs Discord: https://discor

From playlist Algorithm Archive

Affine Transformations — Topic 27 of Machine Learning Foundations

In this video we use hands-on code demos in NumPy to carry out affine transformations, a particular type of matrix transformation that may adjust angles or distances between vectors, but preserves parallelism. These operations can transform the target tensor in a variety of ways including

From playlist Linear Algebra for Machine Learning

Affine polygon rendering (quads, not triangles)

In https://youtu.be/hxOw_p0kLfI I illustrated how affine texture mapping (non perspective-corrected) appears “wonky” when the shape is not an equilateral. Some of this was because it was constructed from triangles. So what happens when we render any convex polygons and not just triangles?

From playlist 3D Rendering Tutorial

Novel Algebraic Operations for Affine Geometry | Algebraic Calculus One | Wild Egg

We introduce some novel conventions to help us set up the foundations of affine geometry. We learn about differences of points, sums of points and vectors, affine combinations and vector proportions. And then use these to state a number of important results from affine geometry, including

From playlist Algebraic Calculus One from Wild Egg

2D Sprite Affine Transformations

This video takes a simple look at using transformation matrices to transform 2D sprites with rotation, translation, scaling and shearing. Source: https://github.com/OneLoneCoder/Javidx9/blob/master/PixelGameEngine/SmallerProjects/OneLoneCoder_PGE_SpriteTransforms.cpp https://discord.gg/W

From playlist Interesting Programming

Affine Cipher on Maple

How to encrypt and decrypt the affine cipher using Maple software. Code from Into to Crypto and Coding Theory 2nd ed. by W. Trappe and LC Washington.

From playlist Cryptography and Coding Theory

Tensor Calculus Episode 10 | Is the Affine Connection a Tensor?

In todays video I look at the transformation properties of the affine connection coefficients to see if they transform as tensor components. This series is based off "Tensor Calculus for Physics" by Dwight Neuenschwander which can be found at: https://www.amazon.com/gp/product/1421415658/

From playlist New To Tensors? Start Here

Shifting a triangle using a transformation vector

👉 Learn how to apply transformations of a figure and on a plane. We will do this by sliding the figure based on the transformation vector or directions of translations. When performing a translation we are sliding a given figure up, down, left or right. The orientation and size of the fi

From playlist Transformations

algebraic geometry 5 Affine space and the Zariski topology

This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the definition of affine space and its Zariski topology.

From playlist Algebraic geometry I: Varieties

Adolfo Guillot: Complete holomorphic vector fields and their singular points - lecture 3

CIRM VIRTUAL EVENT Recorded during the research school "Geometry and Dynamics of Foliations " the May 11, 2020 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on C

From playlist Virtual Conference

Robert Cass: Perverse mod p sheaves on the affine Grassmannian

28 September 2021 Abstract: The geometric Satake equivalence relates representations of a reductive group to perverse sheaves on an affine Grassmannian. Depending on the intended application, there are several versions of this equivalence for different sheaf theories and versions of the a

Tensor Calculus For Physics Ep. 11 | The Covariant Derivative

This video shows how to modify the notion of the derivative to include the affine connection, guaranteeing that the (covariant) derivative of a tensor yields another tensor. This series is based off "Tensor Calculus for Physics" by Dwight Neuenschwander which can be found at: https://www.

From playlist New To Tensors? Start Here

Point Clouds | Student Competition: Computer Vision Training

In this video, you will learn about point clouds and how to work with them in MATLAB. Get files: https://bit.ly/2ZBy0q2 Explore the MATLAB and Simulink Robotics Arena: https://bit.ly/2yIgwfS --------------------------------------------------------------------------------------------------

From playlist Student Competition: Computer Vision Training

Advanced Linear Algebra Full Video Course

Linear algebra is central to almost all areas of mathematics. For instance, #linearalgebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of

From playlist Linear Algebra

Quantum Finite Elements: Lattice Field Theory on Curved Manifolds by Richard Brower

PROGRAM NONPERTURBATIVE AND NUMERICAL APPROACHES TO QUANTUM GRAVITY, STRING THEORY AND HOLOGRAPHY (HYBRID) ORGANIZERS: David Berenstein (University of California, Santa Barbara, USA), Simon Catterall (Syracuse University, USA), Masanori Hanada (University of Surrey, UK), Anosh Joseph (II

From playlist NUMSTRING 2022

Graph transformations 3 of the form af(x)