Transformation (function)

In mathematics, a transformation is a function that maps elements from one set to another, often changing their form or structure in some way. Transformations can be classified into various types depending on their properties and the context in which they are used. Here are a few key types of transformations: 1. **Geometric Transformations**: These are transformations that affect the position, size, and orientation of geometric figures.

An affine transformation is a type of geometric transformation that preserves points, straight lines, and planes. It is a linear mapping method that can translate, scale, rotate, or shear objects in a coordinate space, and it can be represented mathematically using matrices. In an affine transformation, the relationship between original points and transformed points can be expressed in terms of linear algebra.

The Cole–Hopf transformation is a mathematical technique used to transform certain nonlinear partial differential equations into linear ones. It is primarily applied to the Burgers' equation, a fundamental equation in fluid dynamics and various fields of applied mathematics.

Glide reflection is a type of geometric transformation that combines two basic transformations: a translation and a reflection. It can be described in the following steps: 1. **Reflection**: An object is first reflected over a line (in two dimensions) or a plane (in three dimensions). This means that every point of the object is mapped to a corresponding point on the opposite side of the line or plane at an equal distance from it.

Homography is a concept from projective geometry that describes a specific type of transformation between two planes. In the context of computer vision and image processing, homography is often used to relate the coordinates of points in one image to the coordinates of points in another image, typically when those images are of the same scene from different perspectives. ### Mathematical Definition Mathematically, a homography can be represented by a \(3 \times 3\) matrix \(H\).

Homothety, also known as dilation or similarity transformation, is a geometric transformation that alters the size of a figure while maintaining its shape and relative proportions. It can be described as a scaling transformation around a specific point, known as the center of homothety. In formal terms, a homothety can be defined by the following characteristics: 1. **Center of Homothety**: This is a fixed point in the plane from which the scaling occurs.

The term "infinitesimal transformation" is commonly used in the context of differential geometry and mathematical physics, particularly in relation to symmetry operations and the study of continuous groups of transformations (Lie groups). An infinitesimal transformation is a transformation that changes a quantity by an infinitely small amount. Mathematically, it can be thought of as the limit of a transformation as the parameter defining that transformation approaches zero.

Late-stage functionalization (LSF) is a strategic approach in organic synthesis and medicinal chemistry that involves modifying a molecule after it has been synthesized. The primary goal of LSF is to introduce new functional groups or make specific changes to the chemical structure of a drug candidate or organic compound, enhancing its properties or tailoring it for specific applications.

In geometry, scaling refers to the process of resizing an object by a certain factor. This involves changing the dimensions of the object uniformly (maintaining the shape) or non-uniformly (changing shape) while preserving the proportions of the object. The factor by which the object is scaled is called the scale factor. When scaling is uniform, every linear dimension of the object (length, width, height) is multiplied by the same scale factor.