transformation

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Figure 15: Effects of transformations used to construct algebras of points: (left) translation, (right) homothety.

transformation

mathematics

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July 23, 2026

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Figure 15: Effects of transformations used to construct algebras of points: (left) translation, (right) homothety.

In mathematics, a transformation is a rule or process that changes one mathematical object into another. This can involve altering geometric figures, matrices, or vectors. A common type is a linear transformation, where the new components are linear combinations of the original ones, such as flipping, stretching, or rotating a figure.12

Another form is the integral transform, which creates a new function by integrating the product of an existing function and a kernel function. These are useful for simplifying complex problems, particularly differential equations, by converting them into more easily solvable algebraic equations. The process of changing coordinates to represent mathematical objects in a new way is also a form of transformation.345

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trigonometry
Transformation of coordinates
A transformation of coordinates in a plane is a change from one coordinate system to another. Thus, a point in the plane will have two sets of coordinates giving its position with respect to the two coordinate systems used, and a transformation will express the relationship…
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A transformation of coordinates in a plane is a change from one coordinate system to another. Thus, a point in the plane will have two sets of coordinates giving its position with respect to the two coordinate systems used, and a transformation will express the relationship…
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fixed-point theorem
…in mathematics dealing with a transformation of the points of a set into points of the same set where it can be proved that at least one point remains fixed. For example, if each real number is squared, the numbers zero and one remain fixed; whereas the transformation whereby each…
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…in mathematics dealing with a transformation of the points of a set into points of the same set where it can be proved that at least one point remains fixed. For example, if each real number is squared, the numbers zero and one remain fixed; whereas the transformation whereby each…
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mathematics
The foundations of geometry
…“space,” and a group of transformations by means of which figures could be moved around in the space without altering their essential properties. For example, in Euclidean plane geometry the space is the familiar plane, and the transformations are rotations, reflections, translations, and their composites, none of which change either…
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…“space,” and a group of transformations by means of which figures could be moved around in the space without altering their essential properties. For example, in Euclidean plane geometry the space is the familiar plane, and the transformations are rotations, reflections, translations, and their composites, none of which change either…
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Sophus Lie
…on the theory of contact transformations. Appointed extraordinary professor in 1872, he began to research continuous transformation groups in 1873. After working in virtual isolation for more than 10 years, Lie was joined by the German mathematician Friedrich Engel (1861–1941), who had just received his doctorate from the University of…
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…on the theory of contact transformations. Appointed extraordinary professor in 1872, he began to research continuous transformation groups in 1873. After working in virtual isolation for more than 10 years, Lie was joined by the German mathematician Friedrich Engel (1861–1941), who had just received his doctorate from the University of…
Read More