ISOMETRIC EMBEDDING Meaning and
Definition
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Isometric embedding, in mathematics, refers to the process of representing a given metric space within another metric space while preserving the distances between points. In a more detailed dictionary definition, isometric embedding can be described as a mapping or function from one metric space into another such that the distances between any two points in the original space are maintained in the image space.
An isometric embedding is expected to preserve the metric properties of the original space. Specifically, the distances between points in the image space should be equal to the distances between the corresponding points in the original space. This means that the image space is equipped with a metric that induces the same distances as the original metric space.
The concept of isometric embedding is commonly used in various branches of mathematics, including differential geometry, functional analysis, and graph theory. It has applications in studying the geometric properties of surfaces, understanding the structure of metric spaces, and analyzing data represented in high-dimensional spaces.
Isometric embeddings are not always possible or unique for every metric space. The existence and uniqueness depend on the specific properties of the metric spaces involved and the dimensionality of the embedding. However, when an isometric embedding is feasible, it provides a valuable tool for analyzing and comparing geometric structures between different metric spaces.
Etymology of ISOMETRIC EMBEDDING
The term "isometric embedding" comes from the combination of the words "isometric" and "embedding".
1. Isometric: The term "isometric" comes from the Greek words "isos" meaning "equal" and "metron" meaning "measure". In mathematics, "isometric" refers to a transformation or mapping that preserves distances between points. In simpler words, it is a transformation that maintains the same shape and size of objects but can change their position or orientation.
2. Embedding: The term "embedding" in mathematics refers to the process of mapping one mathematical object or structure into another. It represents a way to include or incorporate one thing within another. In the case of "isometric embedding", it specifically denotes the mapping of one mathematical space or manifold into another while preserving the distances between points.
