Isometrisiin

Isometrisiin is a Finnish inflected form of the adjective isometrinen, meaning isometric or related to isometry. It appears in phrases describing concepts or objects connected with isometry, such as isometric transformations or isometric spaces, depending on the grammatical case and usage in a sentence. While isometriseen is not a stand-alone mathematical term in Finnish, it is commonly found in texts discussing isometry in mathematics or Applications.

In mathematics, an isometry is a transformation that preserves distances between points. Formally, a function f between metric spaces satisfies d(f(x), f(y)) = d(x, y) for all points x and y in the domain. In Euclidean space, familiar examples of isometries include translations, rotations, reflections, and glide reflections. The composition of two isometries is again an isometry, and the inverse of an isometry is also an isometry, making the set of all isometries under composition a group.

Key properties of isometries include the preservation of distances and, in Euclidean settings, also the preservation of angles between corresponding vectors. They map lines to lines and preserve straightness and collinearity, though orientation may be preserved or reversed depending on the specific transformation. The study of isometries leads to the concept of isometry groups, such as the Euclidean group E(n), which describes all distance-preserving motions of n-dimensional space. Linear isometries form the orthogonal group O(n) when anchored at the origin.

Applications of isometries span computer graphics, robotics, computer vision, crystallography, and geometric modeling, where preserving shape and size under transformations is essential. In Finnish texts, isometrisiin typically appears as part of a larger phrase describing isometric concepts rather than as a separate mathematical noun.