Euclidean
(adjective)
Adhering to the principles of traditional geometry, in which parallel lines are equidistant.
Examples of Euclidean in the following topics:
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Surfaces in Space
- The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space $R^3$— for example, the surface of a ball.
- On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self-intersections.
- Historically, surfaces were initially defined as subspaces of Euclidean spaces.
- Such a definition considered the surface as part of a larger (Euclidean) space, and as such was termed extrinsic.
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Vectors in Three Dimensions
- A Euclidean vector is a geometric object that has magnitude (i.e. length) and direction.
- A Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra.
- A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point $A$ with a terminal point $B$, and denoted by $\vec{AB}$.
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Parametric Surfaces and Surface Integrals
- A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation.
- A parametric surface is a surface in the Euclidean space $R^3$ which is defined by a parametric equation with two parameters: $\vec r: \Bbb{R}^2 \rightarrow \Bbb{R}^3$.
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Valued relations
- Several "distance" measures are fairly commonly used in network analysis, particularly the Euclidean distance or squared Euclidean distance.
- Figure 13.5 shows the Euclidean distances among the Knoke organizations calculated using Tools>Dissimilarities and Distances>Std Vector dissimilarities/distances.
- The Euclidean distance between two vectors is equal to the square root of the sum of the squared differences between them.
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Clustering similarities or distances profiles
- Depending on how the relations between actors have been measured, several common ways of constructing the actor-by-actor similarity or distance matrix are provided (correlations, Euclidean distances, total matches, or Jaccard coefficients).
- Alternatively, the adjacencies can be turned into a valued measure of dissimilarity by calculating geodesic distances (in which case correlations or Euclidean distances might be chosen as a measure of similarity).
- The first panel shows the structural equivalence matrix - or the degree of similarity among pairs of actors (in this case, dis-similarity, since we chose to analyze Euclidean distances).
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Vector Fields
- A vector field is an assignment of a vector to each point in a subset of Euclidean space.
- In vector calculus, a vector field is an assignment of a vector to each point in a subset of Euclidean space.
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Shape
- In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings.
- Having the same shape is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape.
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Applications of Multiple Integrals
- The gravitational potential associated with a mass distribution given by a mass measure $dm$ on three-dimensional Euclidean space $R^3$ is:
- If there is a continuous function $\rho(x)$ representing the density of the distribution at $x$, so that $dm(x) = \rho (x)d^3x$, where $d^3x$ is the Euclidean volume element, then the gravitational potential is:
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Equivalence of distances: Maxsim
- The distances of each actor re-organized into a sorted list from low to high, and the Euclidean distance is used to calculate the dissimilarity between the distance profiles of each pair of actors.
- The Euclidean distances between these lists are then created as a measure of the non-automorphic-equivalence, and hierarchical clustering is applied.
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Matrix and Vector Norms
- For $p=2$ this is just the ordinary Euclidean norm: $\|\mathbf{x}\| _ 2 = \sqrt{\mathbf{x}^T \mathbf{x}}$ .