Euclidean

Examples of Euclidean in the following topics:

• 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.

• 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 $\vec{AB}$.

• 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: $\vec r: \Bbb{R}^2 \rightarrow \Bbb{R}^3$.

• 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.

• 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).

• 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.

• 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.

• 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:

• 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.

• Matrix and Vector Norms

  • For $p=2$ this is just the ordinary Euclidean norm: $\|\mathbf{x}\| _ 2 = \sqrt{\mathbf{x}^T \mathbf{x}}$ .