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In 2017, over 3 million tests were taken in more than 140 countries, up from 2 million tests in 2012, 1.7 million tests in 2011 and 1.4 million tests in 2009. In 2007, IELTS administered more than one million tests in a single 12-month period for the first time ever, making it the world's most popular English language test for higher education ...
Line–plane intersection; Line–sphere intersection; Intersection of a polyhedron with a line; Line segment intersection; Intersection curve; Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations.
The three possible line-sphere intersections: 1. No intersection. 2. Point intersection. 3. Two point intersection. In analytic geometry, a line and a sphere can intersect in three ways: No intersection at all; Intersection in exactly one point; Intersection in two points.
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are ...
Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs ...
For any natural number n, an n-sphere, often denoted S n, is the set of points in (n + 1)-dimensional Euclidean space that are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular: S 0: a 0-sphere consists of two discrete points, −r and r; S 1: a 1-sphere is a ...
This transformation leaves the sphere in place, but moves points within the sphere according to a Möbius transformation. [15] Any polyhedron with a midsphere, scaled so that the midsphere is the unit sphere, can be transformed in this way into a polyhedron for which the centroid of the points of tangency is at the center of the sphere.
a single line, no points; a single point, a collection of lines, the point is incident with all of the lines; a single line, a collection of points, the points are all incident with the line; a point P incident with a line m, an arbitrary collection of lines all incident with P and an arbitrary collection of points all incident with m;