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The full geometric algebra in three dimensions, Cl 3 (R), has basis (1, e 1, e 2, e 3, e 23, e 31, e 12, e 123). The element e 123 is a trivector and the pseudoscalar for the geometry. Bivectors in three dimensions are sometimes identified with pseudovectors [ 17 ] to which they are related, as discussed below .
Analogous to straight line segments above, one can also define arcs as segments of a curve. In one-dimensional space, a ball is a line segment. An oriented plane segment or bivector generalizes the directed line segment. Beyond Euclidean geometry, geodesic segments play the role of line segments.
A bivector is an oriented plane segment, analogous to directed line segments. [a] A face is a plane segment bounding a solid object. [1] A slab is a region bounded by two parallel planes. A parallelepiped is a region bounded by three pairs of parallel planes.
A vector v (red) represented by • a vector basis (yellow, left: e 1, e 2, e 3), tangent vectors to coordinate curves (black) and • a covector basis or cobasis (blue, right: e 1, e 2, e 3), normal vectors to coordinate surfaces (grey) in general (not necessarily orthogonal) curvilinear coordinates (q 1, q 2, q 3). The basis and cobasis do ...
Its sides are arcs of great circles—the spherical geometry equivalent of line segments in plane geometry. Such polygons may have any number of sides greater than 1. Two-sided spherical polygons— lunes , also called digons or bi-angles —are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a ...
Let be a metric space with distance function .Let be a set of indices and let () be a tuple (indexed collection) of nonempty subsets (the sites) in the space .The Voronoi cell, or Voronoi region, , associated with the site is the set of all points in whose distance to is not greater than their distance to the other sites , where is any index different from .
sum of three equal lengthed vectors. Sylvester's theorem or Sylvester's formula describes a particular interpretation of the sum of three pairwise distinct vectors of equal length in the context of triangle geometry. It is also referred to as Sylvester's (triangle) problem in literature, when it is given as a problem rather than a theorem.
Three line segments with the same direction. In geometry, direction, also known as spatial direction or vector direction, is the common characteristic of all rays which coincide when translated to share a common endpoint; equivalently, it is the common characteristic of vectors (such as the relative position between a pair of points) which can be made equal by scaling (by some positive scalar ...