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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system.
Now for any quaternion vector p, p* = −p, let q = 1 + pε ∈ F, where the required rotation and translation are effected. Evidently the group of units of the ring of dual quaternions is a Lie group. A subgroup has Lie algebra generated by the parameters a r and b s, where a, b ∈ R, and r, s ∈ H. These six parameters generate a subgroup ...
Parallel transport of a vector around a closed loop (from A to N to B and back to A) on the sphere. The angle by which it twists, , is proportional to the area inside the loop. In differential geometry, parallel transport (or parallel translation [a]) is a way of transporting geometrical data along smooth curves in a manifold.
Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group. In the case of glide-reflection symmetry, the symmetry group of an object contains a glide reflection, and hence the group generated by it.
A common problem in computer graphics is to generate a non-zero vector in ℝ 3 that is orthogonal to a given non-zero vector. There is no single continuous function that can do this for all non-zero vector inputs. This is a corollary of the hairy ball theorem. To see this, consider the given vector as the radius of a sphere and note that ...
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal. For such matrices, the half-vectorization is sometimes more useful than the ...
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. [5] [6] Representations of more abstract objects in terms of familiar linear algebra can elucidate properties and simplify calculations within more abstract theories.
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