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Because of the various mathematical components surrounding frames, frame theory has roots in harmonic and functional analysis, operator theory, linear algebra, and matrix theory. [3] The Fourier transform has been used for over a century as a way of decomposing and expanding signals. However, the Fourier transform masks key information ...
In this setting, a frame carries the geometric idea of a basis of a vector space over to other sorts of geometrical spaces (Klein geometries). Some examples of frames are: [3] A linear frame is an ordered basis of a vector space. An orthonormal frame of a vector space is an ordered basis consisting of orthogonal unit vectors (an orthonormal basis).
For example, if is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of . The orthonormal frame bundle is just a reduction of the structure group of F G L ( M ) {\displaystyle F_{\mathrm {GL} }(M)} to the orthogonal group O ( n ) {\displaystyle \mathrm {O} (n)} .
The black dot shows the point with coordinates x = 2, y = 3, and z = 4, or (2, 3, 4). A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines (the axes ) that go through a common point (the origin ), and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all ...
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 ...
These three planes are marked A, B and C, where A is the face side, B is the first datum edge, and C is the second datum edge. In this case, the datum reference frame is A/B/C. A/B/C is shown at the end of feature control frame to show from where the measurement is taken. (See the ASME standard Y14.5M-2009 for more examples and material modifiers.)
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