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The geometric interpretation of curl as rotation corresponds to identifying bivectors ( 2-vectors ) in 3 dimensions with the special orthogonal Lie algebra so ( 3 ) of infinitesimal rotations ( in coordinates, skew-symmetric 3 × 3 matrices ), while representing rotations by vectors corresponds to identifying 1-vectors ( equivalently, 2-vectors ) and so ( 3 ), these all being 3-dimensional spaces.
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geometric and interpretation
Porter and Duff gave a geometric interpretation of the alpha compositing formula by studying orthogonal coverages.
The radius of curvature is introduced completely formally ( without need for geometric interpretation ) as:
However, in this approach the question of the change in radius of curvature with s is handled completely formally, consistent with a geometric interpretation, but not relying upon it, thereby avoiding any questions the image above might suggest about neglecting the variation in ρ.
A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation.
Via this interpretation, geometric operations are realized as algebraic operations in the algebra.
A geometric interpretation of Euler's formula
Picture showing the geometric interpretation of the first iteration of Grover's algorithm.
The interpretation of Bradford's law in terms of a geometric progression was suggested by V. Yatsko who introduced an additional constant and demonstrated that Bradford distribution can be applied to a variety of objects, not only to distribution of articles or citations across journals.
Owing to the geometric interpretation of the dot product, the norm || a || of a vector a in such an inner product space is defined as:
These two requirements show that stochastic vectors have a geometric interpretation: A stochastic vector is a point on the " far face " of a standard orthogonal simplex.
The concept of the complex plane allows a geometric interpretation of complex numbers.
A geometric interpretation to scalar multiplication is a stretching or shrinking of a vector.
Nevertheless it must be stressed that even though it is not a tensor field, it still qualifies as a geometric object with a component-free interpretation.
More generally, the Ricci tensor can be defined in broader class of metric geometries ( by means of the direct geometric interpretation, below ) that includes Finsler geometry.
Principal curves and manifolds give the natural geometric framework for nonlinear dimensionality reduction and extend the geometric interpretation of PCA by explicitly constructing an embedded manifold, and by encoding using standard geometric projection onto the manifold.
( Includes the geometric variational interpretation for the Haralick-Canny edge detector.
This has the geometric interpretation that multiplication by a unit complex number is a proper rotation in the complex plane, and every such rotation is of this form.
A geometric interpretation of the fibration may be obtained using the complex projective line, CP < sup > 1 </ sup >, which is defined to be the set of all complex one dimensional subspaces of C < sup > 2 </ sup >.
The geometric interpretation of De Casteljau's algorithm is straightforward.
In the case that T acts on euclidean space R < sup > n </ sup >, there is a simple geometric interpretation for the singular values: Consider the image by T of the unit sphere ; this is an ellipsoid, and its semi-axes are the singular values of T ( the figure provides an example in R < sup > 2 </ sup >).
A serendipitous encounter with lecture notes by mathematician Marcel Riesz inspired Hestenes to study a geometric interpretation of Dirac matrices.
Loop quantum gravity inherits this geometric interpretation of gravity, and posits that a quantum theory of gravity is fundamentally a quantum theory of spacetime.
geometric and rotation
For a given geometric figure in a given geometric space, consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two images of the figure are the same ( here " the same " does not mean something like e. g. " the same up to translation and rotation ", but it means " exactly the same ").
These components can be modified and manipulated by two-dimensional geometric transformations such as translation, rotation, scaling.
Examples of affine transformations include translation, geometric contraction, expansion, homothety, reflection, rotation, shear mapping, similarity transformation, and spiral similarities and compositions of them.
* through geometric transformations such as scaling, reflection, and rotation ;
Its axis of rotation is observed to return to its original orientation with respect to the earth after one day whatever the latitude, not subject to the unbalanced Coriolis forces acting on the pendulum as a result of its geometric asymmetry.
It is a purely geometric characteristic of the object, as it depends only on its shape and the position of the rotation axis.
They were initially used to accelerate the memory-intensive work of texture mapping and rendering polygons, later adding units to accelerate geometric calculations such as the rotation and translation of vertices into different coordinate systems.
Most common geometric transformations that keep the origin fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection ; if an affine transformation is not a pure translation it keeps some point fixed, and that point can be chosen as origin to make the transformation linear.
A geometric rotation transforms lines to lines, and preserves ratios of distances between points.
a geometric description of the location and rotation of each part, and the exact path of each wire connecting them.
Using the geometric axis as the primary axis of a new body-fixed coordinate system, one arrives at the Euler equation of a gyroscope describing the apparent motion of the rotation axis about the geometric axis of the Earth.
The location of the geometric roll center is solely dictated by the suspension geometry, and can be found using principles of the instant center of rotation.
geometric and corresponds
These reflections generate a Coxeter group W, called the Weyl group of A, and the simplicial complex A corresponds to the standard geometric realization of W. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in A.
A more familiar geometric way to understand the projective transforms is via projective rotations ( the elements of PSO ( n + 1 )), which corresponds to the stereographic projection of rotations of the unit hypersphere, and has dimension Visually, this corresponds to standing at the origin ( or placing a camera at the origin ), and turning one's angle of view, then projecting onto a flat plane.
The development of new forms of geometric projection in the construction of perspective corresponds with the invention of novel pictorial art forms of visual representation in the Italian Renaissance, since the fourteenth century and up till the end of the sixteenth century, and specifically within the circles of architectural and artistic experimentation and design.
Non-singularity ( regularity ) is still stronger — it corresponds to the notion of smoothness of a geometric object at a particular point.
q = 1 corresponds to the geometric mean and q = 2 to the arithmetic mean.
* If we take to be the set of positive real numbers and, then the f-mean corresponds to the geometric mean.
The geometric center frequency corresponds to a mapping of the DC response of the prototype lowpass filter, which is a resonant frequency sometimes equal to the peak frequency of such systems, for example as in a Butterworth filter.
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves ; the dimension of the linear system corresponds to the number of parameters of the family.
To decide in practice when a Cartier divisor D corresponds to an ample line bundle, there are some geometric criteria.
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