Porter and Duff gave a geometric interpretation of the alpha compositing formula by studying orthogonal coverages.

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.

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.

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.

Transformations can be applied to any geometric equation whether or not the equation represents a function.

This can be proved ( for integer t ) by using the formula for geometric series: ( using y = 1-x )

An " elementary " proof can be given using the fact that geometric mean of positive numbers is less than arithmetic mean

Using the Cartesian coordinate system, geometric shapes ( such as curves ) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape.

Smooth manifolds are ' softer ' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology.

For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold — that is, one can smoothly " flatten out " certain manifolds, but it might require distorting the space and affecting the curvature or volume.

( In the last step, the summation is trivial if, where it is 1 + 1 +⋅⋅⋅= N, and otherwise is a geometric series that can be explicitly summed to obtain zero.

Fractals are not limited to geometric patterns, but can also describe processes in time.

This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractional or fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived.

One often cited description that Mandelbrot published to describe geometric fractals is " a rough or fragmented geometric shape that can be split into parts, each of which is ( at least approximately ) a reduced-size copy of the whole "; this is generally helpful but limited.

The values can be substituted into Legendre ’ s identity and the approximations to K, E can be found by terms in the sequences for the arithmetic geometric mean with and.

For example, the geometric mean can give a meaningful " average " to compare two companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0 to 100 for their financial viability.

The geometric mean can also be understood in terms of geometry.

This can be seen easily from the fact that the sequences do converge to a common limit ( which can be shown by Bolzano – Weierstrass theorem ) and the fact that geometric mean is preserved:

For example, the geometric mean of 2 and 8 can be calculated as:

The fundamental property of the geometric mean, which can be proven to be false for any other mean, is

Charges can be animals, objects, or geometric shapes.

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set.

Felix Klein argued in his Erlangen program that one can consider various " geometries " by specifying an appropriate transformation group that leaves certain geometric properties invariant.

The modern study of mineralogy was founded on the principles of crystallography ( the origins of geometric crystallography, itself, can be traced back to the mineralogy practiced in the eighteenth and nineteenth centuries ) and to the microscopic study of rock sections with the invention of the microscope in the 17th century.