[permalink] [id link]
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object.
from
Wikipedia
Some Related Sentences
Euclidean and space
To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product, i. e.
Geometrically, one studies the Euclidean plane ( 2 dimensions ) and Euclidean space ( 3 dimensions ).
This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
So, for example, while R < sup > n </ sup > is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm.
It is simpler to see the notational equivalences between ordinary notation and bra-ket notation, so for now ; consider a vector A as an element of 3-d Euclidean space using the field of real numbers, symbolically stated as.
For example, in three-dimensional complex Euclidean space,
Structures analogous to those found in continuous geometries ( Euclidean plane, real projective space, etc.
Continuum mechanics models begin by assigning a region in three dimensional Euclidean space to the material body being modeled.
Different configurations or states of the body correspond to different regions in Euclidean space.
The Bolzano – Weierstrass theorem gives an equivalent condition for sequential compactness when considering subsets of Euclidean space: a set then is compact if and only if it is closed and bounded.
Euclidean space itself is not compact since it is not bounded.
Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces include spaces consisting not of geometrical points but of functions.
A subset of Euclidean space in particular is called compact if it is closed and bounded.
That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine – Borel theorem.
; Euclidean space
For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:
Some examples of convex subsets of a Euclidean 3-dimensional space are the Archimedean solids and the Platonic solids.
The algebra M < sub > n </ sub >( C ) of n-by-n matrices over C becomes a C *- algebra if we consider matrices as operators on the Euclidean space, C < sup > n </ sup >, and use the operator norm ||.|| on matrices.
The convolution can be defined for functions on groups other than Euclidean space.
Graphing calculators can be used to graph functions defined on the real line, or higher dimensional Euclidean space.
The space R of real numbers and the space C of complex numbers ( with the metric given by the absolute value ) are complete, and so is Euclidean space R < sup > n </ sup >, with the usual distance metric.
Euclidean and object
For nearby astronomical objects ( such as stars in our galaxy ) luminosity distance D < sub > L </ sub > is almost identical to the real distance to the object, because spacetime within our galaxy is almost Euclidean.
For much more distant objects the Euclidean approximation is not valid, and General Relativity must be taken into account when calculating the luminosity distance of an object.
In mathematics, physics, and engineering, a Euclidean vector ( sometimes called a geometric or spatial vector, or — as here — simply a vector ) is a geometric object that has a magnitude ( or length ) and direction and can be added to other vectors according to vector algebra.
This combination of properties cannot be realized by any 3-dimensional object in ordinary Euclidean space.
Such an object can exist in certain Euclidean 3-manifolds.
There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space ( usually a Euclidean space ) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold.
The acceleration Euclidean vector | vectors ( a ) in this visual depict a positively buoyant object which naturally accelerates upward, and upward acceleration of the object is our sign convention.
A 2D geometric model is a geometric model of an object as two-dimensional figure, usually on the Euclidean or Cartesian plane.
A smooth manifold is a mathematical object which looks locally like a smooth deformation of Euclidean space R < sup > n </ sup >: for example a smooth curve or surface looks locally like a smooth deformation of a line or a plane.
The Euclidean division of polynomials has been the object of specific developments.
An object is recognized in a new image by individually comparing each feature from the new image to this database and finding candidate matching features based on Euclidean distance of their feature vectors.
The duocylinder, or double cylinder, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of radius r:
Euclidean and is
In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula
Although Dürer made no innovations in these areas, he is notable as the first Northern European to treat matters of visual representation in a scientific way, and with understanding of Euclidean principles.
The alternated cubic honeycomb is one of 28 space-filling uniform tessellations in Euclidean 3-space, composed of alternating yellow tetrahedron | tetrahedra and red octahedron | octahedra.
Bézout's lemma is a consequence of the Euclidean division defining property, namely that the division by a nonzero integer b has a remainder strictly less than | b |.
In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.
A circle is a simple shape of Euclidean geometry that is the set of points in the plane that are equidistant from a given point, the
A fast way to determine whether two numbers are coprime is given by the Euclidean algorithm.
0.318 seconds.