Geometrically this means that if the origin of reciprocal space is placed at the tip of then diffraction will occur only for reciprocal lattice points that lie on the surface of the Ewald sphere.

Geometrically, this means that a horizontal tangent line to the graph at one point ( x

Geometrically, this means replacing a complicated zonotope P by a simpler zonotope Q that encloses it.

Geometrically, such a matrix is always the composition of a rotation with a scaling, and in particular preserves angles.

Geometrically, ( x < sub > 1 </ sub >, 0 ) is the intersection with the x-axis of a line tangent to f at ( x < sub > 0 </ sub >, f ( x < sub > 0 </ sub >)).

Geometrically two edges meeting at a corner are required to form an angle that is not straight ( 180 °); otherwise, the line segments will be considered parts of a single edge – however mathematically, such corners may sometimes be allowed.

Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. Geometrically, this can be visualized ( at least in the case when r is an integer ) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.

Geometrically, the trace can be interpreted as the infinitesimal change in volume ( as the derivative of the determinant ), which is made precise in Jacobi's formula.

Geometrically, the result asserts that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set ( the solution set of a collection of polynomials in n variables is generally a geometric object ( such as a curve or a surface ) in n-space ).

where v is any fixed vector satisfying Av = b. Geometrically, this says that the solution set to Ax = b is the translation of the null space of A by the vector v. See also Fredholm alternative.

Geometrically, when the state is not expressible as a convex combination of other states, it is a pure state.

Geometrically, the relative position vector R < sub > B / A </ sub > is the vector from point A to point B.

Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image.

Geometrically, the Petersen graph is the graph formed by the vertices and edges of the hemi-dodecahedron, that is, a dodecahedron with opposite points, lines and faces identified together.

Geometrically, four-acceleration is a curvature vector of a world line.

Geometrically, a diagonalizable matrix is an inhomogeneous dilation ( or anisotropic scaling ) – it scales the space, as does a homogeneous dilation, but by a different factor in each direction, determined by the scale factors on each axis ( diagonal entries ).

Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length ; this follows from the Pythagorean theorem.

Geometrically, it is the

Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines x

Geometrically, the theorem states that an integrable module of 1-forms of rank r is the same thing as a codimension-r foliation.

Geometrically, a ribbon is a piece of the envelope of the osculating planes of the curve.

Geometrically, it is possible to " roll " a plane along the ribbon without slipping or twisting so that the regulus always remains within the plane.

Geometrically, the operation consists in cutting each vertex of the polyhedron with a plane cutting all edges adjacent to the vertex at their midpoints ; it is sometimes named rectification.

Geometrically, there is a rough correspondence between Kodaira dimension and curvature: negative Kodaira dimension corresponds to positive curvature, zero Kodaira dimension corresponds to flatness, and maximum Kodaira dimension ( general type ) corresponds to negative curvature.

Geometrically, the Gram determinant is the square of the volume of the parallelotope formed by the vectors.

Geometrically, one studies the Euclidean plane ( 2 dimensions ) and Euclidean space ( 3 dimensions ).

Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection ( from the new position of the sphere ) to the plane.

Geometrically, finitely generated projective modules correspond to vector bundles over affine space, and free modules to trivial vector bundles.

Geometrically, it can be described as an arrow from the origin of the space ( vector tail ) to that point ( vector tip ).

Geometrically, the vector-function () implements an embedding of the comfiguration space of the constrained Newtonian dynamical system into the-dimensional flat comfiguration space of the unconstrained

Geometrically, a sphaleron is simply a saddle point of the electroweak potential energy ( in the infinite dimensional field space ), much like the saddle point of the surface in three dimensional analytic geometry.

Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.

Geometrically incorrect attempt at perspective in a 1614 painting of Old St Paul's Cathedral.

Geometrically u and v are related as having orthogonal trajectories, away from the zeroes of the underlying holomorphic function ; the contours on which u and v are constant cross at right angles.

Geometrically they take the shape of being ovate, shaped like an egg, to elliptic or narrowly elliptic at 3 to 5 times as long as wide.

Geometrically, in singular cohomology of a manifold, one can interpret the product dually in terms of bounding manifolds and intersections, following Poincaré duality: dual to cocycles are cycles, often representable as closed manifolds ( without boundary ), dual to product is intersection, and dual to the subtraction of the bounding products is gluing the two bounding manifolds together along the boundary, obtaining a closed manifold which represents the homology class dual of the Massey product.

Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have K < sub > v </ sub >- rational points for all places v of K. The Selmer group is finite.

Geometrically, the swastika can be regarded as an irregular icosagon or 20-sided polygon.

Geometrically, as we move farther and farther to the right along the x-axis, the value of 1 / x < sup > 2 </ sup > approaches 0.

Geometrically, these subspaces are points, lines, planes, and so on, that pass through the point 0.

Geometrically, the curvature measures how fast the unit tangent vector to the curve rotates.

Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry.

Geometrically, a function and inverse function have graphs that are reflections, in the line.

Geometrically, affine transformations ( affinities ) preserve collinearity.

Geometrically this says that fibres of good mappings may have ' infinitesimal ' structure.

f ( x ) over some interval containing a. Geometrically, the graph defined by R ( x, y ) = 0 will overlap locally with the graph of some equation y = f ( x ).