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Orthogonal trajectories are used in mathematics, for example as curved coordinate systems (i.e. elliptic coordinates) and appear in physics as electric fields and their equipotential curves. If the trajectory intersects the given curves by an arbitrary (but fixed) angle, one gets an isogonal trajectory .
A conformal map acting on a rectangular grid. Note that the orthogonality of the curved grid is retained. While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum ...
The chart is smooth except for a polar coordinate style singularity along β = 0. See charts on SO(3) for a more complete treatment. The space of rotations is called in general "The Hypersphere of rotations ", though this is a misnomer: the group Spin(3) is isometric to the hypersphere S 3 , but the rotation space SO(3) is instead isometric to ...
The trajectory may be viewed as a rotation parametric in time, where the angle of rotation is linear in time: φ = ωt, with ω being an "angular velocity". Analogous to the 3D case, every rotation in 4D space has at least two invariant axis-planes which are left invariant by the rotation and are completely orthogonal (i.e. they intersect at a ...
Cassini ovals and their orthogonal trajectories (hyperbolas) Orthogonal trajectories of a given pencil of curves are curves which intersect all given curves orthogonally. For example the orthogonal trajectories of a pencil of confocal ellipses are the confocal hyperbolas with the same foci. For Cassini ovals one has:
Thus, while the static observers in the cylindrical chart admits a unique family of orthogonal hyperslices =, the Langevin observers admit no such hyperslices. In particular, the spatial surfaces = in the Born chart are orthogonal to the static observers, not to the Langevin observers. This is our second (and much more pointed) indication that ...
In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .
Orthographic projection (also orthogonal projection and analemma) [a] is a means of representing three-dimensional objects in two dimensions.Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, [2] resulting in every plane of the scene appearing in affine transformation on the viewing surface.