mathematics, an orthogonal trajectory is a curve which intersects any curve of a given pencil of (planar) curves orthogonally. For example, the orthogonal trajectories...

represents the entire trajectory as a single high-order orthogonal polynomial. Mesh (Grid) After transcription, the formerly continuous trajectory is now represented...

Orthogonal trajectories of a given pencil of curves are curves which intersect all given curves orthogonally. For example the orthogonal trajectories...

polynomials Orthogonal polyhedron Orthogonal trajectory Orthogonalization Gram–Schmidt process Orthonormal basis Orthonormality Pan-orthogonality occurs in...

of intersections on each orthogonal trajectory. Here, the vertex of the hyperboloid serves as the identity on each trajectory curve. Topologically, a complex...

general orthogonal coordinates may be obtained by starting with some necessary coordinate surfaces and considering their orthogonal trajectories. In Cartesian...

parameterized by eccentricity ε {\displaystyle \varepsilon } . The orthogonal trajectories of this family are given by the differential equation ( 1 y + y...

\varphi =\cdots \ .} ) An orthogonal trajectory of a pencil of curves is a curve which intersects any curve of the pencil orthogonally. For cardioids the following...

− a ) − n {\displaystyle w=z^{m}(z-a)^{-n}} is analytic so the orthogonal trajectories of the family a r g ( w ) = c o n s t . {\displaystyle arg(w)=const...

zeros of f ′(z)). That means that v is a specific solution of the orthogonal trajectory problem for the family of contours given by u (not the only solution...

separation of variables Nicolaus I Bernoulli's s contributions: Orthogonal trajectories Probability and statistics research with the St. Petersburg Paradox...

between P {\displaystyle P} and T ( P ) {\displaystyle T(P)} . The orthogonal trajectories of these curves represent all genus 1 curves with j ≤ 1 {\displaystyle...

mathematician Brook Taylor. In 1720 he posed the problem of reciprocal orthogonal trajectories, which was intended as a challenge for the English Newtonians....

space, and is commonly denoted E(n) or ISO(n), for inhomogeneous special orthogonal group. The Euclidean group E(n) comprises all translations, rotations...

orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite...

is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement...

cylinder in common. The idea of threefold orthogonal systems can be seen as a generalization of orthogonal trajectories. Special examples are systems of confocal...

the Hamilton–Jacobi equation approach to mechanics. The orthogonality of mechanical trajectories characteristic of geometrical optics to the optical wavefronts...

distinguished from concepts such as an "orbit" or a "trajectory" (e.g., a planet's orbit in space or the trajectory of a car on a road) by inclusion of the dimension...

|A\rangle } and | B ⟩ {\displaystyle |B\rangle } are two possible orthogonal trajectories of the particle. The state | Ψ ⟩ {\displaystyle |\Psi \rangle }...

invariant, the rotation angle α {\displaystyle \alpha } for a proper orthogonal 3×3 rotation matrix A {\displaystyle A} is found by α = cos − 1 ⁡ ( A...

charged particles and photons and can give positional information on their trajectory, by tracking the trails of gaseous ionization. The technique was an improvement...

defined this class in terms of equidistant curves which are the orthogonal trajectories of a family of hyperplanes, which also can be seen as solutions...

laws of planetary motion Elliptic orbit Hyperbolic trajectory Parabolic trajectory Radial trajectory Orbit modeling Copernicus. pp 513–514 Bate, Mueller...

the pulse of the English analysts". The problem was to find the orthogonal trajectories of a series of curves represented by a single equation. Newton...

without changing size. To remain orthogonal to uρ while the trajectory r(t) rotates an amount dθ, uθ, which is orthogonal to r(t), also rotates by dθ. See...

⁠ and ⁠ V {\displaystyle \mathbf {V} } ⁠ can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted U Σ V T . {\displaystyle...

twist vector, which vanishes when the Killing vector is hypersurface orthogonal. The latter arises as the spatial components of the twist 4-vector ω μ...

algorithm is sometimes called partial Procrustes superimposition (see also orthogonal Procrustes problem). Let P and Q be two sets, each containing N points...

provide the best visual fit; which usually means trying to minimize the orthogonal distance to the curve (e.g., total least squares), or to otherwise include...