For instance, in the case of the parabola, the distance of any particle from the directrix is equal to its distance from the focus.
"The Romance of Mathematics" by P. Hampson
But General Thomas's plan turned on cavalry work as its directrix.
"Was General Thomas Slow at Nashville?" by Henry V. Boynton
The only surface of revolution having this property is the catenoid formed by the revolution of a catenary about its directrix.
"Encyclopaedia Britannica, 11th Edition, Volume 5, Slice 3" by Various
Thus AB is the directrix of the parabola VED, of which F is the focus.
"The New Gresham Encyclopedia" by Various
Similarly, in an hyperbola a vertex is nearer to the directrix than to the focus.
"Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6" by Various
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We proceed with the technical issue of recovering directrix from coordinates and momenta of strings pieces at a given time t0 in a way which can be used in numerical simulations.
On asimuthal anisotropy in fragmentation of classical relativistic string
Upon integrating (13) and taking into account that y(t0) = x(t0 , σ = 0) is a position of one of the string’s ends, one recovers the directrix on the interval t ∈ (t0 − σmax , t0 + σmax ), and, hence, at any point due to its periodicity.
On asimuthal anisotropy in fragmentation of classical relativistic string
To illustrate the technique described in the previous section, we reconstruct directrix for two simple model initial conditions.
On asimuthal anisotropy in fragmentation of classical relativistic string
To recover directrix we’ll start with drawing a tra jectory of the upper endpoint of the string for (0, σmax) time interval, that is using ﬁrst line of (14).
On asimuthal anisotropy in fragmentation of classical relativistic string
For σ ∈ (p/κ , p/κ + b), we have only the second one of the two differentials in the directrix increment (14) which goes against x2 .
On asimuthal anisotropy in fragmentation of classical relativistic string
Directrix pieces for t ∈ (−σmax , 0) are constructed in exactly the same manner making use of the second line of (14).
On asimuthal anisotropy in fragmentation of classical relativistic string
Finally, combining the differentials which is straightforward, we ﬁnd that the directrix in this case represents a rectangular box with argument σ changing linearly along the rectangle lines.
On asimuthal anisotropy in fragmentation of classical relativistic string
The directrix is shown in ﬁg. 2a together with string positions at some selected time moments.
On asimuthal anisotropy in fragmentation of classical relativistic string
The directrix for the ’invariant’ case and string position at any time can be obtained in the same fashion as in the previous example.
On asimuthal anisotropy in fragmentation of classical relativistic string
Let us write down the directrix for parameterization deﬁned by α and impact parameter in its center of mass frame.
On asimuthal anisotropy in fragmentation of classical relativistic string
The last thing to note is that, the argument of the directrix is equal to its length measured from the point at which we start the directrix construction due to the condition |y′ | = 1 .
On asimuthal anisotropy in fragmentation of classical relativistic string
An example of this type of directrix together with string positions at some time moments is presented in the ﬁgure 2 b.
On asimuthal anisotropy in fragmentation of classical relativistic string
After directrix is reconstructed numerically from the initial conditions as described in previous sections, time evolution of the string arrangement in space is done step by step.
On asimuthal anisotropy in fragmentation of classical relativistic string
Figure 6: Construction of the folding parabola: focus F and directrix D.
Fractal Homeomorphism for Bi-affine Iterated Function Systems
Namely, a parabola is the locus of points from where the distances to a ﬁxed point (focus) and to a ﬁxed line (directrix) are equal.
Old and new about equidistant sets and generalized conics
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