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The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...
In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). [1] [2] A paraxial ray is a ray that makes a small angle (θ) to the optical axis of the system, and lies close to the axis throughout the system. [1]
Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with distance and time coordinates. Using the inverse hyperbolic function artanh , the rapidity w corresponding to velocity v is w = artanh( v / c ) where c is the speed of light.
Geometrical optics is often simplified by making the paraxial approximation, or "small angle approximation". The mathematical behavior then becomes linear , allowing optical components and systems to be described by simple matrices.
3 Planar limit: small angles. 4 History. 5 See also. 6 Notes. ... we will use the small-angle approximation obtained from the Maclaurin series for the cosine and sine ...
Giving the measure of aberration in a plane normal to the optical axis is called a transversal aberration. The size (radius) of the aberration disc in this plane can be shown to be proportional to the cube of the incident angle (θ) under the small-angle approximation, and that the explicit form in this case is
From the large angle analysis it follows that this peak can only extend to about /. The forward peak is thus confined to a small solid angle of approximately π θ c 2 {\displaystyle \pi \theta _{c}^{2}} , and we may conclude that the total small angle cross section decreases with ϵ − 1 {\displaystyle \epsilon ^{-1}} .
The Coriolis force at latitude φ is horizontal in the small angle approximation and is given by , = ,, = , where Ω is the rotational frequency of Earth, F c,x is the component of the Coriolis force in the x direction, and F c,y is the component of the Coriolis force in the y direction.