formulasearchengine - User contributions [en]

Exponential smoothing

2014-02-26T17:27:08Z

128.2.241.62: "formulae" to "formula" (plural to singular); "y" to "x" because x is used everywhere else for the same thing

I'm Sommer (18) from Bregnano, Italy. I'm learning French literature at a local university and I'm just about to graduate. I have a part time job in a backery. Also visit my web page - [http://Ommarelard.Mustafahosny.com/index.php?option=com_content&view=article&id=6988:2012-05-29-05-44-49&catid=5:trials&Itemid=18 Fifa 15 Coin Generator]

Linear equation

2014-01-30T22:47:40Z

128.2.185.124: /* Two-point form */ The right side of Expanding should be reversed(multiple -1).

{{About|the geographical reference system}}
[[File:Sphere filled blue.svg|thumb|200px|right|A graticule on a [[sphere]] or an [[ellipsoid]]. The lines from pole to pole are lines of constant [[longitude]], or '''meridians'''. The circles parallel to the equator are lines of constant latitude, or '''parallels'''. The graticule determines the latitude and longitude of position on the surface.]]

In [[geography]], '''latitude''' (φ) is a [[geographic coordinate]] that specifies the north-south position of a point on the Earth's surface. Latitude is an angle (defined below) which ranges from 0° at the [[Equator]] to 90° (North or South) at the poles. Lines of constant latitude, or '''parallels''', run east–west as circles parallel to the equator. Latitude is used together with [[longitude]] to specify the precise location of features on the surface of the Earth. Since the actual physical surface of the Earth is too complex for mathematical analysis, two levels of abstraction are employed in the definition of these coordinates. In the first step the physical surface is modelled by the [[geoid]], a surface which approximates the [[Sea level|mean sea level]] over the oceans and its continuation under the land masses. The second step is to approximate the geoid by a mathematically simpler reference surface. The simplest choice for the reference surface is a [[sphere]], but the geoid is more accurately modelled by an [[ellipsoid]]. The definitions of latitude and longitude on such reference surfaces are detailed in the following sections. Lines of constant latitude and longitude together constitute a [[Geographic coordinate system|graticule]] on the reference surface. The latitude of a point on the ''actual'' surface is that of the corresponding point on the reference surface, the correspondence being along the [[Normal (geometry)|normal]] to the reference surface which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute a [[geographic coordinate system]] as defined in the specification of the ISO 19111 standard.<ref name=iso19111>The current full documentation of ISO 19111 may be purchased from http://www.iso.org but drafts of the final standard are freely available at many web sites, one such is available at the following
[https://www.seegrid.csiro.au/wiki/pub/Xmml/CoordinateReferenceSystems/19111_FDIS20021107.pdf CSIRO]</ref>

Since there are many different [[reference ellipsoid]]s the latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates (that is latitude and longitude) are ambiguous at best and meaningless at worst". This is of great importance in accurate applications, such as [[GPS]], but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated.

In English texts the latitude angle, defined below, is usually denoted by the Greek lower-case letter [[phi (letter)|phi]] ([[φ]] or [[ɸ]]). It is measured in [[Degree (angle)|degrees]], [[arcminute|minutes and seconds]] or decimal degrees, north or south of the equator.


Measurement of latitude requires an understanding of the gravitational field of the Earth, either for setting up [[theodolite]]s or for determination of [[GPS]] satellite orbits. The study of the [[figure of the Earth]] together with its gravitational field is the science of [[geodesy]]. These topics are not discussed in this article. (See for example the textbooks by Torge<ref name=torge>Torge, W (2001) Geodesy (3rd edition), published by de Gruyter, isbn=3-11-017072-8</ref> and Hofmann-Wellenhof and Moritz.<ref name=wellenhofmoritz>Hofmann-Wellenhof, B and Moritz, H (2006). 'Physical Geodesy (second edition)' ISBN3211-33544-7.</ref>)


This article relates to coordinate systems for the Earth: it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature.

The following lists are available:
* [[List of cities by latitude]]
* [[List of countries by latitude]]


==Latitude on the sphere==

[[File:latitude and longitude graticule on a sphere.svg|thumb|200px|right|A perspective view of the Earth showing how latitude (φ) and longitude (λ) are defined on a spherical model. The graticule spacing is 10 degrees.]]

===The graticule on the sphere===
The graticule formed by the lines of constant latitude and constant longitude is constructed with reference to the rotation axis of the Earth. The primary reference points are the [[Geographical pole|poles]] where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface in the [[Meridian (geography)|meridians]] and the angle between any one meridian plane and that through Greenwich (the [[Prime Meridian]]) defines the longitude: meridians are lines of constant longitude. The plane through the centre of the Earth and orthogonal to the rotation axis intersects the surface in a great circle called the [[equator]]. Planes parallel to the equatorial plane intersect the surface in circles of constant latitude; these are the parallels. The equator has a latitude of 0°, the [[North pole]] has a latitude of 90° north (written 90° N or +90°), and the [[South pole]] has a latitude of 90° south (written 90° S or −90°). The latitude of an arbitrary point is the angle between the equatorial plane and the radius to that point.

The latitude that is defined in this way for the sphere is often termed the spherical latitude to avoid ambiguity with auxiliary latitudes defined in subsequent sections.

===Named latitudes===
[[File:December solstice geometry.svg|thumb|300px|right|The orientation of the Earth at the December solstice.]]
Besides the equator, four other parallels are of significance:
::{| class="wikitable" border="1"
| [[Arctic Circle]] || 66° 33′ 39″ N
|-
| [[Tropic of Cancer]] ||23° 26′ 21″ N
|-
| [[Tropic of Capricorn]] || 23° 26′ 21″ S
|-
| [[Antarctic Circle]] || 66° 33′ 39" S
|}

The plane of the Earth's orbit about the sun is called the [[ecliptic]]. The plane perpendicular to the rotation axis of the Earth is the equatorial plane. The angle between the ecliptic and the equatorial plane is called the inclination of the ecliptic, denoted by <math>i</math> in the figure. The current value of this angle is 23° 26′ 21″.<ref name=almanac>The Astronomical Almanac published annually by the National Almanac Office in the United States (http://asa.usno.navy.mil/) and the United Kingdom (http://astro.ukho.gov.uk/nao/publicat/asa.html).</ref> It is also called the [[axial tilt]] of the Earth since it is equal to the angle between the axis of rotation and the normal to the ecliptic.

The figure shows the geometry of a cross section of the plane normal to the ecliptic and through the centres of the Earth and the Sun at the December [[solstice]] when the sun is overhead at some point of the Tropic of Capricorn. The south polar latitudes below the Antarctic Circle are in daylight whilst the north polar latitudes above the Arctic Circle are in night. The situation is reversed at the June solstice when the sun is overhead at the Tropic of Cancer. The latitudes of the tropics are equal to the inclination of the ecliptic and the polar circles are at latitudes equal to its complement. Only at latitudes in between the two [[tropics]] is it possible for the [[sun]] to be directly overhead (at the [[zenith]]).

The named parallels are clearly indicated on the Mercator projections shown below.

===Map projections from the sphere===
On [[map projections]] there is no simple rule as to how meridians and parallels should appear. For example, on the spherical [[Mercator projection]] the parallels are horizontal and the meridians are vertical whereas on the [[Transverse Mercator projection]] there is no correlation of parallels and meridians with horizontal and vertical, both are complicated curves. The red lines are the named latitudes of the previous section.
{| style="text-align:left" style="margin: 1em auto 1em auto"
|-valign=top
! width="1%" |
! width="36%"|Normal Mercator
! width="3%"|
! width="1%" |
! width="36%" |Transverse Mercator
|-valign=top
|
| align="center" width="200px" | [[Image:MercNormSph enhanced.png|center|thumb|upright|200px]]
|
|
| align="center" width="200px" | [[Image:MercTranSph enhanced.png|center|thumb|upright|200px]]
|}
For map projections of large regions, or the whole world, a spherical Earth model is completely satisfactory since the variations attributable to ellipticity are negligible on the final printed maps.

===Meridian distance on the sphere===

On the sphere the normal passes through the centre and the latitude (φ) is
therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the [[Meridian arc|meridian distance]] is denoted by ''m''(φ) then
::<math> m(\phi)=\frac{\pi}{180}R\phi_{\rm degrees}= R\phi_{\rm radians}.</math>
where R denotes the [[Earth radius#Mean radii|mean radius]] of the Earth. R is equal to 6371 km or 3959 miles. No higher accuracy is appropriate for R since higher precision results necessitate an ellipsoid model. With this value for R the meridian length of 1 degree of latitude on the sphere is 111.2 km or 69 miles. The length of 1 minute of latitude is 1.853 km, or 1.15 miles. (See [[nautical mile]]).

==Latitude on the ellipsoid==

===Ellipsoids===
In 1687 [[Isaac Newton]] published the [[Philosophiæ Naturalis Principia Mathematica|Principia]] in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate [[ellipsoid]].<ref name=newton>Isaac Newton:''Principia'' Book III Proposition XIX Problem III, p. 407 in Andrew Motte translation, available on line at [http://www.archive.org]</ref> (This article uses the term ellipsoid in preference to the older term ''spheroid''). Newton's result was confirmed by geodetic measurements in the eighteenth century. (See [[Meridian arc]].) An oblate ellipsoid is the three dimensional surface generated by the rotation of an ellipse about its shorter axis (minor axis). 'Oblate ellipsoid of revolution' is abbreviated to '''ellipsoid''' in the remainder of this article. (Ellipsoids which do not have an axis of symmetry are termed tri-axial.)

Many different [[Figure of the Earth|reference ellipsoids]] have been used in the history of [[geodesy]]. In pre-satellite days they were devised to give a good fit to the [[geoid]] over the limited area of a survey but, with the advent of [[GPS]], it has become natural to use reference ellipsoids (such as [[WGS84]]) with centres at the centre of mass of the Earth and minor axis aligned to the rotation axis of the Earth. These geocentric ellipsoids are usually within 100m of the geoid. Since latitude is defined with respect to an ellipsoid, the position of a given point is different on each ellipsoid: one can't exactly specify the latitude and longitude of a geographical feature without specifying the ellipsoid used. Many maps maintained by national agencies are based on older ellipsoids so it is necessary to know how the latitude and longitude values are transformed from one ellipsoid to another. GPS handsets include software to carry out [[Datum (geodesy)|datum transformations]] which link WGS84 to the local reference ellipsoid with its associated grid.

===The geometry of the ellipsoid===

The shape of an ellipsoid of revolution is determined by the shape of the [[ellipse]] which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the [[ellipse|semi-major axis]], ''a''. The other parameter is usually (1) the polar radius or [[ellipse|semi-minor axis]], ''b''; or (2) the (first) [[flattening]], ''f''; or (3) the [[ellipse|eccentricity]], ''e''. These parameters are not independent: they are related by
:<math>
\begin{align}
f&=\frac{a-b}{a}, \qquad e^2=2f-f^2,\qquad b=a(1-f)=a\sqrt{1-e^2}.
\end{align}
</math>
Many other parameters (see [[ellipse]], [[ellipsoid]]) appear in the study of geodesy, geophysics and map projections but they can all be expressed in terms of one or two members of the set ''a'', ''b'', ''f'' and ''e''. Both ''f'' and ''e'' are small and often appear in series expansions in calculations; they are of the order 1/300 and 0.08 respectively. Values for a number of ellipsoids are given in [[Figure of the Earth]]. Reference ellipsoids are usually defined by the semi-major axis and the ''inverse '' flattening, ''1/f''. For example, the defining values for the [[WGS84]] ellipsoid, used by all [[Global Positioning System|GPS]] devices, are<ref>[http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350_2.htmlNIMA The WGS84 parameters are listed in the National Geospatial-Intelligence Agency publication TR8350.2] page 3-1.</ref>
:*''a'' (equatorial radius): 6,378,137.0 m exactly
:* ''1/f'' (inverse flattening): 298.257,223,563 exactly
from which are derived
:* ''b'' (polar radius): 6,356,752.3142 m
:* ''e''2 (eccentricity squared): 0.006,694,379,990,14
The difference of the major and minor semi-axes is about 21 km and as fraction of the semi-major axis it equals the flattening; on a computer the ellipsoid could be sized as 300px by 299px. This would be indistinguishable from a sphere shown as 300px by 300px, so illustrations always exaggerate the flattening.

===Geodetic and geocentric latitudes===
[[File:latitude and longitude graticule on an ellipsoid.svg|thumb|200px|right|The definition of geodetic latitude (φ) and longitude (λ) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles.]]
The graticule on the ellipsoid is constructed in exactly the same way as on the sphere. The normal at a point on the surface of an ellipsoid does not pass through the centre, except for points on the equator or at the poles, but the definition of latitude remains unchanged as the angle between the normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing

:'''Geodetic latitude:''' the angle between the normal and the equatorial plane. The standard notation in English publications is φ. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid.
:'''Geocentric latitude:''' the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure [[#Geocentric latitude|below]]). There is no standard notation: examples from various texts include ψ, ''q'', φ', φc, φg. This article uses ψ.
:'''Spherical latitude:''' the angle between the normal to a spherical reference surface and the equatorial plane.
: '''Geographic latitude''' must be used with care. Some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the [[#Astronomical latitude|astronomical latitude]].
:'''Latitude''' (unqualified) should normally refer to the geodetic latitude.

The importance of specifying the reference datum may be illustrated by a simple example. On the reference ellipsoid for WGS84, the centre of the [[Eiffel Tower]] has a geodetic latitude of 48° 51′ 29″ N, or 48.8583° N and longitude of 2° 17′ 40″ E or 2.2944°E. The same coordinates on the datum [[ED50]] define a point on the ground which is 140 m distant from Tower.{{citation needed|date=December 2011}} A web search may produce several different values for the latitude of the Tower; the reference ellipsoid is rarely specified.

===Length of a degree of latitude===

In [[Meridian arc]] and standard texts<ref name=torge/><ref name=osborne/><ref name=rapp/> it is shown that the distance along a meridian from latitude φ to the equator is given by (φ in radians)
::<math>
m(\phi) =\int_0^\phi M(\phi) d\phi
= a(1 - e^2)\int_0^\phi \left (1 - e^2 \sin^2 \phi \right )^{-3/2} d\phi
</math>
The function <math>M(\phi)</math> in the first integral is the meridional [[radius of curvature (applications)|radius of curvature]].

The distance from the equator to the pole is
::<math>
m_p = m(\pi/2)\,
</math>
For [[WGS84]] this distance is 10001.965729 km.

The evaluation of the meridian distance integral is central to many studies in geodesy and map projection. It can be evaluated by expanding the integral by the binomial series and integrating term by term: see [[Meridian arc]] for details. The length of the meridian arc between two given latitudes is given by replacing the limits of the integral by the latitudes concerned. The length of a ''small'' meridian arc is given by <ref name=osborne>Osborne, P (2013)[http://www.mercator99.webspace.virginmedia.com The Mercator Projections] (Chapters 5,6)</ref><ref name=rapp>Rapp, Richard H. (1991). ''Geometric Geodesy, Part I'', Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.[http://hdl.handle.net/1811/24333](Chapter 3)</ref>
::<math>
\begin{align}
\delta m(\phi) &= M(\phi) \delta\phi
= a(1 - e^2) \left (1 - e^2 \sin^2 \phi \right )^{-3/2} \delta\phi\,
\end{align}
</math>
{|class="wikitable" style="margin: 1em auto 1em auto;" border="1" align="right"
!<math>\phi</math>||<math>\Delta^1_{\rm LAT}</math>||<math>\Delta^1_{\rm LONG}</math>
|- style="text-align:right;"
| 0° || 110.574 km|| 111.320 km
|- style="text-align:right;"
| 15° || 110.649 km|| 107.550 km
|- style="text-align:right;"
| 30° || 110.852 km|| 96.486 km
|- style="text-align:right;"
| 45° || 111.132 km|| 78.847 km
|- style="text-align:right;"
| 60° || 111.412 km|| 55.800 km
|- style="text-align:right;"
| 75° || 111.618 km|| 28.902 km
|- style="text-align:right;"
| 90° || 111.694 km|| 0.000 km
|}
When the latitude difference is 1 degree, corresponding to <math>\pi</math>/180 radians, the arc distance is about
::<math>
\Delta^1_{\rm LAT}=
\frac{\pi a(1 - e^2)}{180(1 - e^2 \sin^2 \phi )^{3/2}} \,
</math>

The distance in metres (correct to 0.01 metre) between latitudes (<math> \phi - 0.5</math> deg) and (<math> \phi + 0.5</math> deg) on the WGS84 spheroid is
::<math>
\Delta^1_{\rm LAT}= 111132.954 - 559.822\cos 2\phi + 1.175\cos 4\phi
</math>

The variation of this distance with latitude (on [[WGS84]]) is shown in the table along with the [[Longitude#Length of a degree of longitude|length of a degree of longitude]]:
::<math>
\Delta^1_{\rm LONG}=
\frac{\pi a\cos\phi}{180(1 - e^2 \sin^2 \phi)^{1/2}}\,
</math> 
A calculator for any latitude is provided by (a) the U.S. government's [[National Geospatial-Intelligence Agency]] (NGA),<ref>[http://msi.nga.mil/MSISiteContent/StaticFiles/Calculators/degree.html Length of degree calculator - National Geospatial-Intelligence Agency]</ref> and
(b) CSGnet.<ref>http://www.csgnetwork.com/degreelenllavcalc.html</ref>

==Auxiliary latitudes==

There are six '''auxiliary latitudes''' that have applications to special problems in geodesy, geophysics and the theory of map projections:
:* geocentric latitude,
:* reduced (or parametric) latitude,
:* rectifying latitude,
:* authalic latitude,
:* conformal latitude,
:* isometric latitude.
The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three dimensional [[geographic coordinate system]] as discussed [[#Latitude and coordinate systems|below]]. The remaining latitudes are not used in this way;
they are used ''only'' as intermediate constructs in map projections of the reference ellipsoid to the plane or in calculations of geodesics on the ellipsoid. Their numerical values are not of interest. For example no one would need to calculate the authalic latitude of the Eiffel Tower.

The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, ''a'', and the eccentricity, ''e''. (For inverses see [[#Inverse formulae and series|below]].) The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder.<ref name=snyder>{{Cite book| author=Snyder, John P. | title=Map Projections: A Working Manual. U.S. Geological Survey Professional Paper 1395 | publisher =United States Government Printing Office |location=Washington, D.C. | year=1987}} This paper can be downloaded from [http://pubs.er.usgs.gov/pubs/pp/pp1395 USGS pages.]</ref> Derivations of these expressions may be found in Adams<ref name=adams1921>Adams, Oscar S (1921). ''Latitude Developments Connected With Geodesy and Cartography, (with tables, including a table for Lambert equal area meridional projection)''. Special Publication No. 67 of the US Coast and Geodetic Survey. A facsimile of this publication is available from the US National Oceanic and Atmospheric Administration ([[NOAA]]) at http://docs.lib.noaa.gov/rescue/cgs_specpubs/QB275U35no671921.pdf '''Warning''': Adams uses the nomenclature isometric latitude for the conformal latitude of this article.</ref> and web publications by Osborne<ref name=osborne/> and Rapp.<ref name=rapp/>

===Geocentric latitude===
[[Image:Two-types-of-latitude.png|right|thumb|250px|The definition of geodetic (or geographic) and geocentric latitudes.]]
The '''geocentric latitude''' is the angle between the equatorial plane and the radius from the centre to a point on the surface. The relation between the geocentric latitude (ψ) and the geodetic latitude (φ) is derived in the above references as
::<math>
\psi(\phi)=\tan^{-1}\left[(1-e^2)\tan\phi\right]\;\!.
</math>
The geodetic and geocentric latitudes are equal at the equator and poles. The value of the squared eccentricity is approximately 0.007 (depending on the choice of ellipsoid) and the maximum difference of (φ-ψ) is approximately 11.5 minutes of arc at a geodetic latitude of 45°5′.

===Reduced (or parametric) latitude===
[[File:Ellipsoid reduced angle definition.svg|thumb|200px|right|Definition of the reduced latitude (β) on the ellipsoid.]]
The '''reduced''' or '''parametric latitude''', β, is defined by the radius drawn from the centre of the ellipsoid to that point Q on the surrounding sphere (of radius ''a'') which is the projection parallel to the Earth's axis of a point P on the ellipsoid at latitude <math>\scriptstyle\phi</math>. It was introduced by Legendre<ref name=legendre>A. M. Legendre, 1806, Analyse des triangles tracés sur la surface d'un sphéroïde, Mém. de l'Inst. Nat. de France, 130--161 (1st semester).</ref> and Bessel<ref name=bessel>F. W. Bessel, 1825, Uber die Berechnung der geographischen Langen und Breiten aus geodatischen Vermessungen, Astron.Nachr., 4(86), 241-254, {{doi|10.1002/asna.201011352}}, translated into English by C. F. F. Karney and R. E. Deakin as The calculation of longitude and latitude from geodesic measurements, Astron. Nachr. 331(8), 852-861 (2010), E-print {{arxiv|0908.1824}}, http://adsabs.harvard.edu/abs/1825AN......4..241B.</ref> who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, <math>u(\phi)</math>, is also used in the current literature. The reduced latitude is related to the geodetic latitude by<ref name=osborne/><ref name=rapp/>
::<math>
\beta(\phi)=\tan^{-1}\left[\sqrt{1-e^2}\tan\phi\right]\,\!
</math>
The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates ''p'', the distance from the minor axis, and ''z'', the distance above the equatorial plane, the equation of the [[ellipse]] is
::<math> \frac{p^2}{a^2} + \frac{z^2}{b^2} =1 .</math>
The Cartesian coordinates of the point are parameterized by
::<math> p=a\cos\beta, \qquad z=b\sin\beta; </math>
Cayley<ref name=cayley>A. Cayley, 1870, On the geodesic lines on an oblate spheroid, Phil. Mag. 40 (4th ser.), 329-340.</ref> suggested the term ''parametric latitude'' because of the form of these equations.

The reduced latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics. ([[Vincenty's formulae|Vincenty]], Karney<ref name=Karney>C. F. F. Karney (2013), Algorithms for geodesics, J. Geodesy '''87'''(1), 43–55, DOI: [http://dx.doi.org/10.1007/s00190-012-0578-z 10.1007/s00190-012-0578-z].</ref>).

===Rectifying latitude===
The '''rectifying latitude''', μ, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or π/2 radians:
:: <math>\mu(\phi)={\displaystyle \frac{\pi}{2}\frac{m(\phi)}{m_p}}\,\!</math>
where the meridian distance from the equator to a latitude φ is (see [[Meridian arc]])
::<math>
m(\phi)= a(1 - e^2)\int_0^\phi \left (1 - e^2 \sin^2 \phi \right )^{-3/2} d\phi,
</math>
and the length of the meridian quadrant from the equator to the pole is
::<math>m_p = m(\pi/2).\,</math>
Using the rectifying latitude to define a latitude on a sphere of radius
::<math>R=\frac{2m_p}{\pi}</math>
defines a projection from the ellipsoid to the sphere such that all meridians have true length and uniform scale.
The sphere may then be projected to the plane with an [[equirectangular projection]] to give a double projection from the ellipsoid to the plane such that all meridians have true length and uniform meridian scale. An example of the use of the rectifying latitude is the [[Map projection|Equidistant conic projection]]. (Snyder,<ref name=snyder/> Section 16). The rectifying latitude is also of great importance in the construction of the [[Transverse Mercator projection]].

===Authalic latitude===
The '''authalic''' (Greek for [[wiktionary:authalic|same area]]) latitude, ξ, gives an area-preserving transformation to a sphere.
::<math>\xi(\phi)=\sin^{-1}\left(\frac{q(\phi)}{q_p}\right)\,\!</math>
where
::<math>
\begin{align}
q(\phi)&= \frac{(1 - e^2)\sin\phi}{1 - e^2 \sin^2 \phi}
-\frac{1-e^2}{2e}\ln \left(\frac{1-e\sin\phi}{1+e\sin\phi}\right),\\
&= \frac{(1 - e^2)\sin\phi}{1 - e^2 \sin^2 \phi}
+\frac{1-e^2}{e}\tanh^{-1}(e\sin\phi),
\end{align}
</math>
and
::<math>
q_p = q(\pi/2)
=1-\frac{1-e^2}{2e}\ln \left(\frac{1-e}{1+e}\right)
=1+\frac{1-e^2}{e}\tanh^{-1}e,
\,</math>
and the radius of the sphere is taken as
::<math>R_q=a\sqrt{q_p/2}.\,</math>
An example of the use of the authalic latitude is the [[Albers projection|Albers equal-area conic projection]]. (Snyder,<ref name=snyder/> Section 14).

===Conformal latitude===
The '''conformal latitude''', χ, gives an angle-preserving ([[Conformal map|conformal]]) transformation to the sphere.
::<math>\begin{align}
\chi(\phi)&=2\tan^{-1}\left[
\left(\frac{1+\sin\phi}{1-\sin\phi}\right)
\left(\frac{1-e\sin\phi}{1+e\sin\phi}\right)^{\!\textit{e}}
\;\right]^{1/2}
-\frac{\pi}{2}\\[2ex]
&=2\tan^{-1}\left[
\tan\left(\frac{\phi}{2}+\frac{\pi}{4}\right)
\left(\frac{1-e\sin\phi}{1+e\sin\phi}\right)^{\!\textit{e}/2}
\;\right]
-\frac{\pi}{2}\\
&=\sin^{-1}\left[\tanh\left(\tanh^{-1}(\sin\phi) -e\tanh^{-1}(e\sin\phi)\right)\right]\\
&=\mathrm{gd}\left[\mathrm{gd}^{-1}(\phi)-e\tanh^{-1}(e\sin\phi)\right].

\;\!\end{align}</math>,
where gd(''x'') is the [[Gudermannian function]]. (See also [[Mercator projection#Alternative expressions|Mercator projection]].)
The conformal latitude defines a transformation from the ellipsoid to a sphere of ''arbitrary'' radius such that the angle of intersection between any two lines on the ellipsoid is the same as the corresponding angle on the sphere (so that the shape of ''small'' elements is well preserved). A further conformal transformation from the sphere to the plane gives a conformal double projection from the ellipsoid to the plane. This is not the only way of generating such a conformal projection. For example, the 'exact' version of the [[Transverse Mercator projection]] on the ellipsoid is not a double projection. (It does, however, involve a generalisation of the conformal latitude to the complex plane).

===Isometric latitude===
The '''isometric latitude''' is conventionally denoted by ψ (not to be confused with the geocentric latitude): it is used in the development of the ellipsoidal versions of the normal [[Mercator projection]] and the [[Transverse Mercator projection]]. The name "isometric" arises from the fact that at any point on the ellipsoid equal increments of ψ and longitude λ give rise to equal distance displacements along the meridians and parallels respectively. The [[Geographic coordinate system|graticule]] defined by the lines of constant ψ and constant λ, divides the surface of the ellipsoid into a mesh of squares (of varying size). The isometric latitude is zero at the equator but rapidly diverges from the geodetic latitude, tending to infinity at the poles. The conventional notation is given in Snyder<ref name=snyder/> (page 15):
::<math>
\begin{align}
\psi(\phi)
&=\ln\left[\tan\left( \frac{\pi}{4}+\frac{\phi}{2}\right) \right]
+
\frac{e}{2}\ln\left[ \frac{1-e\sin\phi}{1+e\sin\phi} \right]\\
&=\tanh^{-1}(\sin\phi) -e\tanh^{-1}(e\sin\phi)\\
&=\mathrm{gd}^{-1}(\phi)-e\tanh^{-1}(e\sin\phi).
\end{align}</math>
For the ''normal'' Mercator projection (on the ellipsoid) this function defines the spacing of the parallels: if the length of the equator on the projection is E (units of length or pixels) then the distance, ''y'', of a parallel of latitude φ from the equator is
::<math>
y(\phi)=\frac{E}{2\pi}\psi(\phi).
</math>
The isometric latitude is closely related to the conformal latitude:
::<math>
\begin{align}
\psi(\phi)
&=\mathrm{gd}^{-1} \chi(\phi).
\end{align}</math>

===Inverse formulae and series===
The formulae in the previous sections give the auxiliary latitude in terms of the geodetic latitude. The expressions for the geocentric and reduced latitudes may be inverted directly
but this is impossible in the four remaining cases: the rectifying, authalic,
conformal and isometric latitudes. There are two methods of proceeding. The first is a numerical inversion of the defining equation for each and every particular value of the auxiliary latitude. The methods available are
[[fixed-point iteration]] and [[Newton-Raphson]] root finding. The other, more useful, approach is to express the auxiliary latitude as a series in terms of the geodetic latitude and then invert the series by the method of [[Lagrange reversion]]. Such series are presented by Adams<ref name=adams1921/> who uses Taylor series expansions and gives coefficients in terms of the eccentricity. Osborne<ref name=osborne/> derives series to arbitrary order by using the computer algebra package Maxima<ref>[http://maxima.sourceforge.net/ Maxima computer algebra system]</ref> and expresses the coefficients in terms of both eccentricity and flattening. The series method is not applicable to the isometric latitude and one must use the conformal latitude in an intermediate step.

==Numerical comparison of auxiliary latitudes==
The following plot shows the magnitude of the difference between the geodetic latitude, (denoted as the 'common' latitude on the plot), and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles). In every case the geodetic latitude is the greater. The differences shown on the plot are in arc minutes. The horizontal resolution of the plot fails to make clear that the maxima of the curves are not at 45° but calculation shows that they are within a few arc minutes of 45°. Some representative data points are given in the table following the plot. Note the closeness of the conformal and geocentric latitudes. This was exploited in the days of hand calculators to expedite the construction of map projections. (Snyder,<ref name=snyder/> page 108).

[[Image:Types of latitude difference.png|center|400px]]

:{| class="wikitable" style="margin: 1em auto 1em auto;"
!colspan="6"|Approximate difference from geodetic latitude (<math>\phi\,\!</math> )
|-
!<math>\phi\,\!</math>
!Reduced <math>\phi-\beta\,\!</math>
!Authalic <math>\phi-\xi\,\!</math>
!Rectifying <math>\phi-\mu\,\!</math>
!Conformal <math>\phi-\chi\,\!</math>
!Geocentric <math>\phi-\psi\,\!</math>
|-
|0° || 0.00′|| 0.00′|| 0.00′|| 0.00′|| 0.00′
|-
|15°|| 2.91′|| 3.89′|| 4.37′|| 5.82′|| 5.82′
|-
|30°|| 5.05′|| 6.73′|| 7.57′||10.09′||10.09′
|-
|45°|| 5.84′|| 7.78′|| 8.76′||11.67′||11.67′
|-
|60°|| 5.06′|| 6.75′|| 7.59′||10.12′||10.13′
|-
|75°|| 2.92′|| 3.90′|| 4.39′|| 5.85′|| 5.85′
|-
|90°|| 0.00′|| 0.00′|| 0.00′|| 0.00′|| 0.00′
|}

==Latitude and coordinate systems==
The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional coordinate system on that ellipsoid. To define the position of an arbitrary point it is necessary to extend such a coordinate system into three dimensions. Three latitudes are used in this way: the geodetic, geocentric and reduced latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively.

===Geodetic coordinates===
[[File:Geodetic coordinates.svg|thumb|right|200px| Geodetic coordinates P(ɸ,λ,''h'')]]
At an arbitrary point P consider the line PN which is normal to the reference ellipsoid. The geodetic coordinates P(ɸ,λ,''h'') are the latitude and longitude of the point N on the ellipsoid and the distance PN. This height differs from the height above the geoid or a reference height such as that above mean sea level at a specified location. The direction of PN will also differ from the direction of a vertical plumb line. The relation of these different heights requires knowledge of the shape of the geoid and also the gravity field of the Earth.

===Spherical polar coordinates===

[[File:Geocentric coordinates.svg|thumb|right|200px| Geocentric coordinate related to spherical polar
coordinates P(''r'', θ, λ) ]]
The geocentric latitude ψ is the complement of the polar angle θ in conventional [[spherical polar coordinates]] in which the coordinates of a point are P(''r'', θ, λ) where ''r'' is the distance of P from the centre O, θ is the angle between the radius vector and the polar axis andλ is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points on the normal, which all have the same geodetic latitude, will have differing geocentic latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field.

===Ellipsoidal coordinates===

[[File:Ellipsoidal coordinates.svg|thumb|right|200px|Ellipsoidal coordinates P(''u'',β,λ)]]
The reduced latitude can also be extended to a three dimensional coordinate system. For a point P not on the reference ellipsoid (semi-axes OA and OB) construct an auxiliary ellipsoid which is confocal (same foci F, F') with the reference ellipsoid: the necessary condition is that the product ''ae'' of semi-major axis and eccentricity is the same for both ellipsoids. Let ''u'' be the semi-minor axis (OD) of the auxiliary ellipsoid. Further let β be the reduced latitude of P on the auxiliary ellipsoid. The set (''u'',β,λ) define the ellipsoid coordinates. (Torge<ref name=torge/> Section 4.2.2). These coordinates are the natural choice in models of the gravity field for a uniform distribution of mass bounded by the reference ellipsoid.

===Coordinate conversions===
The relations between the above coordinate systems, and also Cartesian coordinates are not presented here. The transformation between geodetic and Cartesian coordinates may be found in [[Geodetic system#Conversion calculations|Geodetic system]]. The relation of Cartesian and spherical polars is given in [[Spherical coordinate system]]. The relation of Cartesian and ellipsoidal coordinates is discussed in Torge.<ref name=torge/>

==Astronomical latitude==
'''Astronomical latitude''' (Φ) is the angle between the equatorial plane and the true vertical at a point on the surface: the true vertical, the direction of a plumb line, is the direction of the '''gravity field''' at that point. (The gravity field is the resultant of the gravitational acceleration and the centrifugal acceleration at that point. See Torge.<ref name=torge/>) Astronomic latitude is calculated from angles measured between the [[zenith]] and stars whose [[declination]] is accurately known.

In general the true vertical at a point on the surface does not exactly coincide with either the normal to the reference ellipsoid or the normal to the geoid. The angle between the astronomic and geodetic normals is usually a few seconds of arc but it is important in geodesy.<ref name=torge/><ref name=wellenhofmoritz/> The reason why it differs from the normal to the geoid is, because the geoid is an idealized, theoretical shape 'at mean sea level'. Points on the real surface of the earth are usually above or below this idealized geoid surface and here the true vertical can vary slightly. Also, the true vertical at a point at a specific time is influenced by tidal forces, which the theoretical geoid averages out.

Astronomical latitude is not to be confused with [[declination]], the coordinate [[astronomer]]s used in a similar way to describe the locations of stars north/south of the [[celestial equator]] (see [[equatorial coordinates]]), nor with [[ecliptic latitude]], the coordinate that astronomers use to describe the locations of stars north/south of the [[ecliptic]] (see [[ecliptic coordinates]]).

==See also==
<div style="-moz-column-count:3; column-count:3;">
*[[altitude]] ([[sea level|mean sea level]])
*[[American Practical Navigator]]
*[[cardinal direction]]
*[[Degree Confluence Project]]
*[[geodesy]]
*[[geodetic system]]
*[[geographic coordinate system]]
*[[geographical distance]]
*[[geotagging]]
*[[great-circle distance]]
*[[horse latitudes]]
*[[list of cities by latitude]]
*[[list of cities by longitude]]
*[[list of countries by latitude]]
*[[longitude]]
*[[Natural Area Code]]
*[[navigation]]
*[[orders of magnitude (length)]]
*[[World Geodetic System]]
</div>

==Footnotes==
{{reflist}}

==External links==
{{sisterlinks}}
*{{PDFWayback|http://members.verizon.net/~vze2hc4d/proj4/manual.pdf|A Comprehensive Library of Cartographic Projection Functions (Preliminary Draft)|1.88 MB|date=20080625111630}}
*[http://earth-info.nga.mil/gns/html/ GEONets Names Server], access to the [[National Geospatial-Intelligence Agency]]'s (NGA) database of foreign geographic feature names.
*[http://www.bcca.org/misc/qiblih/latlong.html Look-up Latitude and Longitude]
*[http://jan.ucc.nau.edu/~cvm/latlon_find_location.html Resources for determining your latitude and longitude]
*[http://geography.about.com/library/howto/htdegrees.htm Convert decimal degrees into degrees, minutes, seconds] - Info about decimal to [[sexagesimal]] conversion
*[http://www.fcc.gov/mb/audio/bickel/DDDMMSS-decimal.html Convert decimal degrees into degrees, minutes, seconds]
*[http://www.marinewaypoints.com/learn/greatcircle.shtml Distance calculation based on latitude and longitude] - JavaScript version
*{{PDFlink|[https://www.cia.gov/library/publications/the-world-factbook/graphics/ref_maps/pdf/political_world.pdf Zoomable version of the map]|3.47 MB}}
*[http://www.longcamp.com/nav.html Determination of Latitude by Francis Drake on the Coast of California in 1579]
*[http://www.thegpscoordinates.com Longitude and Latitude of Points of Interest]
*[http://www.csgnetwork.com/degreelenllavcalc.html Length Of A Degree Of Latitude And Longitude Calculator]

{{geographical coordinates|state=collapsed}}

[[Category:Lines of latitude|Lines of latitude]]
[[Category:Geodesy]]
[[Category:Navigation]]

Parallel curve

2013-10-10T03:34:36Z

128.2.210.91: spelling fix

In [[optics]] and [[photography]], '''hyperfocal distance''' is a distance beyond which all objects can be brought into an "acceptable" [[focus (optics)|focus]]. There are two commonly used definitions of ''hyperfocal distance'', leading to values that differ only slightly:

''Definition 1:'' The hyperfocal distance is the closest distance at which a [[lens (optics)|lens]] can be focused while keeping [[infinity focus|objects at infinity]] acceptably sharp. When the lens is focused at this distance, all objects at distances from half of the hyperfocal distance out to infinity will be acceptably sharp.

''Definition 2:'' The hyperfocal distance is the distance beyond which all objects are acceptably sharp, for a lens focused at infinity.

The distinction between the two meanings is rarely made, since they have almost identical values. The value computed according to the first definition exceeds that from the second by just one [[focal length]].

As the hyperfocal distance is the focus distance giving the maximum [[depth of field]], it is the most desirable distance to set the focus of a [[fixed focus lens|fixed-focus camera]].<ref name="Kingslake1951">{{cite book |first=Rudolf |last=Kingslake |title=Lenses in Photography: The Practical Guide to Optics for Photographers |location=Garden City, NY |publisher=Garden City Press |year=1951 }}</ref>

==Acceptable sharpness==

The hyperfocal distance is entirely dependent upon what level of sharpness is considered to be acceptable. The criterion for the desired acceptable sharpness is specified through the [[circle of confusion]] (CoC) diameter limit. This criterion is the largest acceptable spot size diameter that an infinitesimal point is allowed to spread out to on the imaging medium (film, digital sensor, etc.).

==Formulae==

For the first definition,

:<math>H = \frac{f^2}{N c} + f</math>

where

:<math>H</math> is hyperfocal distance

:<math>f</math> is [[focal length]]

:<math>N</math> is [[f-number]] (<math>f/D</math> for aperture diameter <math>D</math>)

:<math>c</math> is the [[circle of confusion]] limit

For any practical f-number, the focal length is insignificant in comparison
with the first term, so that

:<math>H \approx \frac{f^2}{N c}</math>

This formula is exact for the second definition, if <math>H</math> is measured from a thin lens, or from the front principal plane of a complex lens; it is also exact for the first definition if <math>H</math> is measured from a point that is one focal length in front of the front principal plane. For practical purposes, there is little difference between the first and second definitions.

==Example==

{{Hyperfocal_distance_depth_of_field_using_EasyTimeline}}
As an example, for a 50 mm lens at <math>f/8</math> using a circle of confusion of 0.03 mm, which is a value typically used in 35 mm photography, the hyperfocal distance according to ''Definition 1'',

:<math>H = \frac{(50)^2}{(8)(0.03)} + (50) = 10467 \mbox{ mm}</math>

If the lens is focused at a distance of 10.5 m, then everything from half that distance (5.2 m) to infinity will be acceptably sharp in our photograph. With the formula for the ''Definition 2'', the result is 10417 mm, a difference of 0.48%.

Note that the second definition assumes that the depth of field is symmetric around ''H'' (which is roughly correct for normal lenses) and therefore assumes a setting in which only half the depth of field is utilized (i.e., for a lens focused at infinity, the depth of field is 5.2 m to infinity).

==Mathematical phenomenon==
The hyperfocal distance is a curious property: While a lens focused at ''H'' will hold a depth of field from ''H''/2 to infinity, if the lens is focused to ''H''/2, the depth of field will extend from ''H''/3 to ''H''; if the lens is then focused to ''H''/3, the depth of field will extend from ''H''/4 to ''H''/2. This continues on through all successive 1/''x'' values of the hyperfocal distance.

Piper (1901) calls this phenomenon "consecutive depths of field" and shows how to test the idea easily. This is also among the earliest of publications to use the word ''hyperfocal''.

The figure on the right illustrates this phenomenon.

==History==

[[Image:Derr Hyperfocal 1906.png|right|thumb|300px|This early use of the term ''hyperfocal distance'', Derr 1906, is by no means the earliest explanation of the concept.]]

The concepts of the two definitions of hyperfocal distance have a long history, tied up with the terminology for depth of field, depth of focus, circle of confusion, etc. Here are some selected early quotations and interpretations on the topic.

===Sutton and Dawson 1867===

Thomas Sutton and George Dawson define ''focal range'' for what we now call ''hyperfocal distance'':<ref>{{cite book |first=Thomas |last=Sutton |first2=George |last2=Dawson |title=A Dictionary of Photography |location=London |publisher=Sampson Low, Son & Marston |year=1867 }}</ref>

{{quote|Focal Range. In every lens there is, corresponding to a given apertal ratio (that is, the ratio of the diameter of the stop to the focal length), a certain distance of a near object from it, between which and infinity all objects are in equally good focus. For instance, in a single view lens of 6 inch focus, with a 1/4 in. stop (apertal ratio one-twenty-fourth), all objects situated at distances lying between 20 feet from the lens and an infinite distance from it (a fixed star, for instance) are in equally good focus. Twenty feet is therefore called the “focal range” of the lens when this stop is used. The focal range is consequently the distance of the nearest object, which will be in good focus when the ground glass is adjusted for an extremely distant object. In the same lens, the focal range will depend upon the size of the diaphragm used, while in different lenses having the same apertal ratio the focal ranges will be greater as the focal length of the lens is increased.

The terms 'apertal ratio' and 'focal range' have not come into general use, but it is very desirable that they should, in order to prevent ambiguity and circumlocution when treating of the properties of photographic lenses. 'Focal range' is a good term, because it expresses the range within which it is necessary to adjust the focus of the lens to objects at different distances from it – in other words, the range within which focusing becomes necessary.}}

Their focal range is about 1000 times their aperture diameter, so it makes sense as a hyperfocal distance with CoC value of ''f''/1000, or image format diagonal times 1/1000 assuming the lens is a “normal” lens. What is not clear, however, is whether the focal range they cite was computed, or empirical.

===Abney 1881===

Sir William de Wivelesley Abney says:<ref>{{cite book |first=W. de W. |last=Abney |title=A Treatise on Photography |edition=First |location=London |publisher=Longmans, Green, and Co |year=1881 }}</ref>

{{quote|The annexed formula will approximately give the nearest point ''p'' which will appear in focus when the distance is accurately focussed, supposing the admissible disc of confusion to be 0.025 cm:
:<math>p = 0.41 \cdot f^2 \cdot a</math>
:when
::<math>f = </math> the focal length of the lens in cm
::<math>a = </math> the ratio of the aperture to the focal length}}

That is, ''a'' is the reciprocal of what we now call the ''f''-number, and the answer is evidently in meters. His 0.41 should obviously be 0.40. Based on his formulae, and on the notion that the ''aperture ratio'' should be kept fixed in comparisons across formats, Abney says:

{{quote|It can be shown that an enlargement from a small negative is better than a picture of the same size taken direct as regards sharpness of detail. ... Care must be taken to distinguish between the advantages to be gained in enlargement by the use of a smaller lens, with the disadvantages that ensue from the deterioration in the relative values of light and shade.}}

===Taylor 1892===

John Traill Taylor recalls this word formula for a sort of hyperfocal distance:<ref>{{cite book |first=J. Traill |last=Taylor |url=http://www.archive.org/details/opticsofphotogra00taylrich |title=The Optics of Photography and Photographic Lenses |location=London |publisher=Whittaker & Co |year=1892 }}</ref>

{{quote|We have seen it laid down as an approximative rule by some writers on optics (Thomas Sutton, if we remember aright), that if the diameter of the stop be a fortieth part of the focus of the lens, the depth of focus will range between infinity and a distance equal to four times as many feet as there are inches in the focus of the lens.}}

This formula implies a stricter CoC criterion than we typically use today.

===Hodges 1895===

John Hodges discusses depth of field without formulas but with some of these relationships:<ref>{{cite book |first=John |last=Hodges |title=Photographic Lenses: How to Choose, and How to Use |location=Bradford |publisher=Percy Lund & Co |year=1895 }}</ref>

{{quote|There is a point, however, beyond which everything will be in pictorially good definition, but the longer the focus of the lens used, the further will the point beyond which everything is in sharp focus be removed from the camera. Mathematically speaking, the amount of depth possessed by a lens varies inversely as the square of its focus.}}

This "mathematically" observed relationship implies that he had a formula at hand, and a parameterization with the f-number or “intensity ratio” in it. To get an inverse-square relation to focal length, you have to assume that the CoC limit is fixed and the aperture diameter scales with the focal length, giving a constant f-number.

===Piper 1901===

C. Welborne Piper may be the first to have published a clear distinction between ''Depth of Field'' in the modern sense and ''Depth of Definition'' in the focal plane, and implies that ''Depth of Focus'' and ''Depth of Distance'' are sometimes used for the former (in modern usage, ''Depth of Focus'' is usually reserved for the latter).<ref>{{cite book |first=C. Welborne |last=Piper |title=A First Book of the Lens: An Elementary Treatise on the Action and Use of the Photographic Lens |location=London |publisher=Hazell, Watson, and Viney |year=1901 }}</ref> He uses the term ''Depth Constant'' for ''H'', and measures it from the front principal focus (i. e., he counts one focal length less than the distance from the lens to get the simpler formula), and even introduces the modern term:

{{quote|This is the maximum depth of field possible, and ''H'' + ''f'' may be styled the distance of maximum depth of field. If we measure this distance extra-focally it is equal to ''H'', and is sometimes called the hyperfocal distance. The depth constant and the hyperfocal distance are quite distinct, though of the same value.}}

It is unclear what distinction he means. Adjacent to Table I in his appendix, he further notes:

{{quote|If we focus on infinity, the constant is the focal distance of the nearest object in focus. If we focus on an extra-focal distance equal to the constant, we obtain a maximum depth of field from approximately half the constant distance up to infinity. The constant is then the hyper-focal distance.}}

At this point we do not have evidence of the term ''hyperfocal'' before Piper, nor the hyphenated ''hyper-focal'' which he also used, but he obviously did not claim to coin this descriptor himself.

===Derr 1906===

Louis Derr may be the first to clearly specify the first definition,<ref>{{cite book |first=Louis |last=Derr |title=Photography for students of physics and chemistry |location=London |publisher=Macmillan |year=1906 }}</ref> which is considered to be the strictly correct one in modern times, and to derive the formula corresponding to it. Using <math>p</math> for hyperfocal distance, <math>D</math> for aperture diameter, <math>d</math> for the diameter that a circle of confusion shall not exceed, and <math>f</math> for focal length, he derives:

:<math>p = \frac{(D + d) f}{d}</math> [http://books.google.com/books?id=AN6d4zTjquwC&pg=PA78]

As the aperture diameter, <math>D</math> is the ratio of the focal length, <math>f</math> to the numerical aperture, <math>N</math>; and the diameter of the circle of confusion, <math>c = d</math>, this gives the equation for the first definition above.

:<math>p = \frac{(\tfrac{f}{N} + c) f}{c} = \frac{f^2}{N c} + f</math>

===Johnson 1909===

George Lindsay Johnson uses the term ''Depth of Field'' for what Abney called ''Depth of Focus,'' and ''Depth of Focus'' in the modern sense (possibly for the first time),<ref>{{cite book |first=George Lindsay |last=Johnson |title=Photographic Optics and Colour Photography |location=London |publisher=Ward & Co |year=1909 }}</ref> as the allowable distance error in the focal plane. His definitions include hyperfocal distance:

{{quote|1=Depth of Focus is a convenient, but not strictly accurate term, used to describe the amount of racking movement (forwards or backwards) which can be given to the screen without the image becoming sensibly blurred, i.e. without any blurring in the image exceeding 1/100 in., or in the case of negatives to be enlarged or scientific work, the 1/10 or 1/100 mm. Then the breadth of a point of light, which, of course, causes blurring on both sides, i.e. 1/50 in = 2''e'' (or 1/100 in = ''e'').}}

His drawing makes it clear that his ''e'' is the radius of the circle of confusion. He has clearly anticipated the need to tie it to format size or enlargement, but has not given a general scheme for choosing it.

{{quote|Depth of Field is precisely the same as depth of focus, only in the former case the depth is measured by the movement of the plate, the object being fixed, while in the latter case the depth is measured by the distance through which the object can be moved without the circle of confusion exceeding 2''e''.

Thus if a lens which is focused for infinity still gives a sharp image for an object at 6 yards, its depth of field is from infinity to 6 yards, every object beyond 6 yards being in focus.

This distance (6 yards) is termed the ''hyperfocal distance'' of the lens, and any allowable confusion disc depends on the focal length of the lens and on the stop used.

If the limit of confusion of half of the disc (i.e. ''e'') be taken as 1/100 in., then the hyperfocal distance
:<math>H = \frac{F d}{e} </math>,
''d'' being the diameter of the stop, ...}}

Johnson's use of ''former'' and ''latter'' seem to be swapped; perhaps ''former'' was here meant to refer to the immediately preceding section title ''Depth of Focus'', and ''latter'' to the current section title ''Depth of Field''. Except for an obvious factor-of-2 error in using the ratio of stop diameter to CoC radius, this definition is the same as Abney's hyperfocal distance.

===Others, early twentieth century===

The term ''hyperfocal distance'' also appears in Cassell's ''Cyclopaedia'' of 1911, ''The Sinclair Handbook of Photography'' of 1913, and Bayley's ''The Complete Photographer'' of 1914.

===Kingslake 1951===

[[Rudolf Kingslake]] is explicit about the two meanings:<ref name="Kingslake1951" />

{{quote|''The Hyperfocal Distance'' – It should be noted that if the camera is focused on a distance ''s'' equal to 1000 times the diameter of the lens aperture, then the far depth <math>D_1</math> becomes infinite. This critical object distance "''h''" is known as the ''Hyperfocal Distance''. For a camera focused on this distance, <math>D_1 = \infty</math> and <math>D_2 = h/2</math>, and we see that the range of distances acceptably in focus will run from just half the hyperfocal distance to infinity. The hyperfocal distance is, therefore, the most desirable distance on which to pre-set the focus of a fixed-focus camera. It is worth noting, too, that if a camera is focused on <math>s = \infty</math>, the closest acceptable object is at <math>L_2 = sh/(h+s) = h/(h/s+1) = h</math> (by equation 21). This is a second important meaning of the hyperfocal distance.}}

Kingslake uses the simplest formulae for DOF near and far distances, which has the effect of making the two different definitions of hyperfocal distance give identical values.

==See also==
{{Portal|Photography}}
*[[Circle of confusion]]
*[[Deep focus]]
*[[Depssi]], depth of field sunrise/sunset indicator
*[[Depth of field]]

==References==
<references/>

==External links==
*http://www.dofmaster.com/dofjs.html to calculate hyperfocal distance and [[depth of field]]
{{photography subject}}

[[Category:Optics]]
[[Category:Length]]
[[Category:Science of photography]]

Rate of convergence

2013-06-28T15:45:58Z

128.2.54.15: /* Basic definition */

{{Unreferenced|date=December 2009}}
In [[mathematics]], the concept of '''irreducibility''' is used in several ways.

* In [[abstract algebra]], '''irreducible''' can be an abbreviation for [[irreducible element]] of an [[integral domain]]; for example an [[irreducible polynomial]].

* In [[representation theory]], an '''[[irreducible representation]]''' is a nontrivial [[representation theory |representation]] with no nontrivial proper subrepresentations. Similarly, an '''irreducible module''' is another name for a [[simple module]].

* [[Absolutely irreducible]] is a term applied to mean [[irreducible]], even after any [[finite extension]] of the [[field (mathematics)|field]] of coefficients. It applies in various situations, for example to irreducibility of a [[linear representation]], or of an [[algebraic variety]]; where it means just the same as ''irreducible over an [[algebraic closure]]''.

* In [[commutative algebra]], a [[commutative ring]] ''R'' is '''irreducible''' if its [[prime spectrum]], that is, the topological space Spec ''R'', is an [[irreducible topological space]].

* A [[matrix (mathematics)|matrix]] is '''irreducible''' if it is not [[similar matrix|similar]] via a [[permutation matrix|permutation]] to a [[block matrix|block]] [[upper triangular matrix]] (that has more than one block of positive size). (Replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a [[directed graph]], the matrix is irreducible if and only if such directed graph is [[Connectivity_(graph_theory)|strongly connected]].)

* Also, a [[Markov chain]] is '''[[Markov chain#Reducibility|irreducible]]''' if there is a non-zero probability of transitioning (even if in more than one step) from any state to any other state.

* In the theory of [[manifold]]s, an ''n''-manifold is '''irreducible''' if any embedded (''n'' − 1)-sphere bounds an embedded ''n''-ball. Implicit in this definition is the use of a suitable [[category (mathematics)|category]], such as the category of differentiable manifolds or the category of piecewise-linear manifolds.

The notions of irreducibility in algebra and manifold theory are related. An ''n''-manifold is called [[Connected sum|prime]], if it cannot be written as a [[connected sum]] of two ''n''-manifolds (neither of which is an ''n''-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the [[3-manifold]] topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over ''S''1 and the twisted 2-sphere bundle over ''S''1. See, for example, [[Prime decomposition (3-manifold)]].

* A [[topological space]] is '''[[irreducible space|irreducible]]''' if it is not the union of two proper closed subsets. This notion is used in [[algebraic geometry]], where spaces are equipped with the [[Zariski topology]]; it is not of much significance for [[Hausdorff space]]s. See also [[irreducible component]], [[algebraic variety]].

* In [[universal algebra]], '''irreducible''' can refer to the inability to represent an [[algebraic structure]] as a composition of simpler structures using a product construction; for example [[subdirectly irreducible]].

* A [[3-manifold]] is [[P²-irreducible]] if it is irreducible and contains no [[2-sided]] <math>\mathbb RP^2</math> ([[real projective plane]]).

* An [[Irreducible fraction]] (or '''fraction in lowest terms''') is a [[vulgar fraction]] in which the [[numerator]] and [[denominator]] are smaller than those in any other equivalent fraction.

{{DEFAULTSORT:Irreducibility (Mathematics)}}
[[Category:Mathematical terminology]]

Default logic

2013-03-07T15:50:55Z

128.2.73.21: /* References */

'''Stephen D. Unwin''' is a [[physicist]] and author best known for his book, ''The Probability of God''. Unwin is a graduate of [[Imperial College London]] and received his doctorate in theoretical physics from the [[University of Manchester]] for his research in the field of [[quantum gravity]]. Formerly the technical attaché to the [[United States Department of Energy]] for the British government, he is president of his own consulting firm, specializing in risk management for various [[Fortune 100]] clients.

In his book, Unwin argues that [[Bayes' theorem|a mathematical equation]] developed by [[Thomas Bayes]] can be used to calculate the [[probability]] that God exists. He does not make the claim that application of this method produces an absolute probability on which everyone would agree, but that it provides a systematic way of ordering one's ideas, weights of belief, and uncertainties in order to determine their implications regarding the probability that God exists.

Unwin employs Bayesian probabilities, a statistical method devised by Reverend Thomas Bayes. He begins with a 50 percent probability that God exists (arguing that 50–50 represents "maximum ignorance"), then applies a modified Bayesian theorem:

:<math>P_{\mathrm{after}} = \frac{P_{\mathrm{before}} \times D}{P_{\mathrm{before}} \times D + 1 - P_{\mathrm{before}}} </math>

In this model, the probability of God's existence after the evidence is considered is a function of the probability before times D ("Divine Indicator Scale"): 10 indicates the evidence is 10 times as likely to be produced if God exists, 2 is two times as likely if God exists, 1 is neutral, 0.5 is moderately more likely if God does not exist, and 0.1 is much more likely if God does not exist. Unwin offers the following figures for six lines of evidence: recognition of goodness (D = 10), existence of moral evil (D = 0.5), existence of natural evil (D = 0.1), intranatural miracles (prayers) (D = 2), extranatural miracles (resurrection) (D = 1), and religious experiences (D = 2).

Plugging these figures into the above formula (in sequence, where the Pafter-figure for the first computation is used for the Pbefore-figure in the second computation, and so on for all six Ds), Unwin concludes: "The probability that God exists is 67%." But then he notes that "this number has a subjective element since it reflects my assessment of the evidence." Unwin's comment refers to his estimates of the various "D" values used to obtain his estimate, whose values would be disputed by many.

==External links==
* [http://www.stephenunwin.com/ Unwin's website, including links to many media interviews and reviews]

{{Persondata 
| NAME = Unwin, Stephen D
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = British mathematician
| DATE OF BIRTH =
| PLACE OF BIRTH =
| DATE OF DEATH =
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{{DEFAULTSORT:Unwin, Stephen D}}
[[Category:Year of birth missing (living people)]]
[[Category:British mathematicians]]
[[Category:Living people]]

Progressively measurable process

2012-03-07T17:08:26Z

128.2.118.172: Fixed by the edit of Dec. 26th 2011

Hi there. Allow me begin by introducing the writer, her title is Myrtle Cleary. Since she was eighteen she's been working as a receptionist but her promotion by no means arrives. Her family lives in Minnesota. The thing she adores most is physique building and now she is attempting to make money with it. Also visit my homepage; [http://product-dev.younetco.com/xuannth/fox_ios1/index.php?do=/profile-1539/info/ home std test kit]

Kernel Fisher discriminant analysis

2012-03-05T20:03:24Z

128.2.194.204: /* Multi-class KFD */

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