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99-404: On its own, the term "latitude" normally refers to the geodetic latitude as defined below. Briefly, the geodetic latitude of a point is the angle formed between the vector perpendicular (or normal ) to the ellipsoidal surface from the point, and the plane of the equator . Two levels of abstraction are employed in the definitions of latitude and longitude. In the first step the physical surface

198-446: A n ) {\displaystyle \mathbb {n} =\left(a_{1},\ldots ,a_{n}\right)} is a normal. The definition of a normal to a surface in three-dimensional space can be extended to ( n − 1 ) {\displaystyle (n-1)} -dimensional hypersurfaces in R n . {\displaystyle \mathbb {R} ^{n}.} A hypersurface may be locally defined implicitly as

297-491: A , 0 , 0 ) , {\displaystyle (a,0,0),} where a ≠ 0 , {\displaystyle a\neq 0,} the rows of the Jacobian matrix are ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} and ( 0 , a , 0 ) . {\displaystyle (0,a,0).} Thus the normal affine space is the plane of equation x =

396-417: A . {\displaystyle x=a.} Similarly, if b ≠ 0 , {\displaystyle b\neq 0,} the normal plane at ( 0 , b , 0 ) {\displaystyle (0,b,0)} is the plane of equation y = b . {\displaystyle y=b.} At the point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)}

495-507: A force , the normal vector , etc. The concept of normality generalizes to orthogonality ( right angles ). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space . The normal vector space or normal space of a manifold at point P {\displaystyle P} is the set of vectors which are orthogonal to the tangent space at P . {\displaystyle P.} Normal vectors are of special interest in

594-483: A geographic coordinate system as defined in the specification of the ISO 19111 standard. Since there are many different reference ellipsoids , the precise 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

693-415: A normal is an object (e.g. a line , ray , or vector ) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the line perpendicular to the tangent line to the curve at the point. A normal vector of length one is called a unit normal vector . A curvature vector is a normal vector whose length is the curvature of the object. Multiplying

792-584: A (possibly non-flat) surface S {\displaystyle S} in 3D space R 3 {\displaystyle \mathbb {R} ^{3}} is parameterized by a system of curvilinear coordinates r ( s , t ) = ( x ( s , t ) , y ( s , t ) , z ( s , t ) ) , {\displaystyle \mathbf {r} (s,t)=(x(s,t),y(s,t),z(s,t)),} with s {\displaystyle s} and t {\displaystyle t} real variables, then

891-560: A 300-by-300-pixel sphere, so illustrations usually exaggerate the flattening. 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: Geographic latitude must be used with care, as some authors use it as

990-455: A day on Earth is therefore 24 hours long rather than the approximately 23-hour 56-minute sidereal day . Again, this is a simplification, based on a hypothetical Earth that orbits at uniform speed around the Sun. The actual speed with which Earth orbits the Sun varies slightly during the year, so the speed with which the Sun seems to move along the ecliptic also varies. For example, the Sun is north of

1089-412: A normal to S is by definition a normal to a tangent plane, given by the cross product of the partial derivatives n = ∂ r ∂ s × ∂ r ∂ t . {\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial s}}\times {\frac {\partial \mathbf {r} }{\partial t}}.} If

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1188-406: A normal vector by −1 results in the opposite vector , which may be used for indicating sides (e.g., interior or exterior). In three-dimensional space , a surface normal , or simply normal , to a surface at point P is a vector perpendicular to the tangent plane of the surface at P . The word normal is also used as an adjective: a line normal to a plane , the normal component of

1287-499: A particular equinox, that is, the equinox of a particular date, known as an epoch ; the coordinates are referred to the direction of the equinox at that date. For instance, the Astronomical Almanac lists the heliocentric position of Mars at 0h Terrestrial Time , 4 January 2010 as: longitude 118°09′15.8″, latitude +1°43′16.7″, true heliocentric distance 1.6302454 AU, mean equinox and ecliptic of date. This specifies

1386-406: A point P , {\displaystyle P,} the normal vector space is the vector space generated by the values at P {\displaystyle P} of the gradient vectors of the f i . {\displaystyle f_{i}.} In other words, a variety is defined as the intersection of k {\displaystyle k} hypersurfaces, and

1485-480: A relatively short time span, perhaps several centuries. J. Laskar computed an expression to order T good to 0.04″ /1000 years over 10,000 years. All of these expressions are for the mean obliquity, that is, without the nutation of the equator included. The true or instantaneous obliquity includes the nutation. Most of the major bodies of the Solar System orbit the Sun in nearly the same plane. This

1584-408: A surface S {\displaystyle S} is given implicitly as the set of points ( x , y , z ) {\displaystyle (x,y,z)} satisfying F ( x , y , z ) = 0 , {\displaystyle F(x,y,z)=0,} then a normal at a point ( x , y , z ) {\displaystyle (x,y,z)} on

1683-415: A surface does not have a tangent plane at a singular point , it has no well-defined normal at that point: for example, the vertex of a cone . In general, it is possible to define a normal almost everywhere for a surface that is Lipschitz continuous . The normal to a (hyper)surface is usually scaled to have unit length , but it does not have a unique direction, since its opposite is also a unit normal. For

1782-417: A surface which is the topological boundary of a set in three dimensions, one can distinguish between two normal orientations , the inward-pointing normal and outer-pointing normal . For an oriented surface , the normal is usually determined by the right-hand rule or its analog in higher dimensions. If the normal is constructed as the cross product of tangent vectors (as described in the text above), it

1881-495: A survey but, with the advent of GPS , it has become natural to use reference ellipsoids (such as WGS84 ) with centre 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 100 m (330 ft) of the geoid. Since latitude is defined with respect to an ellipsoid, the position of a given point is different on each ellipsoid: one cannot exactly specify

1980-555: A synonym for geodetic latitude whilst others use it as an alternative to the 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

2079-460: Is a pseudovector . When applying a transform to a surface it is often useful to derive normals for the resulting surface from the original normals. Specifically, given a 3×3 transformation matrix M , {\displaystyle \mathbf {M} ,} we can determine the matrix W {\displaystyle \mathbf {W} } that transforms a vector n {\displaystyle \mathbf {n} } perpendicular to

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2178-945: Is a given scalar function . If F {\displaystyle F} is continuously differentiable then the hypersurface is a differentiable manifold in the neighbourhood of the points where the gradient is not zero. At these points a normal vector is given by the gradient: n = ∇ F ( x 1 , x 2 , … , x n ) = ( ∂ F ∂ x 1 , ∂ F ∂ x 2 , … , ∂ F ∂ x n ) . {\displaystyle \mathbb {n} =\nabla F\left(x_{1},x_{2},\ldots ,x_{n}\right)=\left({\tfrac {\partial F}{\partial x_{1}}},{\tfrac {\partial F}{\partial x_{2}}},\ldots ,{\tfrac {\partial F}{\partial x_{n}}}\right)\,.} The normal line

2277-399: Is a point on the hyperplane and p i {\displaystyle \mathbf {p} _{i}} for i = 1 , … , n − 1 {\displaystyle i=1,\ldots ,n-1} are linearly independent vectors pointing along the hyperplane, a normal to the hyperplane is any vector n {\displaystyle \mathbf {n} } in

2376-529: Is a point on the plane and p , q {\displaystyle \mathbf {p} ,\mathbf {q} } are non-parallel vectors pointing along the plane, a normal to the plane is a vector normal to both p {\displaystyle \mathbf {p} } and q , {\displaystyle \mathbf {q} ,} which can be found as the cross product n = p × q . {\displaystyle \mathbf {n} =\mathbf {p} \times \mathbf {q} .} If

2475-409: Is also used in the current literature. The parametric latitude is related to the geodetic latitude by: 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: The Cartesian coordinates of

2574-478: Is also used occasionally; the x -axis is directed toward the March equinox, the y -axis 90° to the east, and the z -axis toward the north ecliptic pole; the astronomical unit is the unit of measure. Symbols for ecliptic coordinates are somewhat standardized; see the table. Ecliptic coordinates are convenient for specifying positions of Solar System objects, as most of the planets' orbits have small inclinations to

2673-396: Is by definition a surface normal. Alternatively, if the hyperplane is defined as the solution set of a single linear equation a 1 x 1 + ⋯ + a n x n = c , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=c,} then the vector n = ( a 1 , … ,

2772-484: 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 semi-major axis , a . The other parameter is usually (1) the polar radius or semi-minor axis , b ; or (2) the (first) flattening , f ; or (3) the eccentricity , e . These parameters are not independent: they are related by Many other parameters (see ellipse , ellipsoid ) appear in

2871-453: Is determined with the meridian altitude method. More precise measurement of latitude requires an understanding of the gravitational field of the Earth, either to set up theodolites or to determine GPS satellite orbits. The study of the figure of the Earth together with its gravitational field is the science of geodesy . The graticule is formed by the lines of constant latitude and constant longitude, which are constructed with reference to

2970-490: Is divided into 12 signs of 30° longitude, each of which approximates the Sun's motion in one month. In ancient times, the signs corresponded roughly to 12 of the constellations that straddle the ecliptic. These signs are sometimes still used in modern terminology. The " First Point of Aries " was named when the March equinox Sun was actually in the constellation Aries ; it has since moved into Pisces because of precession of

3069-471: Is likely due to the way in which the Solar System formed from a protoplanetary disk . Probably the closest current representation of the disk is known as the invariable plane of the Solar System . Earth's orbit, and hence, the ecliptic, is inclined a little more than 1° to the invariable plane, Jupiter's orbit is within a little more than ½° of it, and the other major planets are all within about 6°. Because of this, most Solar System bodies appear very close to

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3168-447: Is modeled by the geoid , a surface which approximates the 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 modeled by an ellipsoid of revolution . The definitions of latitude and longitude on such reference surfaces are detailed in

3267-582: Is near an ascending or descending node at the same time it is at conjunction ( new ) or opposition ( full ). The ecliptic is so named because the ancients noted that eclipses only occur when the Moon is crossing it. The exact instants of equinoxes and solstices are the times when the apparent ecliptic longitude (including the effects of aberration and nutation ) of the Sun is 0°, 90°, 180°, and 270°. Because of perturbations of Earth's orbit and anomalies of

3366-466: Is of great importance in accurate applications, such as a Global Positioning System (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 ( ϕ or φ ). It is measured in degrees , minutes and seconds or decimal degrees , north or south of

3465-789: Is perpendicular to }}M\mathbb {t} \quad \,&{\text{ if and only if }}\quad 0=(W\mathbb {n} )\cdot (M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=(W\mathbb {n} )^{\mathrm {T} }(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\left(\mathbb {n} ^{\mathrm {T} }W^{\mathrm {T} }\right)(M\mathbb {t} )\\&{\text{ if and only if }}\quad 0=\mathbb {n} ^{\mathrm {T} }\left(W^{\mathrm {T} }M\right)\mathbb {t} \\\end{alignedat}}} Choosing W {\displaystyle \mathbf {W} } such that W T M = I , {\displaystyle W^{\mathrm {T} }M=I,} or W = ( M − 1 ) T , {\displaystyle W=(M^{-1})^{\mathrm {T} },} will satisfy

3564-535: Is the September equinox or descending node . The orientation of Earth's axis and equator are not fixed in space, but rotate about the poles of the ecliptic with a period of about 26,000 years, a process known as lunisolar precession , as it is due mostly to the gravitational effect of the Moon and Sun on Earth's equatorial bulge . Likewise, the ecliptic itself is not fixed. The gravitational perturbations of

3663-451: Is the angle between the equatorial plane and the normal to the surface at that point: the normal to the surface of the sphere is along the radial vector. The latitude, as defined in this way for the sphere, is often termed the spherical latitude, to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article. Besides the equator, four other parallels are of significance: The plane of

3762-421: Is the meridional radius of curvature . The quarter meridian distance from the equator to the pole is For WGS84 this distance is 10 001 .965 729  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

3861-831: Is the one-dimensional subspace with basis { n } . {\displaystyle \{\mathbf {n} \}.} A differential variety defined by implicit equations in the n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} is the set of the common zeros of a finite set of differentiable functions in n {\displaystyle n} variables f 1 ( x 1 , … , x n ) , … , f k ( x 1 , … , x n ) . {\displaystyle f_{1}\left(x_{1},\ldots ,x_{n}\right),\ldots ,f_{k}\left(x_{1},\ldots ,x_{n}\right).} The Jacobian matrix of

3960-521: The Philosophiæ Naturalis Principia Mathematica , in which he proved that a rotating self-gravitating fluid body in equilibrium takes the form of an oblate ellipsoid. (This article uses the term ellipsoid in preference to the older term spheroid .) Newton's result was confirmed by geodetic measurements in the 18th century. (See Meridian arc .) An oblate ellipsoid is the three-dimensional surface generated by

4059-414: The Sun (actually of Earth in its orbit) cause short-term small-amplitude periodic oscillations of Earth's axis, and hence the celestial equator, known as nutation . This adds a periodic component to the position of the equinoxes; the positions of the celestial equator and (March) equinox with fully updated precession and nutation are called the true equator and equinox ; the positions without nutation are

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4158-454: The celestial equator , it crosses the ecliptic at two points known as the equinoxes . The Sun, in its apparent motion along the ecliptic, crosses the celestial equator at these points, one from south to north, the other from north to south. The crossing from south to north is known as the March equinox , also known as the first point of Aries and the ascending node of the ecliptic on the celestial equator. The crossing from north to south

4257-455: The foot of a perpendicular ) can be defined at the point P on the surface where the normal vector contains Q . The normal distance of a point Q to a curve or to a surface is the Euclidean distance between Q and its foot P . The normal direction to a space curve is: where R = κ − 1 {\displaystyle R=\kappa ^{-1}} is

4356-422: The mean equator and equinox . Obliquity of the ecliptic is the term used by astronomers for the inclination of Earth's equator with respect to the ecliptic, or of Earth's rotation axis to a perpendicular to the ecliptic. It is about 23.4° and is currently decreasing 0.013 degrees (47 arcseconds) per hundred years because of planetary perturbations. The angular value of the obliquity is found by observation of

4455-407: The mean equinox of 4 January 2010 0h TT as above , without the addition of nutation. Because the orbit of the Moon is inclined only about 5.145° to the ecliptic and the Sun is always very near the ecliptic, eclipses always occur on or near it. Because of the inclination of the Moon's orbit, eclipses do not occur at every conjunction and opposition of the Sun and Moon, but only when the Moon

4554-446: The null space of the matrix P = [ p 1 ⋯ p n − 1 ] , {\displaystyle P={\begin{bmatrix}\mathbf {p} _{1}&\cdots &\mathbf {p} _{n-1}\end{bmatrix}},} meaning P n = 0 . {\displaystyle P\mathbf {n} =\mathbf {0} .} That is, any vector orthogonal to all in-plane vectors

4653-438: The radius of curvature (reciprocal curvature ); T {\displaystyle \mathbf {T} } is the tangent vector , in terms of the curve position r {\displaystyle \mathbf {r} } and arc-length s {\displaystyle s} : For a convex polygon (such as a triangle ), a surface normal can be calculated as the vector cross product of two (non-parallel) edges of

4752-580: The zenith ). On map projections there is no universal rule as to how meridians and parallels should appear. The examples below show the named parallels (as red lines) on the commonly used Mercator projection and the Transverse Mercator projection . On the former the parallels are horizontal and the meridians are vertical, whereas on the latter there is no exact relationship of parallels and meridians with horizontal and vertical: both are complicated curves. \ In 1687 Isaac Newton published

4851-401: The Earth's orbit about the Sun is called the ecliptic , and 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 variously the axial tilt, the obliquity, or the inclination of the ecliptic, and it is conventionally denoted by i . The latitude of the tropical circles is equal to i and the latitude of

4950-468: The Earth–Moon barycenter wobbles slightly around a mean position in a complex fashion. Because Earth's rotational axis is not perpendicular to its orbital plane , Earth's equatorial plane is not coplanar with the ecliptic plane, but is inclined to it by an angle of about 23.4°, which is known as the obliquity of the ecliptic . If the equator is projected outward to the celestial sphere , forming

5049-691: 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. Only at latitudes in between the two tropics is it possible for the Sun to be directly overhead (at

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5148-447: The Sun, the apparent position of the Sun takes one year to make a complete circuit of the ecliptic. With slightly more than 365 days in one year, the Sun moves a little less than 1° eastward every day. This small difference in the Sun's position against the stars causes any particular spot on Earth's surface to catch up with (and stand directly north or south of) the Sun about four minutes later each day than it would if Earth did not orbit;

5247-570: The WGS84 spheroid is The variation of this distance with latitude (on WGS84 ) is shown in the table along with the length of a degree of longitude (east–west distance): A calculator for any latitude is provided by the U.S. Government's National Geospatial-Intelligence Agency (NGA). The following graph illustrates the variation of both a degree of latitude and a degree of longitude with latitude. There are six auxiliary latitudes that have applications to special problems in geodesy, geophysics and

5346-439: The above equation, giving a W n {\displaystyle W\mathbb {n} } perpendicular to M t , {\displaystyle M\mathbb {t} ,} or an n ′ {\displaystyle \mathbf {n} ^{\prime }} perpendicular to t ′ , {\displaystyle \mathbf {t} ^{\prime },} as required. Therefore, one should use

5445-425: The angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by m ( ϕ ) then where R denotes the mean radius of the Earth. R is equal to 6,371 km or 3,959 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

5544-461: The calendar , the dates of these are not fixed. The ecliptic currently passes through the following thirteen constellations : There are twelve constellations that are not on the ecliptic, but are close enough that the Moon and planets can occasionally appear in them. The ecliptic forms the center of the zodiac , a celestial belt about 20° wide in latitude through which the Sun, Moon, and planets always appear to move. Traditionally, this region

5643-422: The case of smooth curves and smooth surfaces . The normal is often used in 3D computer graphics (notice the singular, as only one normal will be defined) to determine a surface's orientation toward a light source for flat shading , or the orientation of each of the surface's corners ( vertices ) to mimic a curved surface with Phong shading . The foot of a normal at a point of interest Q (analogous to

5742-467: The celestial equator for about 185 days of each year, and south of it for about 180 days. The variation of orbital speed accounts for part of the equation of time . Because of the movement of Earth around the Earth–Moon center of mass , the apparent path of the Sun wobbles slightly, with a period of about one month . Because of further perturbations by the other planets of the Solar System ,

5841-410: The celestial equator. Spherical coordinates , known as ecliptic longitude and latitude or celestial longitude and latitude, are used to specify positions of bodies on the celestial sphere with respect to the ecliptic. Longitude is measured positively eastward 0° to 360° along the ecliptic from the March equinox, the same direction in which the Sun appears to move. Latitude is measured perpendicular to

5940-638: The centre of the Earth and perpendicular to the rotation axis intersects the surface at 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

6039-407: The datum ED50 define a point on the ground which is 140 metres (460 feet) distant from the tower. A web search may produce several different values for the latitude of the tower; the reference ellipsoid is rarely specified. The length of a degree of latitude depends on the figure of the Earth assumed. On the sphere the normal passes through the centre and the latitude ( ϕ ) is therefore equal to

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6138-406: The ecliptic in the sky. The invariable plane is defined by the angular momentum of the entire Solar System, essentially the vector sum of all of the orbital and rotational angular momenta of all the bodies of the system; more than 60% of the total comes from the orbit of Jupiter. That sum requires precise knowledge of every object in the system, making it a somewhat uncertain value. Because of

6237-412: The ecliptic, and therefore always appear relatively close to it on the sky. Because Earth's orbit, and hence the ecliptic, moves very little, it is a relatively fixed reference with respect to the stars. Because of the precessional motion of the equinox , the ecliptic coordinates of objects on the celestial sphere are continuously changing. Specifying a position in ecliptic coordinates requires specifying

6336-436: The ecliptic, to +90° northward or −90° southward to the poles of the ecliptic, the ecliptic itself being 0° latitude. For a complete spherical position, a distance parameter is also necessary. Different distance units are used for different objects. Within the Solar System, astronomical units are used, and for objects near Earth , Earth radii or kilometers are used. A corresponding right-handed rectangular coordinate system

6435-400: 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 ϕ . It was introduced by Legendre and Bessel 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, u ( ϕ ) ,

6534-465: The equator. For navigational purposes positions are given in degrees and decimal minutes. For instance, The Needles lighthouse is at 50°39.734′ N 001°35.500′ W. This article relates to coordinate systems for the Earth: it may be adapted to cover the Moon, planets and other celestial objects ( planetographic latitude ). For a brief history, see History of latitude . In celestial navigation , latitude

6633-438: The following sections. Lines of constant latitude and longitude together constitute a 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 to the reference surface, which passes through the point on the physical surface. Latitude and longitude together with some specification of height constitute

6732-399: The geocentric latitude ( θ ) and the geodetic latitude ( ϕ ) is: For points not on the surface of the ellipsoid, the relationship involves additionally the ellipsoidal height h : where N is the prime vertical radius of curvature. The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of

6831-1857: The graph of a function z = f ( x , y ) , {\displaystyle z=f(x,y),} an upward-pointing normal can be found either from the parametrization r ( x , y ) = ( x , y , f ( x , y ) ) , {\displaystyle \mathbf {r} (x,y)=(x,y,f(x,y)),} giving n = ∂ r ∂ x × ∂ r ∂ y = ( 1 , 0 , ∂ f ∂ x ) × ( 0 , 1 , ∂ f ∂ y ) = ( − ∂ f ∂ x , − ∂ f ∂ y , 1 ) ; {\displaystyle \mathbf {n} ={\frac {\partial \mathbf {r} }{\partial x}}\times {\frac {\partial \mathbf {r} }{\partial y}}=\left(1,0,{\tfrac {\partial f}{\partial x}}\right)\times \left(0,1,{\tfrac {\partial f}{\partial y}}\right)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right);} or more simply from its implicit form F ( x , y , z ) = z − f ( x , y ) = 0 , {\displaystyle F(x,y,z)=z-f(x,y)=0,} giving n = ∇ F ( x , y , z ) = ( − ∂ f ∂ x , − ∂ f ∂ y , 1 ) . {\displaystyle \mathbf {n} =\nabla F(x,y,z)=\left(-{\tfrac {\partial f}{\partial x}},-{\tfrac {\partial f}{\partial y}},1\right).} Since

6930-1038: The inverse transpose of the linear transformation when transforming surface normals. The inverse transpose is equal to the original matrix if the matrix is orthonormal, that is, purely rotational with no scaling or shearing. For an ( n − 1 ) {\displaystyle (n-1)} -dimensional hyperplane in n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} given by its parametric representation r ( t 1 , … , t n − 1 ) = p 0 + t 1 p 1 + ⋯ + t n − 1 p n − 1 , {\displaystyle \mathbf {r} \left(t_{1},\ldots ,t_{n-1}\right)=\mathbf {p} _{0}+t_{1}\mathbf {p} _{1}+\cdots +t_{n-1}\mathbf {p} _{n-1},} where p 0 {\displaystyle \mathbf {p} _{0}}

7029-451: 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 one must know how the latitude and longitude values are transformed from one ellipsoid to another. GPS handsets include software to carry out datum transformations which link WGS84 to the local reference ellipsoid with its associated grid. The shape of an ellipsoid of revolution

7128-536: 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 When the latitude difference is 1 degree, corresponding to ⁠ π / 180 ⁠ radians, the arc distance is about The distance in metres (correct to 0.01 metre) between latitudes ϕ {\displaystyle \phi }  − 0.5 degrees and ϕ {\displaystyle \phi }  + 0.5 degrees on

7227-400: The motions of Earth and other planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, and from these ephemerides various astronomical values, including the obliquity, are derived. Until 1983 the obliquity for any date was calculated from work of Newcomb , who analyzed positions of

7326-458: The normal vector space at a point is the vector space generated by the normal vectors of the hypersurfaces at the point. The normal (affine) space at a point P {\displaystyle P} of the variety is the affine subspace passing through P {\displaystyle P} and generated by the normal vector space at P . {\displaystyle P.} These definitions may be extended verbatim to

7425-426: The other bodies of the Solar System cause a much smaller motion of the plane of Earth's orbit, and hence of the ecliptic, known as planetary precession . The combined action of these two motions is called general precession , and changes the position of the equinoxes by about 50 arc seconds (about 0.014°) per year. Once again, this is a simplification. Periodic motions of the Moon and apparent periodic motions of

7524-412: The perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the background of stars . The ecliptic is an important reference plane and is the basis of the ecliptic coordinate system . The ecliptic is the apparent path of the Sun throughout the course of a year . Because Earth takes one year to orbit

7623-515: The planets until about 1895: ε = 23°27′08.26″ − 46.845″ T − 0.0059″ T + 0.00181″ T where ε is the obliquity and T is tropical centuries from B1900.0 to the date in question. From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac . Obliquity based on DE200, which analyzed observations from 1911 to 1979,

7722-493: The point are parameterized by Cayley suggested the term parametric latitude because of the form of these equations. The parametric latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics, ( Vincenty , Karney). 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: Normal (geometry) In geometry ,

7821-396: The points where the variety is not a manifold. Let V be the variety defined in the 3-dimensional space by the equations x y = 0 , z = 0. {\displaystyle x\,y=0,\quad z=0.} This variety is the union of the x {\displaystyle x} -axis and the y {\displaystyle y} -axis. At a point (

7920-462: The polar circles is its complement (90° - i ). The axis of rotation varies slowly over time and the values given here are those for the current epoch . The time variation is discussed more fully in the article on axial tilt . The figure shows the geometry of a cross-section of the plane perpendicular to the ecliptic and through the centres of the Earth and the Sun at the December solstice when

8019-662: The polygon. For a plane given by the general form plane equation a x + b y + c z + d = 0 , {\displaystyle ax+by+cz+d=0,} the vector n = ( a , b , c ) {\displaystyle \mathbf {n} =(a,b,c)} is a normal. For a plane whose equation is given in parametric form r ( s , t ) = r 0 + s p + t q , {\displaystyle \mathbf {r} (s,t)=\mathbf {r} _{0}+s\mathbf {p} +t\mathbf {q} ,} where r 0 {\displaystyle \mathbf {r} _{0}}

8118-509: 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 below .) The forms given are, apart from notational variants, those in

8217-473: The rotation axis of the Earth. The primary reference points are the poles where the axis of rotation of the Earth intersects the reference surface. Planes which contain the rotation axis intersect the surface at the 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

8316-417: 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 triaxial .) Many different 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

8415-440: The rows of the Jacobian matrix are ( 0 , 0 , 1 ) {\displaystyle (0,0,1)} and ( 0 , 0 , 0 ) . {\displaystyle (0,0,0).} Thus the normal vector space and the normal affine space have dimension 1 and the normal affine space is the z {\displaystyle z} -axis. The normal ray is the outward-pointing ray perpendicular to

8514-518: The semi-major axis and the inverse flattening, ⁠ 1 / f ⁠ . For example, the defining values for the WGS84 ellipsoid, used by all GPS devices, are from which are derived The difference between the semi-major and semi-minor axes is about 21 km (13 miles) and as fraction of the semi-major axis it equals the flattening; on a computer monitor the ellipsoid could be sized as 300 by 299 pixels. This would barely be distinguishable from

8613-426: The set of points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} satisfying an equation F ( x 1 , x 2 , … , x n ) = 0 , {\displaystyle F(x_{1},x_{2},\ldots ,x_{n})=0,} where F {\displaystyle F}

8712-468: The sky's distant background. The ecliptic forms one of the two fundamental planes used as reference for positions on the celestial sphere, the other being the celestial equator . Perpendicular to the ecliptic are the ecliptic poles , the north ecliptic pole being the pole north of the equator. Of the two fundamental planes, the ecliptic is closer to unmoving against the background stars, its motion due to planetary precession being roughly 1/100 that of

8811-417: The sphere is 111.2 km (69.1 statute miles) (60.0 nautical miles). The length of one minute of latitude is 1.853 km (1.151 statute miles) (1.00 nautical miles), while the length of 1 second of latitude is 30.8 m or 101 feet (see nautical mile ). In Meridian arc and standard texts it is shown that the distance along a meridian from latitude ϕ to the equator is given by ( ϕ in radians) where M ( ϕ )

8910-402: The squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of ϕ − θ {\displaystyle \phi {-}\theta } may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′. The parametric latitude or reduced latitude , β , is defined by the radius drawn from the centre of

9009-454: The standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder. Derivations of these expressions may be found in Adams and online publications by Osborne and Rapp. The geocentric latitude is the angle between the equatorial plane and the radius from the centre to a point of interest. When the point is on the surface of the ellipsoid, the relation between

9108-472: 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 / 298 ⁠ and 0.0818 respectively. Values for a number of ellipsoids are given in Figure of the Earth . Reference ellipsoids are usually defined by

9207-450: The surface is given by the gradient n = ∇ F ( x , y , z ) . {\displaystyle \mathbf {n} =\nabla F(x,y,z).} since the gradient at any point is perpendicular to the level set S . {\displaystyle S.} For a surface S {\displaystyle S} in R 3 {\displaystyle \mathbb {R} ^{3}} given as

9306-407: The surface of an optical medium at a given point. In reflection of light , the angle of incidence and the angle of reflection are respectively the angle between the normal and the incident ray (on the plane of incidence ) and the angle between the normal and the reflected ray . Ecliptic The ecliptic or ecliptic plane is the orbital plane of Earth around the Sun . From

9405-1175: The tangent plane t {\displaystyle \mathbf {t} } into a vector n ′ {\displaystyle \mathbf {n} ^{\prime }} perpendicular to the transformed tangent plane M t , {\displaystyle \mathbf {Mt} ,} by the following logic: Write n′ as W n . {\displaystyle \mathbf {Wn} .} We must find W . {\displaystyle \mathbf {W} .} W n  is perpendicular to  M t  if and only if  0 = ( W n ) ⋅ ( M t )  if and only if  0 = ( W n ) T ( M t )  if and only if  0 = ( n T W T ) ( M t )  if and only if  0 = n T ( W T M ) t {\displaystyle {\begin{alignedat}{5}W\mathbb {n} {\text{

9504-407: The theory of map projections: 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 below . The remaining latitudes are not used in this way; they are used only as intermediate constructs in map projections of

9603-399: The uncertainty regarding the exact location of the invariable plane, and because the ecliptic is well defined by the apparent motion of the Sun, the ecliptic is used as the reference plane of the Solar System both for precision and convenience. The only drawback of using the ecliptic instead of the invariable plane is that over geologic time scales, it will move against fixed reference points in

9702-434: The variety is the k × n {\displaystyle k\times n} matrix whose i {\displaystyle i} -th row is the gradient of f i . {\displaystyle f_{i}.} By the implicit function theorem , the variety is a manifold in the neighborhood of a point where the Jacobian matrix has rank k . {\displaystyle k.} At such

9801-445: Was calculated: ε = 23°26′21.45″ − 46.815″ T − 0.0006″ T + 0.00181″ T where hereafter T is Julian centuries from J2000.0 . JPL's fundamental ephemerides have been continually updated. The Astronomical Almanac for 2010 specifies: ε = 23°26′21.406″ − 46.836769″ T − 0.0001831″ T + 0.00200340″ T − 0.576×10 ″ T − 4.34×10 ″ T These expressions for the obliquity are intended for high precision over

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