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1.ANALYTICAL GEOMETRYAnalytic geometry, oranalytical geometry has two differentmeanings in mathematics. The modern and advanced meaning

refers to the geometry ofanalytic varieties. This article focuseson the classical and elementary meaning.

In classical mathematics, analytic geometry, also known ascoordinate geometry, orCartesian geometry, is the study ofgeometry using a coordinate system and the principles ofalgebra and analysis. This contrasts with the synthetic approachofEuclidean geometry, which treats certain geometric notions as

primitive, and uses deductive reasoning based on axioms andtheorems to derive truth. Analytic geometry is widely used inphysics and engineering, and is the foundation of most modernfields of geometry, including algebraic, differential, discrete,and computational geometry.

Usually the Cartesian coordinate system is applied to manipulateequations forplanes, straightlines, and squares, often in two andsometimes in three dimensions of measurement. Geometrically,one studies the Euclidean plane (2 dimensions) and Euclideanspace (3 dimensions). As taught in school books, analyticgeometry can be explained more simply: it is concerned withdefining geometrical shapes in a numerical way and extractingnumerical information from that representation. The numericaloutput, however, might also be a vectoror a shape. That thealgebra of the real numbers can be employed to yield results

about the linear continuum of geometry relies on the CantorDedekind axiom.

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Cartesian coordinates.

COORDINATESIn analytic geometry, theplane is given a

coordinate system, by which everypoint has a pair ofrealnumbercoordinates. The most common coordinate system to

use is the Cartesian coordinate system, where each point has anx-coordinate representing its horizontal position, and ay-coordinate representing its vertical position. These are typicallywritten as an ordered pair(x,y). This system can also be used

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for three-dimensional geometry, where every point in Euclideanspace is represented by an ordered triple of coordinates (x,y,z).

Other coordinate systems are possible. On the plane the most

common alternative ispolar coordinates, where every point isrepresented by its radiusrfrom the origin and its angle. Inthree dimensions, common alternative coordinate systemsinclude cylindrical coordinates and spherical coordinates.

In analytic geometry, any equation involving the coordinatesspecifies a subset of the plane, namely the solution set for the

equation. For example, the equationy =x corresponds to the setof all the points on the plane whosex-coordinate andy-coordinate are equal. These points form a line, andy =x is saidto be the equation for this line. In general, linear equationsinvolvingx andy specify lines, quadratic equations specifyconic sections, and more complicated equations describe morecomplicated figures.

Usually, a single equation corresponds to a curve on the plane.

This is not always the case: the trivial equationx =x specifiesthe entire plane, and the equationx2 +y2 = 0 specifies only thesingle point (0, 0). In three dimensions, a single equation usuallygives a surface, and a curve must be specified as the intersectionof two surfaces (see below), or as a system ofparametricequations. The equationx2 +y2 = r2 is the equation for any circlewith a radius of r.

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The distance formula on the plane follows from the Pythagoreantheorem.

In analytic geometry, geometric notions such as distance andangle measure are defined using formulas. These definitions aredesigned to be consistent with the underlying Euclideangeometry. For example, using Cartesian coordinates on theplane, the distance between two points (x1,y1) and (x2,y2) isdefined by the formula

which can be viewed as a version of the Pythagorean theorem.Similarly, the angle that a line makes with the horizontal can bedefined by the formula

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where m is the slope of the line.

Transformations

Transformations are applied to parentfunctions to turn it into a new function with similarcharacteristics. For example, the parent function y=1/x has ahorizontal and a vertical asymptote, and occupies the first and

third quadrant, and all of its transformed forms have onehorizontal and vertical asymptote,and occupies either the 1st and3rd or 2nd and 4th quadrant. In general, ify =f(x), then it can betransformed intoy = af(b(x k)) + h. In the new transformedfunction, a is the factor that vertically stretches the function if itis greater than 1 or vertically compresses the function if it is lessthan 1, and for negative a values, the function is reflected in thex-axis. The b value compresses the graph of the function

horizontally if greater than 1 and stretches the functionhorizontally if less than 1, and like a, reflects the function in they-axis when it is negative. The k and h values introducetranslations, h, vertical, and khorizontal. Positive h and kvaluesmean the function is translated to the positive end of its axis andnegative meaning translation towards the negative end.

Intersections:

Problem: In a convex pentagonABCDE, the sideshave lengths 1, 2, 3, 4, and 5, though not necessarily in thatorder. LetF, G,H, andIbe the midpoints of the sidesAB,BC,CD, andDE, respectively. LetXbe the midpoint of segmentFH,

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and Ybe the midpoint of segment GI. The length of segmentXYis an integer. Find all possible values for the length of sideAE.

Solution:Without loss of generality, letA,B, C,D, andEbe

located atA = (0,0),B = (a,0), C= (b,e),D = (c,f), andE= (d,g).Using the midpoint formula, the pointsF, G,H,I,X, and Yarelocated at

, , , ,

, and

Using the distance formula,

and

SinceXYhas to be an integer,

soAE= 4.

Modern analytic geometry:An analytic variety is defined locally as the set of common

solutions of several equations involving analytic functions. It isanalogous to the included concept of real or complex algebraic

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variety. Any complex manifold is an analytic variety. Sinceanalytic varieties may have singular points, not all analytic

varieties are manifolds.

Analytic geometry is essentially equivalent to real and complexAlgebraic geometry as it has been shown by Jean-Pierre Serre inhis paperGAGA, whose name is, in French,Algebraic geometryand analytic geometric. Nevertheless, the two fields remaindistinct, as the methods of proof are quite different and algebraicgeometry includes also geometry in finite characteristic.

Line:

A line is a straight one-dimensional figure having no thicknessand extending infinitely in both directions. A line is sometimescalled a straight line or, more archaically, a right line (Casey1893), to emphasize that it has no "wiggles" anywhere along itslength. While lines are intrinsically one-dimensional objects,they may be embedded in higher dimensional spaces.

Harary (1994) called an edge of a graph a "line."

A line is uniquely determined by two points, and the line passing

through points and is denoted . Similarly, the length of the

finite line segment terminating at these points may be denoted. A line may also be denoted with a single lower-case letter(Jurgensen et al. 1963, p. 22).

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Euclid defined a line as a "breadthless length," and a straightline as a line that "lies evenly with the points on itself" (Kline1956, Dunham 1990).

Consider first lines in a two-dimensionalplane. Two lines lyingin the same plane that do not intersect one another are said to beparallel lines. Two lines lying in different planes that do notintersect one another are said to be skew lines.

The line withx-intercept andy-intercept is given by theintercept form

(1)

The line through with slope is given by thepoint-slopeform

(2)

The line with -intercept and slope is given by the slope-intercept form

(3)

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(13)

The angle between lines

(14)

(15)

is

(16)

The line joining points with trilinear coordinates and

is the set of point satisfying

(17)

(18

)The line through in the direction and the line through

in direction intersectiff

(19)

The line through a point parallel to

(20)

is

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(21)

The lines

(22)

(23)

areparallel if

(24)

for all , andperpendicularif

(25)

for all (Sommerville 1961, Kimberling 1998, p. 29).

The line through a point perpendicularto () is given by

(26)

In three-dimensional space, the line passing through the point

andparallel to the nonzerovector hasparametricequations

(27)

(28)

(29)

written concisely as

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(30)

Similarly, the line in three dimensions passing through and

has parametric vector equation

(31)

where this parametrization corresponds to and .

Vertical Line:

A vertical line is one which runs up and down the page.

In geometry, a vertical line is one which runs from up and downthe page. Its cousin is the horizontal line which runs left to rightacross the page. A vertical line isperpendicularto a horizontalline.

Vertical Angles:A pair of non-adjacent angles formed by the intersection of two

straight lines

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when two lines intersect, four angles are formed.

Each opposite pair are called vertical angles and

are always congruent.

Horizontal Line:

A horizontal line is one which runs left-to-right

across the page.

In geometry, a horizontal line is one which runs

from left to right across the page. It comes from

the word 'horizon', in the sense that horizontal

lines are parallel to the horizon.

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Drawing line:You can draw a line that just goes off the edges of the page, as

in the figure above. More commonly it is shown as a line withan arrow head on each end as shown below. The arrow headsmean that the line goes off to infinity in both directions.

Lines are commonly named in two ways:1. By any two points on the line. In the figure above, the line

would be called JK because it passes through the two pointsJ and K. Recall that points are usually labelled with singleupper-case (capital) letters. There is a symbol for thiswhich looks like

JK

. This is read as "line JK". The two arrow heads indicatethat this is a line which passes through J and K but goes onforever in both directions.

2. By a single letter.The line above could also be called simply "y". Byconvention, this is usually a single lower case (small) letter.This method is sometimes used when the line does not have

two points on it to define it.

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Problem:

Find the angle between the line parallel to

the vector [2, -1, 0] and the plane given bythe vector equation r.[3, 0, 4] = 5

Diagram

v is the vector to which the line is parallel.

P is the point of intersection of the line and plane.

N is the normal to the plane through P.

is the angle between the line and plane.

is the angle between the normal and the line.

The angle, , between the normal and the line can be easilyfound using 'the angle between two lines' method.

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The required angle, , is then the difference between and onerightangle.

From the equation to the given plane, r.[3, 0, 4] = 5, the normalto the plane is parallel to the vector [3, 0, 4]. The line is parallelto the vector [2, -1, 0].

The angle between the normal to the plane and line is therefore,the angle between the two vectors [3, 0, 4] and [2, -1, 0];

[3, 0, 4] . [2, -1, 0] = | [3, 0, 4] | | |cos

cos =

=

= 1 radianThe angle between the line and the plane, is given by;

= - 1

= 0.57radians

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Note, if the value of had been greater than one right angle, thatis, the obtuse angle between the line and plane then, would be

- .

Plane

A flat surface that is infinitely large and with zero thicknessClearly, when you read the above definition, such a thing cannotpossibly really exist. Imagine a flat sheet of metal. Now make itinfinitely large in both directions. This means that no matterhow far you go, you never reach its edges. Now imagine that itis so thin that it actually has no thickness at all. In spite of this, itremains completely rigid and flat. This is the 'plane' in

geometry.It fits into a scheme that starts with a point, which has nodimensions and goes up through solids which have threedimensions:

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It is difficult to draw planes, since the edges have to be drawn.When you see a picture that represents a plane, alwaysremember that it actually has no edges, and it is infinitely large.

The plane has two dimensions: length and width. But since theplane is infinitely large, the length and width cannot bemeasured.

Just as a line is defined by two points, a plane is defined by threepoints. Given three points that are not collinear, there is just oneplane that contains all three.

Parallel planes:

You can think of parallel planes as sheets of cardboard oneabove the other with a gap between them. Parallel planes are thesame distance apart everywhere, and so they never touch.

Intersecting planes:If two planes are not parallel, then they will intersect (crossover) each other somewhere. Two planes always intersect ata line, as shown on the right.

This is similar to the way two lines intersect at a point.

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The term "cylinder" has a number of related meanings. In itsmost general usage, the word "cylinder" refers to a solid

bounded by a closed generalized cylinder(a.k.a. cylindricalsurface) and two parallel planes (Kern and Bland 1948, p. 32;Harris and Stocker 1998, p. 102). A cylinder of this sort havinga polygonal base is therefore aprism Harris and use the term"general cylinder" to refer to the solid bounded a closedgeneralized cylinder.

Unfortunately, the term "cylinder" is commonly used not only to

refer to the solid bounded by a cylindrical surface, but to thecylindrical surface itself To make matters worse, according totopologists, a cylindrical surface is not even a true surface, butrather a so-called surface with boundary .

As if this were not confusing enough, the term "cylinder" whenused without qualification commonly refers to the particularcase of a solid of circularcross section in which the centers ofthe circles all lie on a single line (i.e., a circular cylinder). A

cylinder is called a right cylinder if it is "straight" in the sensethat its cross sections lie directly on top of each other; otherwise,the cylinder is said to be oblique. The unqualified term"cylinder" is also commonly used to refer to a right circularcylinder and this is the usage followed in this work.

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The right cylinder of radius with axis given by the line segment

with endpoints and is implemented in Mathematica

as Cylinder[ x1,y1,z1 , x2,y2,z2 , r].

The illustrations above show a circular right cylinder of heightand radius .

If a plane inclined with respect to the caps of a right circularcylinderintersects a cylinder, it does so in an ellipse. Thecylinder was extensively studied by Archimedes in his two-volume workOn the Sphere and Cylinderin ca. 225 BC.

As illustrated above, a cylinder can be described topologically asa square in which top and bottom edges are given parallelorientations and the left and right edges are joined to place thearrow heads and tails into coincidence . The cylindrical surfaceof a circular cylinder has Euler characteristic .

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The lateral surface of a cylinder of height and radius can bedescribed parametrically by

(

1)

(

2

)

(

3

)

for and .

These are the basis forcylindrical coordinates. The lateral

surface area and volume of the cylinder of height and radiusare

(

4

)

(

5

)

The formula for the volume of a cylinder leads to the

mathematical joke: "What is the volume of a pizza of thickness

and radius ?" Answer: pi z z a. This result is sometimes knownas the secondpizza theorem.

21.
http://mathworld.wolfram.com/Lateral.htmlhttp://mathworld.wolfram.com/Radius.htmlhttp://mathworld.wolfram.com/CylindricalCoordinates.htmlhttp://mathworld.wolfram.com/Lateral.htmlhttp://mathworld.wolfram.com/SurfaceArea.htmlhttp://mathworld.wolfram.com/Volume.htmlhttp://mathworld.wolfram.com/Radius.htmlhttp://mathworld.wolfram.com/PizzaTheorem.htmlhttp://mathworld.wolfram.com/Lateral.htmlhttp://mathworld.wolfram.com/Radius.htmlhttp://mathworld.wolfram.com/CylindricalCoordinates.htmlhttp://mathworld.wolfram.com/Lateral.htmlhttp://mathworld.wolfram.com/SurfaceArea.htmlhttp://mathworld.wolfram.com/Volume.htmlhttp://mathworld.wolfram.com/Radius.htmlhttp://mathworld.wolfram.com/PizzaTheorem.html


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If the top and bottom caps are added, the total surface area of acylinder is given by

(

6)

(

7

)

The interior of the cylinder of radius , height , and mass hasmoment of inertia tensor about its centroid is

(

8

)

The volume-to-total surface area ratio for a cylindrical solid is

(9

)

which is related to the harmonic mean of the radius and height .The fact that

(1

0)

was known to Archimedes .

Using the parametrization

(1

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1)

(1

2)

(1

3)

gives coefficients of the first fundamental form

(1

4)

(15)

(1

6)

the coefficients of the second fundamental form

(1

7)

(1

8)

(1

9)

area element

(2

0)

Gaussian curvature

23.
http://mathworld.wolfram.com/FirstFundamentalForm.htmlhttp://mathworld.wolfram.com/SecondFundamentalForm.htmlhttp://mathworld.wolfram.com/AreaElement.htmlhttp://mathworld.wolfram.com/GaussianCurvature.htmlhttp://mathworld.wolfram.com/FirstFundamentalForm.htmlhttp://mathworld.wolfram.com/SecondFundamentalForm.htmlhttp://mathworld.wolfram.com/AreaElement.htmlhttp://mathworld.wolfram.com/GaussianCurvature.html


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(2

1)

mean curvature

(2

2)

andprincipal curvatures

(2

3)

(24)

It is possible to arrange seven finite cylinders so that each is

tangent to the other six, as illustrated above.

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http://mathworld.wolfram.com/MeanCurvature.htmlhttp://mathworld.wolfram.com/PrincipalCurvatures.htmlhttp://mathworld.wolfram.com/MeanCurvature.htmlhttp://mathworld.wolfram.com/PrincipalCurvatures.html


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ConeDefinition of Cone:

Cone is a three-dimensional figure that has onecircular base and one vertex.

More about Cone: Right Cone: A right cone is a cone in which the

vertex is aligned directly above the center ofthe base. The base need not be a circle here.

Right Circular Cone: When the base of a rightcone is a circle, it is called a right circular cone.In a right circular cone, each point on the circleis equidistant from the vertex of the cone.

Oblique Cone: When the vertex of a cone is notaligned directly above the center of its base, itis called an oblique cone.

Solved Example on ConeFind the base area of a cone of height 5 units and base radius 2units.Choices:

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A. 5B. 2C. 4D. 12.57Correct Answer: DSolution:Step 1: The base radius is 2 units.Step 2: So, the area of the base is 2 2 = 12.57 sq units.

Related Terms for Cone

1. Circle 5. Oblique Cone

2. Vertex 6 Right Circular Cone

3. Base Right Cone 8 Three-Dimensional F

Sphere

Sphere

Sphere Facts

Notice these

interesting

things:

It is perfectly

symmetrical

It has no edges or

vertices (corners)

It is not a polyhedronGlass Sphere.

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All points on the

surface are the same

distance from the

center

And for

reference:

Surface Area = 4

r2

Volume = (4/3)

r3

Balls and marbles are

shaped like spheres.

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Largest Volume forSmallest Surface

Of all the shapes, a sphere has thesmallest surface area for a volume.Or put another way it can contain thegreatest volume for a fixed surfacearea.

Example: if you blow up a balloon itnaturally forms a sphere because it istrying to hold as much air as possible

with as small a surface as possible.Press the Play button to see.

IN NATURE:

The sphere appears in naturewhenever a surface wants to be assmall as possible. Examples includebubbles and water drops, can youthink of more?

THE EARTH:

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The Planet

Earth, our

home, is nearlya sphere, except

that it is

squashed a little

at the poles.

It is a spheroid,

which means it just

misses out on beinga sphere because it

isn't perfect in one

direction (in the

Earth's case: North-

South)

Other Cool Spheres:

Other Cool

Spheres

Other Cool

Spheres

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30.


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Cylindrical coordinate system

A cylindrical coordinate system with origin O, polar

axisA, and longitudinal axis L. The dot is the point

with radial distance = 4, angular coordinate

= 130, and heightz= 4.

A cylindrical coordinate system is a three-dimensional

coordinate system that specifies point positions by the distancefrom a chosen reference axis, the direction from the axis relativeto a chosen reference direction, and the distance from a chosenreference plane perpendicular to the axis. The latter distance isgiven as a positive or negative number depending on which sideof the reference plane faces the point.

The origin of the system is the point where all three coordinatescan be given as zero. This is the intersection between thereference plane and the axis.

The axis is variously called the cylindricalorlongitudinalaxis,to differentiate it from thepolar axis, which is the ray that lies inthe reference plane, starting at the origin and pointing in thereference direction.

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The distance from the axis may be called the radial distance orradius, while the angular coordinate is sometimes referred to asthe angular position or as the azimuth. The radius and theazimuth are together called thepolar coordinates, as theycorrespond to a two-dimensionalpolar coordinate system in theplane through the point, parallel to the reference plane. The thirdcoordinate may be called the heightoraltitude (if the referenceplane is considered horizontal), longitudinal position, oraxialposition.

Cylindrical coordinates are useful in connection with objects andphenomena that have some rotational symmetry about the

longitudinal axis, such as water flow in a straight pipe withround cross-section, heat distribution in a metal cylinder, and soon.

The coordinate surfaces of the cylindrical

coordinates (, ,z). The red cylinder shows thepoints with =2, the blue plane shows the points

withz=1, and the yellow half-plane shows the

points with =60. Thez-axis is vertical and the

x-axis is highlighted in green. The three surfaces

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http://en.wikipedia.org/wiki/Polar_coordinateshttp://en.wikipedia.org/wiki/Symmetryhttp://en.wikipedia.org/wiki/Cylinder_(geometry)http://en.wikipedia.org/wiki/Cylinder_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/File:Cylindrical_coordinate_surfaces.pnghttp://en.wikipedia.org/wiki/File:Cylindrical_coordinate_surfaces.pnghttp://en.wikipedia.org/wiki/Polar_coordinateshttp://en.wikipedia.org/wiki/Symmetryhttp://en.wikipedia.org/wiki/Cylinder_(geometry)http://en.wikipedia.org/wiki/Cylinder_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)


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intersect at the point P with those coordinates

(shown as a black sphere); the Cartesian

coordinates ofP are roughly .

Cylindrical Coordinate Surfaces. The three

orthogonal components, (green), (red), andz

(blue), each increasing at a constant rate. The

point is at the intersection between the three

colored surfaces.

In concrete situations, and in many mathematical illustrations, apositive angular coordinate is measured counterclockwise asseen from any point with positive height.

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Coordinate system conversions:The cylindrical coordinate system is one of many three-dimensional coordinate systems. The following formulae may be

used to convert between them.

Cartesian coordinates:

For the conversion between cylindrical and Cartesian coordinatesystems, it is convenient to assume that the reference plane ofthe former is the Cartesianxy plane (with equationz= 0) , andthe cylindrical axis is the Cartesianzaxis. Then thezcoordinate

is the same in both systems, and the correspondence betweencylindrical (,) and Cartesian (x,y) are the same as for polarcoordinates, namely

in one direction, and

in the other. The arcsin function is the inverse of the sine

function, and is assumed to return an angle in the range[/2,+/2] = [90,+90]. These formulas yield an azimuth inthe range [90,+270).

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Many modern programming languages provide a function thatwill compute the correct azimuth , in the range (, ], givenxandy, without the need to perform a case analysis as above.

Spherical coordinates:Spherical coordinates (radius r, elevation or inclination ,azimuth ), may be converted into cylindrical coordinates by:

is

elevation:

is

inclination:

Cylindrical coordinates may be convertedinto spherical coordinates by:

iselevation:

isinclination:

Line and volume elements:

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In many problems involving cylindrical polar coordinates, it isuseful to know the line and volume elements; these are used inintegration to solve problems involving paths and volumes.

The line element is

The volume element is

The surface element in a surface of constant radius (a verticalcylinder) is

The surface element in a surface of constant azimuth (a verticalhalf-plane) is

The surface element in a surface of constant heightz(a

horizontal plane) is

The del operator in this system is written as

and the Laplace operator is defined by

Cylindrical harmonics:36.
http://en.wikipedia.org/wiki/Line_elementhttp://en.wikipedia.org/wiki/Volume_elementhttp://en.wikipedia.org/wiki/Surface_elementhttp://en.wikipedia.org/wiki/Delhttp://en.wikipedia.org/wiki/Laplace_operatorhttp://en.wikipedia.org/wiki/Line_elementhttp://en.wikipedia.org/wiki/Volume_elementhttp://en.wikipedia.org/wiki/Surface_elementhttp://en.wikipedia.org/wiki/Delhttp://en.wikipedia.org/wiki/Laplace_operator


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A spherical coordinate system with origin O, zenith

directionZand azimuth axisA. The point has

radius r= 4, inclination = 70, and azimuth

= 130.

An alternate spherical coordinate system, using

elevation from the reference plane instead of

inclination from the zenith. The point has radius

r= 4, elevation = 50, and azimuth = 130.

The system above is an example of a left-handed

coordinate system.

In mathematics, a spherical coordinate system is a coordinatesystem forthree-dimensional space where the position of a point

is specified by three numbers: the radial distance of that pointfrom a fixed origin, its inclination angle measured from a fixedzenith direction, and the azimuth angle of its orthogonalprojection on a reference plane that passes through the originand is orthogonal to the zenith, measured from a fixed reference

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http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Coordinate_systemhttp://en.wikipedia.org/wiki/Coordinate_systemhttp://en.wikipedia.org/wiki/Dimensionhttp://en.wikipedia.org/wiki/Zenithhttp://en.wikipedia.org/wiki/Azimuthhttp://en.wikipedia.org/wiki/Orthogonal_projectionhttp://en.wikipedia.org/wiki/Orthogonal_projectionhttp://en.wikipedia.org/wiki/File:Coord_system_SE_0.svghttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Coordinate_systemhttp://en.wikipedia.org/wiki/Coordinate_systemhttp://en.wikipedia.org/wiki/Dimensionhttp://en.wikipedia.org/wiki/Zenithhttp://en.wikipedia.org/wiki/Azimuthhttp://en.wikipedia.org/wiki/Orthogonal_projectionhttp://en.wikipedia.org/wiki/Orthogonal_projection


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direction on that plane. The inclination angle is often replacedby the elevation angle measured from the reference plane.

The radial distance is also called the radius orradial

coordinate, and the inclination may be called colatitude, zenithangle, normal angle, orpolar angle.

In geography and astronomy, the elevation and azimuth (orquantities very close to them) are called the latitude andlongitude, respectively; and the radial distance is usuallyreplaced by an altitude (measured from a central point or from asea level surface).

The concept of spherical coordinates can be extended to higherdimensional spaces and are then referred to as hypersphericalcoordinates.

Illustration of spherical coordinates. The red sphere

shows the points with r= 2, the blue cone shows

the points with inclination (or elevation) = 45,

and the yellow half-plane shows the points with

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azimuth = 60. The zenith direction is vertical,

and the zero-azimuth axis is highlighted in green.

The spherical coordinates (2,45,60) determine

the point of space where those three surfacesintersect, shown as a black sphere.

Definition:To define a spherical coordinate system, one must choosetwo orthogonal directions, the zenith and the azimuthreference, and an origin point in space. These choicesdetermine a reference plane that contains the origin and isperpendicular to the zenith. The spherical coordinates of apointPare then defined as follows:

the radius or radial distance is the Euclideandistance from the origin O to P.

the inclination (orpolar angle) is the anglebetween the zenith direction and the linesegment OP.

the azimuth (or azimuthal angle) is the signedangle measured from the azimuth reference

direction to the orthogonal projection of theline segment OP on the reference plane.

The sign of the azimuth is determined by choosing what is apositive sense of turning about the zenith. This choice isarbitrary, and is part of the coordinate system's definition.

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The elevation angle is 90 degrees (/2 radians) minus theinclination angle.

If the inclination is zero or 180 degrees (radians), the azimuth

is arbitrary. If the radius is zero, both azimuth and inclinationare arbitrary.

In linear algebra, the vectorfrom the origin O to the pointPisoften called theposition vectorofP.

Several different conventions exist for

representing the three coordinates, and for

the order in which they should be written.

The use of (r, , ) to denote, respectively,

radial distance, inclination (or elevation), and

azimuth, is common practice in physics, and

is specified by ISO standard 33-11.

However, some authors (including mathematicians) use forinclination (or elevation) and for azimuth, which "provides a

logical extension of the usual polar coordinates notation. Someauthors may also list the azimuth before the inclination (orelevation), and/or use instead ofrfor radial distance. Somecombinations of these choices result in a left-handed coordinatesystem. The standard convention (r, , ) conflicts with theusual notation for the two-dimensionalpolar coordinates, whereis often used for the azimuth. It may also conflict with thenotation used for three-dimensional cylindrical coordinates. [1]

The angles are typically measured in degrees () orradians (rad),where 360 = 2 rad. Degrees are most common in geography,astronomy, and engineering, whereas radians are commonlyused in mathematics and theoretical physics. The unit for radialdistance is usually determined by the context.

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When the system is used for physical three-space, it iscustomary to use positive sign for azimuth angles that aremeasured in the counter-clockwise sense from the referencedirection on the reference plane, as seen from the zenith side ofthe plane. This convention is used, in particular, forgeographical coordinates, where the "zenith" direction is northand positive azimuth (longitude) angles are measured eastwardsfrom someprime meridian.

Unique coordinates:

Any spherical coordinate triplet (r, , ) specifies a single point

of three-dimensional space. On the other hand, every point hasinfinitely many equivalent spherical coordinates. One can add orsubtract any number of full turns to either angular measurewithout changing the angles themselves, and therefore withoutchanging the point. It is also convenient, in many contexts, toallow negative radial distances, with the convention that (r, ,) is equivalent to (r, +180, ) for any r, , and . Moreover,(r, , ) is equivalent to(r, , +180).

If it is necessary to define a unique set of spherical coordinatesfor each point, one may restrict their ranges. A common choiceis:

r 0

0 180 ( rad)

0 < 360 (2 rad)However, the azimuth is often restricted to the interval (180,+180], or (, +] in radians, instead of [0, 360). This is thestandard convention for geographic longitude.

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The range [0, 180] for inclination is equivalent to [90, +90]for elevation (latitude).

Even with these restrictions, ifis zero or 180 (elevation is 90

or -90) then the azimuth angle is arbitrary; and ifris zero, bothazimuth and inclination/elevation are arbitrary. To make thecoordinates unique one can use the convention that in thesecases the arbitrary coordinates are zero.

Plotting:

To plot a point from its spherical coordinates (r, , ), where isinclination, move runits from the origin in the zenith direction,rotate by about the origin towards the azimuth referencedirection, and rotate by about the zenith in the properdirection.

Applications:The geographic coordinate system uses the azimuth andelevation of the spherical coordinate system to express locations

on Earth, calling them respectively longitude and latitude. Justas the two-dimensional Cartesian coordinate system is useful onthe plane, a two-dimensional spherical coordinate system isuseful on the surface of a sphere. In this system, the sphere istaken as a unit sphere, so the radius is unity and can generally beignored. This simplification can also be very useful whendealing with objects such as rotational matrices.

Spherical coordinates are useful in analyzing systems that have

some degree of symmetry about a point, such as volumeintegrals inside a sphere, the potential energy field surrounding aconcentrated mass or charge, or global weather simulation in aplanet's atmosphere. A sphere that has the Cartesian equationx2

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+y2 +z2 = c2 has the simple equation r= c in sphericalcoordinates.

Two importantpartial differential equations that arise in many

physical problems, Laplace's equation and the Helmholtzequation, allow a separation of variables in sphericalcoordinates. The angular portions of the solutions to suchequations take the form ofspherical harmonics.

Another application is ergonomic design, where ris the armlength of a stationary person and the angles describe thedirection of the arm as it reaches out.

The output pattern of an industrial loudspeaker

shown using spherical polar plots taken at six

frequencies

Three dimensional modeling ofloudspeakeroutput patterns canbe used to predict their performance. A number of polar plotsare required, taken at a wide selection of frequencies, as the

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pattern changes greatly with frequency. Polar plots help to showthat many loudspeakers tend toward omnidirectionality at lowerfrequencies.

The spherical coordinate system is also commonly used in 3Dgame development to rotate the camera around the player'sposition.

Coordinate system conversionsAs the spherical coordinate system is only one of many three-dimensional coordinate systems, there exist equations forconverting coordinates between the spherical coordinate system

and others.

Cartesian coordinates

The spherical coordinates (r, , ) of a point can be obtainedfrom its Cartesian coordinates (x,y,z) by the formulae

The inverse tangent denoted in = tan-1(y/x) must be suitablydefined, taking into account the correct quadrant of (x,y). See

article atan2.These formulas assume that the two systems have the sameorigin, that the spherical reference plane is the Cartesianxyplane, that is inclination from thezdirection, and that theazimuth angles are measured from the Cartesianx axis (so that

44.
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they axis has =+90). Ifmeasures elevation from thereference plane instead of inclination from the zenith the arccosabove becomes an arcsin, and the cos and sin below becomeswitched.

Conversely, the Cartesian coordinates may be retrieved from thespherical coordinates (r, , ), where r [0, ), [0, ], [0, 2 ), by:

ss raza.docx2

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