Spherical Coordinates

9 Key excerpts on "Spherical Coordinates"

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  Calculus in 3D

  Geometry, Vectors, and Multivariate Calculus

  • Zbigniew Nitecki(Author)
  • 2018(Publication Date)
  • American Mathematical Society

    (Publisher)

  Spherical Coordinates. Another coordinate system in space, which is particularly useful in problems involving rotations around various axes through the origin (for ex-ample, astronomical observations, where the origin is at the center of the earth) is the system of Spherical Coordinates . Here, a point ? is located relative to the origin ? 1.1. Locating Points in Space 5 by measuring the distance 𝜌 = |??| of ? from the origin together with two angles: the angle 𝜃 between the ?? -plane and the plane containing the ? -axis and the line ?? , and the angle 𝜙 between the (positive) ? -axis and the line ?? (Figure 1.4). Of course, the ∙ Figure 1.4. Spherical Coordinates spherical coordinate 𝜃 of ? is identical to the cylindrical coordinate 𝜃 , and we use the same letter to indicate this identity. 3 While 𝜃 is sometimes allowed to take on all real values, it is customary in Spherical Coordinates to restrict 𝜙 to 0 ≤ 𝜙 ≤ 𝜋 . The relation between the cylindrical coordinates (?, 𝜃, ?) and the Spherical Coordinates (𝜌, 𝜃, 𝜙) of a point ? is illustrated in Figure 1.5 (which is drawn in the vertical plane determined by 𝜃 ): ∙ Figure 1.5. Spherical vs. Cylindrical Coordinates ? = 𝜌 sin 𝜙, 𝜃 = 𝜃, ? = 𝜌 cos 𝜙. (1.6) To invert these relations, we note that, since 𝜌 ≥ 0 and 0 ≤ 𝜙 ≤ 𝜋 by convention, ? and ? completely determine 𝜌 and 𝜙 : 𝜌 = √ ? 2 + ? 2 , 𝜃 = 𝜃, 𝜙 = arccos ? 𝜌 . (1.7) The ambiguities in Spherical Coordinates are the same as those for cylindrical coordi-nates: the origin has 𝜌 = 0 and both 𝜃 and 𝜙 arbitrary; any other point on the ? -axis 3 Be warned that in some of the engineering and physics literature the names of the two spherical angles are reversed, leading to potential confusion when converting between spherical and cylindrical coordinates. 6 Chapter 1. Coordinates and Vectors ( 𝜙 = 0 or 𝜙 = 𝜋 ) has arbitrary 𝜃 , and for points off the ? -axis, 𝜃 can (in principle) be augmented by arbitrary even multiples of 𝜋 .

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  Coordinate Systems and Map Projections

  • D.H. Maling(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Reference has already been made to the generalised system of polar coordinates in three dimensions, of which the (φ, λ) system is a special case. Another system of spherical polar coordinates which are, again, suitably simplified for representation of the curved surface of a sphere are the bearing and distance coordinates, which have particular value in the construction of map projections. These are studied in Chapter 7. A third and quite diñ'erent system of location is by three-dimensional car-tesian coordinates which differ from plane rectangular coordinates by the addition of a third, Z, axis which is perpendicular to the other two and, for both sphere and spheroid, corresponds to the axis of rotation. This is described in Chapter 4. Angles on the sphere Having established the properties of geographical coordinates as the primary method of location, it is now desirable to introduce some additional concepts about the geometry of the sphere. A spherical angle is the inclination, at their point of intersection, of two arcs of great circles measured on the curved surface of the sphere. It is also equal to the plane angle formed between two tangents, drawn at the point of intersection, one to each great circle. Thus in Fig. 3.03, the spherical angle between the two great circles PA and PD is the angle DPA, which is equal to the plane angle KPJ, For the purpose of the present study spherical angles are encountered in two forms. The first is to permit an alternative definition of longitude. That given on page 51 describing longitude as an angle measured at the centre of the sphere in the plane of the equator is so worded to emphasise that geographical coordinates are a form of polar coordinates and the vectorial angles should therefore be measured at the origin of the system. However, we can see from Figs 3.01, 3.02 and 3.03 that the angle λ can be measured anywhere on the earth's axis of rotation, provided that this angle is measured in a plane parallel to the equator.

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  Introduction to Precalculus Analysis, An

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of circular and orbital motion. Position and navigation Polar coordinates are used often in navigation, as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft ________________________ WORLD TECHNOLOGIES ________________________ use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively. Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic control). Modeling Systems displaying radial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the groundwater flow equation when applied to radially symmetric wells. Systems with a radial force are also good candidates for the use of the polar coordinate system.

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  Electric Field Analysis

  • Sivaji Chakravorti(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  Figure 3.5 .

  As shown in Figure 3.5 , r is the distance from the pole to the projection of the point P on the polar plane, that is, the x–y plane passing through the pole, θ is the azimuthal angle, that is, the angle from the polar axis spinning around the z -axis in counter-clockwise direction, and z is the vertical height from the polar plane. The ranges of the values of the three coordinates are 0 ≤ r < ∞, 0 ≤ θ ≤ 2π and −∞ < z < ∞.

  FIGURE 3.5 Depiction of cylindrical coordinates of a point.

  In the cylindrical coordinate system, the three constant coordinate surfaces are defined by Equation 3.9

  f 1

  = r ,

  f 2

  = θ ,

  f 3

  = z

  ⁢ ( 3.9 )

  Figure 3.6 shows the three constant coordinate surfaces in the cylindrical coordinate system. Out of these three surfaces, the first and the third surfaces, namely, f 1 = r and f 3 = z , are constant distance surfaces, whereas the second one, that is, f 2 = θ, is a constant angle surface. As shown in Figure 3.6 , the surfaces θ = constant and z = constant are planes, whereas the surface r = constant is a cylindrical surface.

  In this coordinate system, two unit vectors are defined on the x–y plane. The unit vector ûr points in the direction of increasing r , that is, radially outwards from the z -axis and the unit vector û θ points in the direction of increasing θ, that is, it points in the direction of the tangent to the circle of radius r in the counter-clockwise sense. The third unit vector ûz points in the direction of increasing z , that is, vertically upwards from the x–y plane. The unit vectors are shown in Figure 3.5 . The orthogonality of cylindrical coordinate system is defined by Equation 3.10

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  Spherical Geometry and Its Applications

  • Marshall Whittlesey(Author)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  π is a matter of convenience. So a point in space does not have a unique set of Spherical Coordinates. However, we have the following theorem.

  Theorem 8.1 Suppose that (ϕ, θ) are polar coordinates for a sphere of radius r. Suppose that x, y, and z axes are chosen for space in the following manner. Let the center of the sphere be the origin. Let the positive z axis point in the direction of the pole of the coordinate system. Let the positive x axis point in the direction of the point (

  π 2

  , 0) and let the positive y axis point in the direction of the point (

  π 2

  ,

  π 2

  ). Then a point A in space on the sphere of radius r with xyz coordinates (x, y, z) and polar coordinates (ϕ, θ) satisfies

  ( x , y , z ) = ( r sin ϕ cos θ , r sin ϕ sin θ , r cos ϕ ) .

  (2.3)

  Proof.

  Figure 2.13: Theorem 8.1.

  The discussion above shows how every (ϕ, θ) determines a point A on the sphere. Let X, Y, and Z be the projections of the point A to the corresponding axes. Then ΔAZO is a right triangle with a right angle at Z. By definition of arc measure, m∠AOZ = m

  A P

  ⌢

  , which is ϕ by definition. Since ΔAZO is a right triangle and the length of the segment

  O A

  ¯

  is r, the coordinate of Z is r cos ϕ and the length of

  A Z

  ¯

  is r sin ϕ. Thus z = r cos ϕ, as desired. Next, let W be the projection of A to the XY plane. Then the length of

  O W

  ¯

  is the same as the length of

  A Z

  ¯

  , which is r sin ϕ. The projections of W to the x and y axes are X and Y, respectively. Next, the absolute value of θ is the measure of planar angle ∠XOW. Then since W lies at radius r sin ϕ from O and

  O W

  ¯

  forms a signed angle of θ with the positive x axis, its xy coordinates must be x = r sin ϕ cos θ and y = r sin ϕ sin θ.

  Exercises §8

  1.  Extend §6, Exercise 1c as follows: if a point X has latitude-longitude coordinates (a, b) with respect to some system of cooordinates on the sphere, then the distance c from X to the point with coordinates (0, 0) satisfies cos(c) = cos(a) cos(b

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  Coordinates

  • Wolfram Neutsch(Author)
  • 2011(Publication Date)
  • De Gruyter

    (Publisher)

  1068 29. Gauß-Krllger Coordinates 29. Gaue-Krüger Coordinates 29.1. Soldner's Parameterization of the Spheroid Many geometric problems like surveying or the precise cartography of (parts o f ) the terrestrial surface depend strongly on the introduction of suitable coordinates. For almost all purposes it will be sufficient to approximate the Earth's figure by a spheroid (= ellipsoid with rotational symmetry). Several different systems have been described in the relevant literature, but we cannot do more than select two of the most important and at the same time mathematically most interesting approaches. Before we can come to a detailed description, we have to introduce a few (often used) technical terms and explain the elementary geometric connections between them. The object we are going to investigate is a flattened rotationally symme-tric ellipsoid § which we shall in what follows call the Earth, although the theory evidently can be applied without change to all oblate spheroids. Only the special formulas to be developed later, which contain the numeri-cal values of the Earth's dimensions, are exceptions to this rule. The halved principal axes of $ will be denoted as a and b, silently assum-ing a a b. The Earth is symmetric around its figure axis which intersects the terres-trial surface in two antipodal locations (North and South pole). We intro-duce positively oriented Cartesian coordinates x,y,z, whose origin 0 lies at the centre of the Earth. The z-axis points to the North pole. The orientation of the x- and y-axes is irrelevant in the present situation as the configuration is rotationally symmetric. In geography, one fixes a base point Β on the surf ace of the Earth (the observatory of Greenwich or, more precisely: a certain position on its grounds) where one sets y = 0 and χ positive. The equations ζ = const, define a family of parallel planes which form cir-cular lines of intersection with S, the circles of latitude.

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  Classical Mechanics

  • A. Douglas Davis(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  6.3 Cylindrica l Pola r Coordinate s We ca n easil y exten d ou r notatio n t o thre e dimension s b y addin g a z-componen t an d creatin g cylindrical pola r coordinate s a s sketche d in Figur e 6.3.1 . æ Figur e 6.3. 1 Cylindrica l pola r coordinates . CHAPTE R 6 / COORDINAT E SYSTEM S 6.4 Spherica l Pola r Coordinate s On e o f th e mos t usefu l coordinat e system s is spherical polar coordinates (ofte n simpl y calle d Spherical Coordinates), a s show n in Figur e 6.4.1 . Suc h a syste m is usefu l wheneve r ther e is spherica l symmetry—fo r example , describin g th e force s bindin g a n electro n t o it s nucleus , givin g th e locatio n o f th e spac e shuttl e in orbi t aroun d Earth , o r chartin g th e cours e o f th e U.S.S. Enterprise throug h th e galaxy . In Figur e 6.4.1 , poin t Ñ is locate d b y givin g it s radial distance fro m th e origi n r, an d azimuthal angle ö locatin g a plan e whos e angl e of rotatio n is measure d fro m th e x-axis , an d a polar angle 0 , givin g it s angula r locatio n measure d down fro m th e z-axis . Not e tha t 0 ca n var y fro m 0 t o ð , an d ö fro m 0 t o 2ð . Thi s is sketche d in Figur e 6.4.1 . We ca n the n defin e mutuall y perpendicula r uni t vector s f, È, an d a s show n in Figur e 6.4. 1 b y givin g th e direction s tha t Ñ move s a s r, 0, an d ö increase , respectively . Thes e uni t vector s ar e sketche d agai n in Figur e 6.4.2 . Not e tha t f an d è bot h lie in a plan e containin g th e z-axi s an d rotate d a n amoun t ö fro m th e x-axis . Th e uni t vecto r cji lie s in th e xy-plane . Figur e 6.4. 3 Th e locatio n o f a poin t Ñ is give n b y it s radia l distanc e fro m th e z-axis , p , its angula r rotatio n fro m th e x-axis , ö (jus t a s in th e previou s cas e wit h r an d 0), and it s elevatio n abov e th e xy plane , z.

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  Fundamental Maths

  For Engineering and Science

  • Mark Breach(Author)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  20 Coordinate systems What this chapter is about In this chapter you will investigate spatial relationships in a two-dimensional plane. By the end of this chapter you will be able to calculate distance and orientation in a two-dimensional plane, the areas of straight-sided figures, find the intersection of lines, and calculate parameters and formulae of circles. You will be able to change from Cartesian to polar coordinates and vice versa and to apply Pythagoras’ theorem in three dimensions. 20-01 Cartesian coordinate system applications The Cartesian coordinate system was introduced in Chapter 6 and it is now appropriate to look at some of its implications. The first is that not everyone uses it in the same way as mathematicians. In the Cartesian coordinate system the primary reference direction is the x -axis and orientation is measured anticlockwise from it. Geographers, and particularly land and engineering surveyors, use a system in which the primary direction is north and orientation, such as a compass bearing, is measured clockwise from it. Although the fundamental mathematical relationships are not affected by this difference, their applications may be. 20-02 Distance and orientation between two points Two points in a coordinate frame may be joined by a straight line. The length of that line will be the distance between the two points. The orientation of the line is measured with 209 Figure 20.1 north east α P ( E P , N P ) x -axis y -axis α P ( x P , y P ) respect to the x -direction within the Cartesian coordinate system and with respect to the north direction in the geographical frame. An application of Pythagoras’ theorem can be used to find the length of the line, and the tangent of its orientation relates the sides of the right angled triangles in Figure 20.2.

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  Solid Geometry with MATLAB Programming

  • Nita H. Shah, Falguni S. Acharya(Authors)
  • 2023(Publication Date)
  • River Publishers

    (Publisher)

  3 Sphere 3.1 Definition The sphere is a locus of a point that moves so that its distance from a fixed point always remains constant. The fixed point is called the center of the sphere and the constant distance is called the radius of the sphere. 3.2 Equation of Sphere in Vector Form Consider a sphere S with center C and radius ‘ a ’. Let P be any point on the sphere. Let c → and r → be the position vectors of C and P respectively with respect to O; the origin of reference. ∴ C P → = O P → − O C → ∴ C P → = r → − c → But. radius C P = | C P → | = a. (3.1) ∴ | r → − c → | = a (3.2) ∴ | r → − c → | 2 = a 2 ∴ (r → − c →). (r → − c →) = a 2 ∴ r →. r → − r →. c → − c →. r → + c →. c → = a 2 ∴. | r → | 2 − 2 r →. c → + | c → | 2 = a 2 ∴ | r → | 2 − 2 r →. c → + d = 0 ∴ | r → | 2 − 2 r →. c → = | c → | 2 = a 2 where d = | c → | 2 − a 2 is the equation of sphere in vector form. Case 1): If O lies on the sphere, then OC = OP; i.e., c. = a ⇒ d = 0 The Equation of sphere is | r → | 2 − 2 r →. c → = 0 Case 2): If the center C is at the origin O, then c = 0 and d = − a 2. ∴ Equation of sphere is | r → | 2 − 2 r →. c → = 0 | r → | 2 = a 2 | r → | = a. Equation of sphere in the cartesian form: Consider a sphere S with center C (α, β, γ), and radius ‘ a ’. Let P (x, y, z) be any point on the sphere. Let x → and c → be the position vector of P and C respectively with respect to O; the origin of reference. ∴ r → = x i → + y j → + z k → c → = α i → + β j → + γ k →. The equation in vector form. is | r → − c → | = a. ∴ | (x − α) i → + (y − β) j → + (x − γ) k → | = a ∴ (x − α) 2 + (y − β) 2 + (z − γ) 2 = a 2 which is the equation of sphere S in cartesian form with center C (α, β, γ), and radius ‘ a ’. Remark: When center C is origin. i.e., α = β = γ = 0 then the equation of the sphere is x 2 + y 2 + z 2 = a 2. 3.3 General Equation of the Sphere The general equation of the sphere

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