Polar Coordinates

8 Key excerpts on "Polar Coordinates"

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  eBook - ePub

  Geographical Information Systems and Computer Cartography

  • Chris B. Jones(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  Figure 4.1c ). This type of coordinate system is of use in cartography when plotting certain types of projection in which position can conveniently be defined relative to a single, central point, rather than a pair of axes. It is also relevant in general when it is appropriate to retain a sense of relative direction.

  In mathematical usage, polar coordinate angles are conventionally measured anticlockwise from a horizontal axis (Figure 4.1c ). In surveying and cartography, the angles are usually measured clockwise from a vertical axis, the angular units being either degrees (0-360) or grads (0-400), where 90° and 100 grads are equivalent to π/2 radians.

  For a given origin, a cartographic polar coordinate of r,θ can be expressed in rectangular coordinates using the trigonometric formulae

  x = ? r ⁢ sin ⁡ θ , ? y = r ⁢ cos ⁡ θ ⁢ ?

  Conversely, the Polar Coordinates can be expressed in terms of the rectangular coordinates by finding θ from the relationship

  tan ⁡ θ = x / y

  so that θ can be found using the appropriate inverse tan function on a computer. Knowing θ,

  r ⁢ ? = ? y / cos ⁡ θ ⁢ ?? ? ? ?

  o r

  ⁢ ? ? ? r = x / sin ⁡ θ ⁢ ?

  We may also note that

  r 2

  ⁢ ? ? = ? ?

  x 2

  ⁢ ? ? + ? ?

  y 2

  ⁢

  Spherical coordinates

  We have seen that though planar coordinate systems are essential for constructing maps on flat surfaces, they cannot be used for representing extensive regions of the earth without introducing serious distortion in measurements such as distance and area. When high accuracy is not required these problems of distortion can be avoided by the use of a spherical coordinate system. This provides a single, consistent and relatively undistorted reference frame for recording positions and making measurements of the earth's surface. The coordinates can then be projected to a suitable planar coordinate system when a small-scale map of a particular region or aspect of the earth is required.

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  eBook - ePub

  Fundamentals of Modern Mathematics

  A Practical Review

  • David B. MacNeil(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  x.

  76. Spherical Polar Coordinates.In place of the perpendicular axes of the Cartesian system, spherical Polar Coordinates are based upon two angular measurements and one distance measurement. Whereas the three-dimensional Cartesian system is based on the measurements x, y, and z, which give the perpendicular distances from the YZ-, XZ-, and XY-planes respectively, the spherical polar system is based on r, θ and ϕ. As is illustrated in Figure 7-2 , r is the distance from the origin to the point P. Similarly the angle θ is the angle between OX and OP′, where P′ is the projection* of P on the XY-plane and ϕ is the angle between OZ and OP.

  FIG . 7-2.

  It is evident that the variables in the spherical polar coordinate system are subject to the following restrictions.

  Since r, the radius vector, measures the absolute distance between the origin and a point, it cannot assume a value less than zero. Similarly, θ may be thought of, in geographical terms, as a measure of longitude and as such may assume any value in the interval from 0° to 360°. Finally, ϕ may be considered as the colatitude of the given point. Thus, provided that a point is not on the OZ line, it can be represented by a unique set of spherical Polar Coordinates just as it can be represented by a unique set of Cartesian coordinates.

  Noting that r sin ϕ = OP′, we obtain the following relationship between the spherical polar and Cartesian systems.

  If the r is held constant in the spherical polar coordinate system and θ and ϕ are allowed to range over all possible values, the resulting locus of points is a sphere of radius r. Similarly, if θ is held constant and r and ϕ are allowed to assume all possible values, the resulting locus of points is a half plane from the Z-axis. And if ϕ is held constant while r and θ are allowed to assume all possible values, the resulting locus of points is a right circular cone with its apex at the origin and axis along the Z

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  eBook - ePub

  Mathematics for Scientific and Technical Students

  • H. Davies, H.G. Davies, G.A. Hicks(Authors)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  coordinates, are required to locate a point in a plane. Two systems are used:

  (a) Cartesian coordinates (x , y )

  This is the most commonly used system. Two perpendicular datum lines are used, the horizontal line is called the x -axis, the vertical line is called the y -axis, as shown in Fig. 9.1 . The point of intersection of the two axes is called the origin O. Any point P is located by its perpendicular distance from the two axes.

  Fig. 9.1

  (b) Polar Coordinates (r , θ )

  In this system a point P is located at a distance r along a line OP from a fixed point O , called the pole, as shown in Fig 9.2 . θ is the angle that the line OP makes with the reference + x -axis. It is important to remember that θ is positive when OP rotates anticlockwise.

  Fig. 9.2

  Fig. 9.3

  Fig. 9.3 shows the relationship between both systems. Pythagoras’s theorem and trigonometry can be used to change from one system to another.

  (c) Conversion from Cartesian to Polar Coordinates

  From Pythagoras

  and

  The smallest value of θ is usually quoted and can be positive or negative. The value of θ obtained must be checked so that it places P in the correct quadrant. This can be done by using a sketch to check the results, as shown in Example 9.1 .

  (d) Conversion from polar to Cartesian coordinates

  From Fig. 9.3 , using trigonometrical ratios in a right-angled triangle:

  Example 9.1 The Cartesian coordinates of a point P are (–4, –6). Convert these to Polar Coordinates.

  The point P with these two coordinates is shown in Fig. 9.4 .

  Fig. 9.4

  The value of 56.3° is not in agreement with the position of P in the third quadrant. The smallest magnitude of θ is 180 – 56.3 = 123.7°, and this is seen to be negative because it is in a clockwise direction.

  The Polar Coordinates are (7.2, –124°).

  Example 9.2 Convert the Polar Coordinates (18, 125°) to Cartesian coordinates.

  Fig. 9.5

  From Fig. 9.5 ,

  The Cartesian coordinates are (–10.3, 14.7).

  9.3 The distance between two points

  Fig. 9.6 shows two points P (x 1 , y 1 ) and Q (x 2 , y 2 )

  Horizontal distance

  Fig. 9.6

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  eBook - ePub

  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  section 2.6 ).

  The mathematical methods in a two-dimensional coordinate system that we introduce in this chapter can be generalized to a three-dimensional coordinate system (although we will not do it in this book). A three-dimensional coordinate system can be used as a mathematical representation of the three-dimensional physical space in which we humans move around; and it is in this coordinate representation of three-dimensional space that mathematicians and scientists can carry out advanced mathematical simulations of dynamic processes like ocean currents, planetary weather patterns, motions of projectiles, and exploding stars.

  2.2WORKING IN A COORDINATE SYSTEM

   

  Any given point in the Cartesian plane can be labeled by means of two numbers called coordinates. The first coordinate is called the x-coordinate, and the second is called the y-coordinate. The x- and y-coordinates together form a coordinate pair. The x-coordinate can be found by following a vertical line from the given point to a point on the x-axis and reading its position on the number line. Similarly, the y-coordinate can be found by following a horizontal line from the given point to a point on the y-axis and reading its position on the number line. Points in the Cartesian plane are usually labeled using uppercase letters with the coordinate values following in parentheses, as shown in figure 2.2 . The axes separate the Cartesian plane into four quadrants (first, second, third, and fourth quadrants) that can be labeled as I, II, III, and IV, respectively, in a counterclockwise order, starting with the upper right quadrant.

  FIGURE 2.2. Points in the Cartesian plane.

  2.3LINEAR EQUATIONS AND STRAIGHT LINES

   

  Much of mathematics involves the study of related quantities in the form of equations. For this reason, we begin with a study of linear equations and the graphs (infinite straight lines) of linear equations in the Cartesian plane because these quantify the simplest kind of relationship (linear) that related quantities can have. We will refer to an “infinite straight line” (in the Cartesian plane) as a “line.”

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  eBook - ePub

  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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  eBook - ePub

  Engineering Mathematics

  A Programmed Approach, 3th Edition

  • C W. Evans(Author)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  Fig. 3.1 ). The quadrants so formed are labelled anticlockwise as the first quadrant, second quadrant, third quadrant and fourth quadrant respectively.

  Fig. 3.1 The cartesian system.

  Given any point P, the absolute values of x and y are then obtained from the shortest distance of P to the y-axis and the x-axis respectively. The following conventions then hold:

  First quadrant x ⩾ 0, y ⩾ 0

  Second quadrant x ⩽ 0, y ⩾ 0

  Third quadrant x ⩽ 0, y ⩽ 0

  Fourth quadrant x ⩾ 0, y ⩽ 0

  In this way, given any point in the plane we obtain a unique ordered pair (x, y)of real numbers. Conversely, given any ordered pair (x, y) of real numbers we obtain a unique point in the plane. We therefore identify the point P with the ordered pair (x, y) and refer to the point (x, y).

  If P is the point (x, y), x and y are known as the cartesian coordinates of the point P; x is called the abscissa and y is called the ordinate.

  This simple idea was initially due to the famous French philosopher Descartes and enabled algebra and geometry, two hitherto separate branches of mathematics, to be united. It is difficult to overestimate the benefits of this unification for science and technology, but Descartes threw it out almost as an afterthought to his philosophical treatise. The name ‘cartesian system’ comes from the latinized form of Descartes.

  The cartesian system is not the only system which can be used to represent points in the plane. Another is the polar coordinate system.

  In the polar coordinate system there is a fixed point O, called the origin, and a fixed line emanating from O called the initial line OX (Fig. 3.2 ). It is convenient to identify the initial line with the positive x-axis, although this identification is by no means essential. A point P is then determined by r, its distance from O, and by θ, the angle XOP measured anticlockwise. In this way, given any point in the plane we obtain an ordered pair of real numbers (r, θ) where r

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  eBook - ePub

  A First Course in Geometry

  • Edward T Walsh(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  Similarly, we can characterize the points Q, R, and S by writing 2(4, − 3), R (−5, 1) and S (−2, 0). It should be clear that the order in which we write the coordinates of a point is important. We will always write the x -coordinate first and the y -coordinate second. Thus we call these number pairs, ordered pairs. Given a point in the plane of and, the previous paragraph outlines the process for finding an ordered pair of real numbers associated with it. Conversely, if we are given a pair of coordinates, (−1, −2), for example, we can locate the point corresponding to that pair of numbers by drawing perpendiculars to the x - and y -axes at the points where x = − 1 and y = − 2 respectively. The point of intersection of these two perpendiculars is the point we are seeking. To draw a set of coordinate axes and to draw a set of points which are located in the way that we have been discussing is a task often performed in various branches of applied mathematics. To draw such a set of points is to plot that set of points. The previous paragraphs informally establish the existence of a one-to-one correspondence between the points of a plane and the set of all ordered pairs of real numbers. We call such a correspondence a Cartesian coordinate system (from Cartesius, the Latin form of Descartes’ name). Since it is often awkward to refer to “the point with coordinates (a, b),” we will often find it convenient to adopt the more widespread though inexact usage “the point (a, b).” The coordinate axes separate the plane into four distinct regions known as quadrants. The quadrants are named using Roman numerals I–IV, beginning with the region in which both coordinates are positive and then moving around the origin in a counterclockwise direction. Thus the points (2, 1), (−1, −2), (π, − π), and (−, 7) are in quadrants I, III, IV, and II, respectively. Chapter 2 provided a method for computing the distance between two points on a line

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  eBook - ePub

  Matrix Vector Analysis

  • Richard L. Eisenman(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  chapter IV      COORDINATE SYSTEMS

  Mature use of algebraic methods in vector analysis requires you to select appropriate algebraic tools and then to use these tools to express and develop physical and geometric ideas through algebraic computations. Analytic computations are the computing machine for physical and geometric vector ideas, and geometric and physical ideas are the visual aids for representing and inspiring algebraic ideas.

  4.1 | SELECTING COORDINATES

  In this section we shall see why we want a slave coordinate system, what a coordinate system is, when to change coordinate systems, and where a coordinate system shows up geometrically. Why do we want a coordinate system? Physical and geometric applications motivate the concept of a space of points. An algebraic representation of points through coordinates is desirable as a setting for expressing and developing ideas through computations. Conversely, an algebraic manipulation is clarified through its geometric and physical interpretations.

  What is a coordinate system?

  Definition: A (curvilinear) coordinate system is a set of scalar point functions whose values designate a point.

       For example, in the rectangular coordinate system x = x(P) is a scalar point function defined as the horizontal displacement of the point P from the origin. In the spherical coordinate system, r = r(P) is a scalar point function defined as the radial displacement of the point P from the origin.

  When should one change coordinate systems? The natural symmetry of a physical problem at hand usually suggests an appropriate coordinate system. For example, if elliptical waves generate from two foci, you naturally choose one coordinate which holds a constant value on any given ellipse. If temperature or electrostatic potential has constant values on certain cylinders, you naturally select cylindrical coordinates. In loran (long range navigation) there is a natural grid of hyperbolas. In a central force field you naturally select Polar Coordinates. From the algebraic viewpoint, there may be a natural change of variable which motivates the geometric coordinates; e.g., to integrate , you would naturally select the scalar point function r = (x2 + y2 + z2 )½ as the new (spherical) coordinate. How does one change coordinate systems? The rectangular coordinate system (x, y, z) is usually considered basic, and another coordinate system (u, v, w

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