Cartesian to Polar Coordinates

6 Key excerpts on "Cartesian to Polar Coordinates"

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  eBook - PDF

  Physics for Scientists and Engineers with Modern Physics

  • Raymond Serway, John Jewett(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  In two dimensions, this description is accomplished with the use of the Cartesian coor- dinate system, in which perpendicular axes intersect at a point defined as the origin O (Fig. 3.1). Cartesian coordinates of a point in space, representing the x and y values of the point, and expressed as ( x , y), are also called rectangular coordinates. Sometimes it is more convenient to represent a point in a plane by its plane polar coordinates ( r , u) as shown in Figure 3.2a. In this polar coordinate system, r is the dis- tance from the origin to the point having Cartesian coordinates ( x, y) and u is the angle between a fixed axis and a line drawn from the origin to the point. The fixed axis is often the positive x axis, and u is usually measured counterclockwise from it. From the right triangle in Figure 3.2b, we find that sin u 5 y/ r and that cos u 5 x/ r. (A review of trigonometric functions is given in Appendix B.4.) Therefore, starting with the plane polar coordinates of any point, we can obtain the Cartesian coordi- nates by using the equations x 5 r cos u (3.1) y 5 r sin u (3.2) Conversely, if we know the Cartesian coordinates, the definitions of trigonometry tell us that the polar coordinates are given by tan u 5 y x (3.3) r 5 Ïx 2 1 y 2 (3.4) Equation 3.4 is the familiar Pythagorean theorem. These four expressions relating the coordinates ( x, y) to the coordinates ( r , u) apply only when u is defined as shown in Figure 3.2a—in other words, when pos- itive u is an angle measured counterclockwise from the positive x axis. (Some sci- entific calculators perform conversions between Cartesian and polar coordinates based on these standard conventions.) If the reference axis for the polar angle u is chosen to be one other than the positive x axis or if the sense of increasing u is cho- sen differently, the expressions relating the two sets of coordinates will be different from those above.

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  Fundamental Maths

  For Engineering and Science

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

    (Publisher)

  The Cartesian x -and y -coordinates of a point are measurements from the y -and x -axis respectively. Polar coordinates of a point are the distance of the point from the origin and the orientation of the line joining the origin and the point, with the x -axis, ð r , Þ . There is a direct connection between Cartesian coordinates and polar coordinates, as can be seen from Figure 20.20. The x -coordinate is r cos and the y -coordinate is r sin . Therefore ð x , y Þ¼ð r cos , r sin Þ and so ð r , Þ¼ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y 2 q , tan 1 y x Þ . Note that specifying values for x and y defines only one point, as does specifying r and . Also, a given point only has one set of values for x and y . However the same point can have four different sets of values in polar coordinates. For example if the Cartesian coordinates of a point are ð ffiffiffi 3 p , 1 Þ then the polar coordinates would be ð 2, 30 8 Þ . Coordinate systems 227 x -axis y -axis θ r r cos θ r sin θ ( r , θ ) Figure 20.20 However the polar coordinates could also be expressed as ð 2, 330 8 Þ because 30 8 clockwise from the x -axis brings you to the same orientation as 330 8 anticlockwise from the x -axis. Also, a rotation of 180 8 gives orientation in the opposite direction, as does multiplying the value of r by 1. So, if orientation to the opposite direction is applied twice, that brings us back to where we started. Therefore, since 30 8 þ 180 8 ¼ 210 8 and 330 8 þ 180 8 ¼ 150 8 , then: ð 2, 30 8 Þ is the same point as ð 2, 210 8 Þ and ð 2, 330 8 Þ is the same point as ð 2, 150 8 Þ . Therefore all four sets of polar coordinates describe the same point. Your calculator should have a Polar-to-Rectangular function and vice versa. Different calculators will work differently so read your manual to find how to convert polar coordinates to rectangular (Cartesian) coordinates and vice versa. CAUTION Make sure your calculator is set to the correct angular mode – degrees, radians or grads – before starting your calculations.

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  No longer available |Learn more

  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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  eBook - ePub

  3D Math Primer for Graphics and Game Development

  • Fletcher Dunn, Ian Parberry(Authors)
  • 2011(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  Chapter 7

  Polar Coordinate Systems

  First of all, we must note that the universe is spherical. — Nicolaus Copernicus (1473–1543)

  The Cartesian coordinate system isn’t the only system for mapping out space and defining locations precisely. An alternative to the Cartesian system is the polar coordinate system, which is the subject of this chapter. If you’re not very familiar with polar coordinates, it might seem like an esoteric or advanced topic (especially because of the trig), and you might be tempted to gloss over. Please don’t make this mistake. There are many very practical problems in areas such as AI and camera control whose solutions (and inherent difficulties!) can be readily understood in the framework of polar coordinates.

  This chapter is organized into the following sections:

  • Section 7.1 describes 2D polar coordinates.
  • Section 7.2 gives some examples where polar coordinates are preferable to Cartesian coordinates.
  • Section 7.3 shows how polar space works in 3D and introduces cylindrical and spherical coordinates.
  • Finally, Section 7.4 makes it clear that polar space can be used to describe vectors as well as positions.

  7.1   2D Polar Space

  This section introduces the basic idea behind polar coordinates, using two dimensions to get us warmed up. Section 7.1.1 shows how to use polar coordinates to describe position. Section 7.1.2 discusses aliasing of polar coordinates. Section 7.1.3 shows how to convert between polar and Cartesian coordinates in 2D.

  7.1.1   Locating Points by Using 2D Polar Coordinates

  Remember that a 2D Cartesian coordinate space has an origin, which establishes the position of the coordinate space, and two axes that pass through the origin, which establish the orientation of the space. A 2D polar coordinate space also has an origin (known as the pole ), which has the same basic purpose—it defines the “center” of the coordinate space. A polar coordinate space has only one axis, however, sometimes called the polar axis , which is usually depicted as a ray from the origin. It is customary in math literature for the polar axis to point to the right in diagrams, and thus it corresponds to the +x axis in a Cartesian system, as shown in Figure 7.1

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  eBook - PDF

  Engineering Mechanics

  Dynamics

  • Benson H. Tongue, Daniel T. Kawano(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Assume that the wheel is attached to a car that is traveling to the 52 CH 2 MOTION OF TRANSLATING BODIES 2.3 POLAR AND CYLINDRICAL COORDINATES Learning Objective: Describe a particle’s motion in polar and cylindrical coordinates. The next set of coordinates we will examine are called polar coordinates because they are radially based, like a map of the world would be if it were centered on the North or South Pole, showing lines of latitude and longitude. A typical polar plot is shown in Figure 2.3.1. Rather than determining the position of a body P by going forward some amount and then right some amount, as we did in Figure 2.2.2, the polar approach determines P’s position by the distance (r) it is from the origin and the angle  that P’s position vector makes with the positive horizontal axis. The radial unit vector e r always points from the origin to the body. The total position vector is given by r P  P O   = r e r (2.41) which says that to get to P, you must go a distance r in the e r direction. Unlike the Cartesian case, in which we needed two terms to define position, in the polar case we need just one. You would use this viewpoint, for example, if you were an airport traffic controller. Because you were using radar, the information you would have about an airplane would be its straight-line distance from you (r) and its vertical inclination (). As just mentioned, e r is not oriented permanently vertical or per- manently horizontal as a Cartesian unit vector would be. Rather, it is determined by where P is with respect to the origin O. The unit vector e r always points from the origin to P, and therefore if P moves around O, the orientation of e r will change. This is going to make life difficult when we try to find velocities and accelerations. But before we do that, let’s introduce e  , the other unit vector we’ll be using (and which we’ll call the angular unit vector). This vector is found by imagining that the length r is fixed and then increasing .

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  eBook - PDF

  Calculus in 3D

  Geometry, Vectors, and Multivariate Calculus

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

    (Publisher)

  Cylindrical Coordinates a (vertical) plane, which can be regarded as the ?? -plane rotated so that the ? -axis moves 𝜃 radians counterclockwise (in the horizontal plane); we take as our coordinates the angle 𝜃 together with the coordinates of ? in this plane, which equal the distance ? of the point from the ? -axis and its (signed) distance ? from the ?? -plane. We can think of this as a hybrid: combine the polar coordinates (?, 𝜃) of the projection ? ?? with the vertical rectangular coordinate ? of ? to obtain the cylindrical coordinates (?, 𝜃, ?) of ? . Even though in principle ? could be taken as negative, in this system it is customary to confine ourselves to ? ≥ 0 . The relation between the cylindrical coordinates (?, 𝜃, ?) and the rectangular coordinates (?, ?, ?) of a point ? is essentially given by Equation (1.3): ? = ? cos 𝜃, ? = ? sin 𝜃, ? = ?. (1.4) We have included the last relation to stress the fact that this coordinate is the same in both systems. The inverse relations are given by ? 2 = ? 2 + ? 2 , cos 𝜃 = ? ? , sin 𝜃 = ? ? (1.5) and, for cylindrical coordinates, the trivial relation ? = ? . The name “cylindrical coordinates” comes from the geometric fact that the locus of the equation ? = ? (which in polar coordinates gives a circle of radius ? about the origin) gives a vertical cylinder whose axis of symmetry is the ? -axis, with radius ? . Cylindrical coordinates carry the ambiguities of polar coordinates: a point on the ? -axis has ? = 0 and 𝜃 arbitrary, while a point off the ? -axis has 𝜃 determined up to adding even multiples of 𝜋 (since ? is taken to be positive). 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 ?

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