Cylindrical Coordinates

3 Key excerpts on "Cylindrical Coordinates"

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  eBook - ePub

  Applied Metrology for Manufacturing Engineering

  • Ammar Grous(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  7.7.1.2.1. Cartesian coordinates

  In Cartesian coordinates, the axes X, Y, and Z define the position of a point in space as shown in Figure 7.21 .

  Figure 7.21.. Cartesian coordinates system

  7.7.1.2.2. Cylindrical Coordinates system

  In Cylindrical Coordinates, a point in space is defined by: – the projected distance from the origin;

  – the angle φ formed with the axis (first axis) XX′; and

  – the value of the axis ZZ′, as shown in Figure 7.22 .

  Figure 7.22.. Cylindrical Coordinates system

  7.7.1.2.3. Spherical coordinates system

  In spherical coordinates, the point in the space is defined by: – the distance from the origin, in the space;

  – the angle φ formed with the axis (first axis) XX′; and

  – the angle θ formed (according to the GEOPAK-Win) by the axis ZZ′ to the vector of the point (Figure 7.23 ).

  Figure 7.23.. Spherical coordinates system

  According to the GEOPAK-Win, one should pay attention to the fact that the angle θ can be interpreted with the classical mathematical sense from literature to literature. Sometimes, this means an elevation above XY relative to a plane.

  7.7.1.3. Measures via the dialog box of Cosmos

  We can measure an entity listed in the repertoire of the GEOPAK-Win [MIT 00]: point, line, circle, ellipse, plane, cone, sphere, cylinder, contour, calculation of angles, and distance. To measure a circle, e.g. simply click on the icon on the top left of the dialog box (Figure 7.24 ).

  Figure 7.24.. Dialog box to measure a circle (source: Mitutoyo Cosmos)

  The dialog box is identical for all elements being measured via the GEOPAK-WIN. The example of measurement of a circle is a clearly demonstrative. We notice that there are five distinctly distributed data on:

  1. the icons located on the first horizontal row are the elements of construction such as, measurement, connecting elements, calculations, etc.;

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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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  Geographical Information Systems and Computer Cartography

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

    (Publisher)

  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.

  Any point on the surface of a sphere of given radius can be uniquely defined by the angles which the radius passing through the point makes with two reference planes passing through the centre (Figure 4.5 ). This is equivalent to a 3D polar coordinate system in

  Figure 4.5 Cutaway view of the earth showing the angular relationship between a point P at latitude ϕ and longitude λ and the planes of the equator and the datum meridian. Note that ϕ is measured in the plane of the meridian which passes through P, and λ is measured in the plane of the equator.

  which the distance from the point to the origin is fixed and hence the locus of all possible positions describes a sphere. On the earth, when it is treated as a sphere, the reference planes of the geographical coordinate system are the horizontal one perpendicular to the axis of rotation, which intersects the surface at the equator, and the vertical one which includes the rotation axis and intersects the surface on an arc called a meridian

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