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Module 2

The document discusses different coordinate systems, including circular cylindrical coordinates and spherical coordinates. Circular cylindrical coordinates locate a point in three-dimensional space using the distance (ρ) from the origin, the angle (φ) between the line from the point to the origin and an arbitrary reference line, and the distance (z) from an arbitrary reference plane perpendicular to the line ρ = 0. Spherical coordinates locate a point using its distance (r) from the origin, its polar angle (θ) from the positive z-axis, and its azimuthal angle (φ) in the xy-plane measured from the positive x-axis. Conversion formulas between rectangular and spherical coordinates are provided.

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74 views17 pages

Module 2

The document discusses different coordinate systems, including circular cylindrical coordinates and spherical coordinates. Circular cylindrical coordinates locate a point in three-dimensional space using the distance (ρ) from the origin, the angle (φ) between the line from the point to the origin and an arbitrary reference line, and the distance (z) from an arbitrary reference plane perpendicular to the line ρ = 0. Spherical coordinates locate a point using its distance (r) from the origin, its polar angle (θ) from the positive z-axis, and its azimuthal angle (φ) in the xy-plane measured from the positive x-axis. Conversion formulas between rectangular and spherical coordinates are provided.

Uploaded by

John Rivas
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© © All Rights Reserved
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Download as PDF, TXT or read online on Scribd
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CHAPTER I

OTHER
COORDINATE
SYSTEMS
CIRCULAR CYLINDRICAL COORDINATES
The circular cylindrical coordinate system is the three-dimensional version of
the polar coordinates of analytic geometry. In polar coordinates, a point is located
in a plane by giving both its distance ρ from the origin and the angle φ between the
line from the point to the origin and an arbitrary radial line, taken as φ = 0. In
circular cylindrical coordinates, we also specify the distance z of the point from an
arbitrary z = 0 reference plane that is perpendicular to the line ρ = 0. For simplicity,
we usually refer to circular cylindrical coordinates simply as cylindrical coordinates.
CIRCULAR CYLINDRICAL COORDINATES
CIRCULAR CYLINDRICAL COORDINATES
CIRCULAR CYLINDRICAL COORDINATES
CIRCULAR CYLINDRICAL COORDINATES
CIRCULAR CYLINDRICAL COORDINATES

Solution for D.1.a


1. Plot the given coordinates C(ρ = 4.4, Φ = −115◦, z = 2) , in order to check the
exact location of the point and to counter check the sign convention of the
conversion later
CIRCULAR CYLINDRICAL COORDINATES

Solution for D.1.a


2. Use the following formula to calculate for the rectangular coordinates

x = 4.4 cos(-115°) = - 1.860


y = 4.4 sin(-115°) = - 3.990
z=2

Then, the answer should be written as C(- 1.860, - 3.990, 2)


CIRCULAR CYLINDRICAL COORDINATES

Solution for D.1.b


1. Plot the given coordinates
CIRCULAR CYLINDRICAL COORDINATES

Solution for D.1.b


2. Use the following formula to calculate for the circular cylindrical coordinates

ρ= −3.1 2 + 2.6 2 = 4.046

2.6
Φ = tan-1 = -39.99° ≈ -40° , the -40° is the angle from the –x-axis to the
−3.1
+y-axis but the azimuth should always start from the +x-axis then
Φ = (180-40)° = 140°, this is positive since rotation is from +x-axis - +y-axis
z = -3

Then the answer should be written as D(ρ = 4.05, Φ = 140.0◦, z = −3);


CIRCULAR CYLINDRICAL COORDINATES

Solution for D.1.c


Since the rectangular coordinates were shown/calculated on the previous problem
we can now directly calculate the distance from point C to point D with this
formula,

d= 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2 + 𝑧2 − 𝑧1 2

= −3.1 − (−1.860) 2 + 2.6 − (−3.99) 2 + −3 − 2 2

d = 8.364 units
CIRCULAR CYLINDRICAL COORDINATES

NOTE: Be vigilant in substituting for the values with its corresponding subscripts, Example for
D1.5.c since the problem is to specify the distance from C to D then point C should carry the
values of x1, y1 and z1 then point D for x2, y2 and z2.
SPHERICAL COORDINATES

NOTE: As per our reference Range R is equal to r


SPHERICAL COORDINATES

The three spherical coordinates


SPHERICAL COORDINATES
The transformation of scalars from the rectangular to the spherical coordinate system is
easily made by using the figure shown to relate the two sets of variables:

The transformation in the reverse direction is achieved with the help of


SPHERICAL COORDINATES
SPHERICAL COORDINATES

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