Machinery's Handbook, 31st Edition
CHANGING COORDINATE SYSTEMS 47 Furthermore, by the Pythagorean theorem and trigonometry, r x 2 y 2 = + and θ tan –1 y x - = ( ) . See TRIGONOMETRY on page 94 for a discussion of the related princi- ples of trigonometry. Example 1: Convert the Cartesian coordinate (3, 2) into polar coordinates. r 3 2 2 2 + 9+4 13 3.6 ≈ = = = θ tan –1 2 3 -- 33.69 ° = = ( ) Therefore the point is located approximately 3.6 units from the origin at an angle of about 33.69°. Thus, (3.6, 33.69º) is the polar form of the Cartesian point (3, 2). Graphically, the polar and Cartesian coordinates are related in the following figure:
θ tan –1 y x - = ( )
x 2 y 2 = + and
Y
(3, 2)
x
2
P
r = 3.6
1
y
θ = 33.69°
0
X
0
1
2
3
Example 2: Convert the polar form (5, 608°) to Cartesian coordinates. First note that this point lies 5 units from the origin at an angle of 608°. As explained on page 105, in Trigonometric Functions , this locates the point in Quadrant IV. By trigonometry, x = r cos q and y = r sin q . Then, x = 5cos(608º) = - 1.873 and y = 5sin(608°) = - 4.636. Therefore, the Cartesian point equivalent is ( - 1.873, - 4.636). This point lies in the fourth quadrant, where both coordinates are negative. Spherical Coordinates.— It is convenient in certain problems, for example, those con cerned with spherical surfaces and therefore three-dimensional, to introduce spherical co- ordinates. In three-dimensional space, as the figure on the right shows, the x,y -plane is like a floor in a room, and the third dimension is given by the z -axis, which is where the walls of the room meet. An arbitrary point P in this space is described by three rectangular coordinates ( x , y , z ), converted to the following spherical coordinates : the distance r between point P and the origin O , the angle f that OP ′ makes with the x , y -plane, and the angle l that the projec- tion OP ′ (the “shadow” of the segment OP on the x, y -plane) makes with the positive x -axis.
z
z
pole
P
P
r
r
O
O
e
y
x
x
y
The rectangular coordinates of a point in space can therefore be calculated from the spherical coordinates, and vice versa, by use of the formulas in the following table.
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