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If you ve forgotten what the Arctangent function is, and why we use a capital A to denote it, you can check in Chap 3 to refresh your memory Notice that the Cartesian z value is irrelevant when we want to find the direction angle in cylindrical coordinates
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Cartesian to cylindrical: finding r When we want to calculate the r coordinate in cylindrical three-space on the basis of a point in Cartesian xyz space, we use the Cartesian two-space distance formula, exactly as we would in the polar plane The radius depends only on the values of x and y; the z coordinate is irrelevant The r coordinate is therefore equal to the distance between the projection point P and the origin in the xy plane, which is
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r = (x2 + y2)1/2
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Cartesian to cylindrical: finding h When we want to change the Cartesian z value to the cylindrical h value in three-space, we can make the direct substitution
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An example Let s convert the Cartesian point (x,y,z) = (1,1,1) to cylindrical three-space coordinates In this situation, x = 1 and y = 1 To find the angle, we should use the formula
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q = Arctan ( y /x) because x > 0 and y > 0 When we plug in the values for x and y, we get q = Arctan (1/1) = Arctan 1 = p /4 When we input the values for x and y to the formula for r, we get r = (12 + 12)1/2 = 21/2 Because z = 1, we know that h=z=1 We ve just found that the cylindrical equivalent point is (q,r,h) = (p /4,21/2,1)
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Spherical Coordinates
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We must pay close attention to the meaning of the radius in cylindrical coordinates The cylindrical radius goes from the origin to the reference-plane projection of the point whose coordinates we re interested in It does not go straight through space to the point of interest, which is usually outside the reference plane
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Here s a challenge!
Convert the Cartesian point (x,y,z) = ( 5, 12,8) to cylindrical coordinates Using a calculator, approximate all irrational values to four decimal places
Solution
We have x = 5 and y = 12 To find the angle, we should use the formula q = p + Arctan ( y /x) because x < 0 and y < 0 When we plug in x = 5 and y = 12, we get q = p + Arctan [( 12)/( 5)] = p + Arctan (12/5) That is a theoretically exact answer, but it s an irrational number A calculator set to work in radians (not degrees) allows us to approximate this to four decimal places as q 43176 When we input x = 5 and y = 12 to the formula for r, we get r = [( 5)2 + ( 12)2]1/2 = (25 + 144)1/2 = 1691/2 = 13 Because z = 8, we know that h=z=8 We ve found that the cylindrical equivalent point is (q,r,h) (43176,13,8) The value of q is approximate to four decimal places, while r and h are exact values
Spherical Coordinates
Figure 9-5 illustrates a system of spherical coordinates for defining points in three-space Instead of one angle and two displacements as in cylindrical coordinates, we now use two angles and one displacement
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+z Reference axis P r x P
Reference plane
Figure 9-5
Spherical coordinates define points in threespace according to a horizontal angle, a vertical angle, and a radius
How it works In the spherical coordinate arrangement, we start with a horizontal Cartesian reference plane, just as we do when we set up cylindrical coordinates The positive Cartesian x axis forms the reference axis Suppose that we want to define the location of a point P Consider its projection, P , onto the reference plane:
The horizontal angle, which we call q, turns counterclockwise in the reference plane from the reference axis to the ray that goes out from the origin through P The vertical angle, which we call f, turns downward from the vertical axis to the ray that goes out from the origin through P The radius, which we call r, is the straight-line distance from the origin to P These three coordinates, taken all together, provide us with sufficient information to uniquely define the location of P in three-space We can express the spherical coordinates as an ordered triple P = (q,f,r)
Strange values In spherical three-space, we can have nonstandard horizontal direction angles, but it s always best to add or subtract whatever multiple of 2p will keep us within the preferred range of 0 q < 2p If q 2p, it represents at least one complete counterclockwise rotation from the reference axis If q < 0, it represents clockwise rotation from the reference axis
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