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cos (p /3) = 1/2 and cos p = 1
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Spherical Conversions
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For things to work without ambiguity when we go the other way, we want the arccosine to be a true function To do that, we must restrict its range (output) to an interval where we don t get into trouble with ambiguity By convention, mathematicians specify the closed interval [0,p] for this purpose That happens to be the ideal range of values for our vertical angle f in spherical coordinates When we make this restriction, we capitalize the A and write Arccosine or Arccos to indicate that we re working with a true function Then we can state the above facts in reverse using the Arccosine function, getting Arccos 1/2 = p /3 and Arccos ( 1) = p For any real number u, we can be sure that Arccos (cos u) = u Going the other way, for any real number v such that 1 v 1, we know that cos (Arccos v) = v We restrict v because the Arccosine function is not defined for input values less than 1 or larger than 1
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Cartesian to spherical: finding e We ve learned how to find the vertical angle on the basis of the Cartesian coordinate z That formula is
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z = r cos f We can use algebra to rearrange this, getting cos f = z /r provided r 0 When we examine Fig 9-9, we can see that for any given point P, the absolute value of z can never exceed r, so we can be sure that 1 z /r 1 Therefore, we can take the Arccosine of both sides of the preceding equation, getting Arccos (cos f) = Arccos (z /r) Simplifying, we obtain f = Arccos (z /r)
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This formula works nicely if we know the value of r But we sometimes want to find the vertical angle in terms of x, y, and z exclusively We ve found that r = (x2 + y2 + z2)1/2 so we can substitute to obtain f = Arccos [z / (x2 + y2 + z2)1/2]
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An example Consider a point P in spherical three-space whose coordinates are given by
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P = (q,f,r) = (3p /2,p /2,5) Let s find the equivalent coordinates in Cartesian xyz space We ll start by calculating the x value The formula is x = r sin f cos q When we plug in the spherical values, we get x = 5 sin (p /2) cos (3p /2) = 5 1 0 = 0 The formula for y is y = r sin f sin q Plugging in the spherical values yields y = 5 sin (p /2) sin (3p /2) = 5 1 1 = 5 The formula for z is z = r cos f When we put in the spherical values, we get z = 5 cos (p /2) = 5 0 = 0 In xyz space, our point can be specified as P = (0, 5,0)
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Another example Let s convert the xyz space point ( 1, 1,1) to spherical coordinates To find the radius, we use the formula
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r = (x2 + y2 + z2)1/2
Spherical Conversions
Plugging in the values, we get r = [( 1)2 + ( 1)2 + 12]1/2 = (1 + 1 + 1)1/2 = 31/2 To find the horizontal angle, we use the formula q = p + Arctan ( y /x) because x < 0 and y < 0 When we plug in the values for x and y, we get q = p + Arctan [ 1/( 1)] = p + Arctan 1 = p + p /4 = 5p /4 To find the vertical angle, we can use the formula f = Arccos (z /r) We already know that r = 31/2, so f = Arccos (1/31/2) = Arccos 3 1/2 Our spherical ordered triple, listing the coordinates in the order P = (q,f,r), is P = [5p /4,(Arccos 3 1/2),31/2]
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