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This situation doesn t resemble anything we ve seen so far in Cartesian, polar, or cylindrical coordinates If we hold the vertical angle constant in a spherical coordinate system, we get the set of points formed by a line passing through the origin and rotated with respect to the vertical axis If k is a real-number constant, then the graph of f=k is a double cone whose axis corresponds to the vertical axis and whose apex is at the origin, as shown in Fig 9-8
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When we set the vertical angle equal to a constant in spherical coordinates, we get a double cone whose axis corresponds to the vertical axis
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Spherical Conversions
Converting coordinates between xyz space and spherical three-space is a little tricky, but not too difficult Let s think about a point P whose spherical coordinates are (q,f,r) and whose Cartesian coordinates are (x,y,z)
Spherical to Cartesian: finding x In spherical coordinates, the radius is usually outside of the reference plane, so we can t use it directly in the same formulas as the cylindrical radius But we can construct a projection radius identical to the cylindrical radius: the distance from the origin to the projection point P in the reference plane In Fig 9-9, the projection radius is called r From this geometry, we can see that r is equal to the true spherical radius times the sine of the vertical angle As an equation, we have
r = r sin f
Spherical Conversions
The x value conversion formula from cylindrical coordinates, which we learned earlier in this chapter, tells us that x = r cos q where q is the horizontal direction angle, which is the same in spherical and cylindrical coordinates Substituting the quantity (r sin f) for r gives us x = r sin f cos q
Spherical to Cartesian: finding y When we found the cylindrical equivalent of the Cartesian y value, we took the radius in the reference plane and multiplied by the sine of the direction angle in that plane In the spherical-coordinate situation of Fig 9-9, that translates to
y = r sin q where q is the horizontal direction angle We can substitute (r sin f) for r to get y = r sin f sin q
Spherical to Cartesian: finding z Let s look again at Fig 9-9, and locate the projection point P on the z axis, such that the z values of P and P are equal We can see that P , P, P , and the origin form the vertices of a
+z Reference axis
x P
Reference plane
Figure 9-9
Conversion between spherical and Cartesian three-space coordinates involves several geometric variables
Alternative Three-Space
rectangle perpendicular to the reference plane It follows that P , P, and the origin are at the vertices of a right triangle By trigonometry, the z value of P is equal to the spherical radius r times the cosine of the vertical angle f Because the z values of P and P are the same, we can deduce that the z value of P is given by z = r cos f
Cartesian to spherical: finding r Now let s figure out how to get from Cartesian xyz space to spherical three-space The radius is the easiest coordinate to find, so let s do it first Recall that the spherical radius of a point is its distance from the origin Therefore, when we want to find the spherical radius r for point P in terms of its xyz space coordinates, we can apply the Cartesian three-space distance formula to get
r = (x2 + y2 + z2)1/2
Cartesian to spherical: finding q The horizontal angle in spherical coordinates is identical to its counterpart in cylindrical coordinates, so we can use the conversion table from earlier in this chapter
q=0 by default q=0 q = Arctan ( y /x) q = p/2 q = p + Arctan ( y /x) q=p q = p + Arctan ( y /x) q = 3p/2 q = 2p + Arctan ( y /x) When x = 0 and y = 0 that is, at the origin When x > 0 and y = 0 When x > 0 and y > 0 When x = 0 and y > 0 When x < 0 and y > 0 When x < 0 and y = 0 When x < 0 and y < 0 When x = 0 and y < 0 When x > 0 and y < 0
The Arccosine Before we can find the vertical spherical angle for a point that s given to us in Cartesian coordinates, we must be familiar with the arccosine relation It s abbreviated arccos (cos 1 in some texts), and it undoes the work of the cosine function For example, we know that