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Distorted Spheres
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Spheres can be made out of the round by increasing or decreasing the axial radii in the x, y, and z directions individually
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Alternative equation for a sphere centered at the origin Once again, consider the general equation of a perfect sphere centered at the origin That equation, in standard form, is
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x2 + y2 + z2 = r2 where r is the radius If we divide through by r2, we get x2/r2 + y2/r2 + z2/r2 = 1 This equation tells us that the radius is always the same, whether we measure it in the direction of the x axis, y axis, or z axis To emphasize the fact that we can, if desired, change any of all of these axial radii, let s rewrite the above equation as x2/a2 + y2/b2 + z2/c2 = 1 where a, b, and c are positive real numbers representing the radii along the x, y, and z axes, respectively In the case of a perfect sphere, we have a=b=c If these three positive real-number constants a, b, and c are not all the same, then we have a distorted sphere
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Distorted Spheres
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Oblate sphere centered at the origin Suppose that we take a perfect sphere and then shorten one of the three axial radii This process gives us an object called an oblate sphere It s flattened, like a soft rubber ball when pressed between our hands Figure 17-5 shows an example This is what we get if we take the sphere from Fig 17-3 and reduce the axial radius b (the one that goes along the y axis), while leaving the axial radii a and c unchanged The center of the object is still at the origin, but we can no longer say that all the points on its surface are equidistant from the origin The general equation for an oblate sphere centered at the origin is
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x2/a2 + y2/b2 + z2/c2 = 1 where a is the x-axial radius, b is the y-axial radius, c is the z-axial radius, and exactly one of the following relationships holds true among them: a<b=c b<a=c c<a=b
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Alternative equation for a sphere centered away from the origin Earlier in this chapter, we learned that the general equation of a sphere centered at some point other than the origin in Cartesian xyz space is
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(x x0)2 + (y y0)2 + (z z0)2 = r2
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Center is at (0, 0, 0)
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Radius in y direction =b
Radius in z direction =c y
Radius in x direction =a
Figure 17-5
An oblate sphere in Cartesian xyz space, centered at the origin
Surfaces in Three-Space
where r is the radius, and (x0,y0,z0) are the coordinates of the center Dividing through by r2, we obtain (x x0)2/r2 + (y y0)2/r2 + (z z0)2/r2 = 1 As we did with the sphere centered at the origin, we can rewrite this equation, getting (x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 where a, b, and c are the radii parallel to the x, y, and z axes, respectively As before, with a perfect sphere, we have a=b=c If a, b, and c are not all the same, then the sphere is distorted
Oblate sphere centered away from the origin If we take a sphere that s centered at (x0,y0,z0) and shorten one of the axial radii, we get an oblate sphere defined by the general equation
(x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 where exactly one of the following is true: a<b=c b<a=c c<a=b Figure 17-6 should give you a general idea of what happens in a case like this Imagine a sphere centered at (x0,y0,z0) that has been squashed in the direction defined by a line parallel to the y axis
Ellipsoid centered at the origin Again, imagine that we have a perfect sphere centered at the origin in Cartesian xyz space Let s lengthen one of the axial radii while leaving the other two unchanged This stretching process produces an ellipsoid It s elongated, like a football with blunted ends Figure 17-7 shows an example Imagine that we take the sphere from Fig 17-3 and then stretch it in the z direction The general equation for an ellipsoid centered at the origin is
x2/a2 + y2/b2 + z2/c2 = 1 where a is the x-axial radius, b is the y-axial radius, c is the z-axial radius, and exactly one of the following relationships is true: a>b=c b>a=c c>a=b