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Figure 17-6 An oblate sphere in Cartesian xyz space, centered
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Figure 17-7
An ellipsoid in Cartesian xyz space, centered at the origin
Surfaces in Three-Space
Ellipsoid centered away from the origin Consider a sphere centered at (x0,y0,z0) If we make one of the axial radii longer while leaving the other two unchanged, we get an ellipsoid 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 17-8 portrays a situation in which a sphere centered at (x0,y0,z0) has been stretched along a line parallel to the z axis to obtain an ellipsoid
Oblate ellipsoid centered at the origin One more time, imagine a sphere centered at the origin We start out with all three axial radii equal in measure Then we lengthen one of them, shorten another, and leave the third one unchanged This process gives us an oblate ellipsoid Figure 17-9 shows an example where we take the sphere from Fig 17-3, squash the radius in the y direction, stretch the radius in the z direction, and leave the radius unchanged in the x direction The general equation for an oblate ellipsoid centered at the origin is
x2/a2 + y2/b2 + z2/c2 = 1
+y Center is at (x0, y0, z0) Radius in y direction =b
Radius in z direction =c y
Radius in x direction =a
Figure 17-8
An ellipsoid in Cartesian xyz space, centered at (x0,y0,z0)
Distorted Spheres
Center is at (0, 0, 0)
Radius in y direction =b
Radius in z direction =c y
Radius in x direction =a
Figure 17-9
An oblate ellipsoid in Cartesian xyz space, centered at the origin
where a is the x-axial radius, b is the y-axial radius, c is the z-axial radius, and all of the following are true: a b b c a c
Oblate ellipsoid centered away from the origin Finally, imagine a sphere that s centered at (x0,y0,z0) If we lengthen one of the axial radii, shorten another, and leave the third one unchanged, we get an oblate ellipsoid defined by
(x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 where all of the following are true: a b b c a c Figure 17-10 shows an example of what happens when we move the center of the oblate ellipsoid from Fig 17-9 away from the origin
Surfaces in Three-Space
+y Center is at (x0, y0, z0) Radius in y direction =b
Radius in z direction =c
Radius in x direction =a
Figure 17-10
An oblate ellipsoid in Cartesian xyz space, centered at (x0,y0,z0)
An example Suppose that the coordinates of the center of a certain oblate sphere in Cartesian xyz space are (1,2,3) The axial radius in the x direction is 4, the axial radius in the y direction is 4, and the axial radius in the z direction is 2 The general equation is
(x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 where (x0,y0,z0) are the coordinates of the center, a is the is the axial radius in the x direction, b is the axial radius in the y direction, and c is the axial radius in the z direction We know that (x0,y0,z0) = (1,2,3) a=4 b=4 c=2 Plugging these values into the general equation, we conclude that our oblate sphere can be represented by the following equation: (x 1)2/42 + (y 2)2/42 + (z 3)2/22 = 1 which simplifies to (x 1)2/16 + (y 2)2/16 + (z 3)2/4 = 1
Distorted Spheres
Another example The coordinates of the center of an ellipsoid are ( 3, 2, 6) The axial radius in the x direction is 3, the axial radius in the y direction is 7, and the axial radius in the z direction is 3 The general equation is
(x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 This time, we have (x0,y0,z0) = ( 3, 2, 6) a=3 b=7 c=3 Plugging these values into the general equation, we obtain [x ( 3)]2/32 + [y ( 2)]2/72 + [z ( 6)]2/32 = 1 which simplifies to (x + 3)2/9 + (y + 2)2/49 + (z + 6)2/9 = 1
Still another example The coordinates of the center of an oblate ellipsoid are (0, 3,11) The axial radius in the x direction is 5, the axial radius in the y direction is 8, and the axial radius in the z direction is 1 The general equation is
(x x0)2/a2 + (y y0)2/b2 + (z z0)2/c2 = 1 In this case, we have (x0,y0,z0) = (0, 3,11) a=5 b=8 c=1 Plugging these values into the general equation gives us [x 0]2/52 + [y ( 3)]2/82 + (z 11)2/12 = 1 which simplifies to x2/25 + (y + 3)2/64 + (z 11)2 = 1
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