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Surfaces in Three-Space
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A hyperboloid of one sheet in Cartesian xyz space, centered at the origin
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Figure 17-11 shows a perspective on Cartesian xyz space that we haven t seen before We re looking down on the yz plane from somewhere near the positive x axis Nevertheless, the axes are correctly oriented with respect to each other, as you can verify by referring back to Chap 7 Let s stay with this axis orientation as we look at the next couple of objects
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Hyperboloid of two sheets Figure 17-12 shows a hyperboloid of two sheets, which is the graph in Cartesian xyz space of an equation having the form
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x2/a2 + y2/b2 z2/c2 = 1 where a, b, and c are positive real-number constants Here, we have two surfaces that resemble bowls facing in opposite directions In theory, the bowls extend infinitely toward the left and the right in this illustration Both surfaces share a common straight-line axis that coincides with the coordinate y axis, and the two sheets are exact mirror images of each other The center of the entire hyperboloid is at the origin The contours of the surfaces depend on the values of a, b, and c
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Other Surfaces
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A hyperboloid of two sheets in Cartesian xyz space, centered at the origin
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If we make the term containing x2 positive instead of the term containing y2, we get the general equation x2/a2 y2/b2 z2/c2 = 1 which produces a hyperboloid of two sheets whose axis lies along the coordinate x axis, and whose center is at the origin If we move the plus sign so it s in front of the term containing z2, the general equation becomes x2/a2 y2/b2 + z2/c2 = 1 This maneuver gives us a hyperboloid of two sheets whose axis lies along the coordinate z axis, and whose center is at the origin
Elliptic cone Figure 17-13 shows an elliptic cone It s what we get when we graph an equation of the form
x2/a2 + y2/b2 z2/c2 = 0 where a, b, and c are positive real-number constants The perpendicular cross sections of the cone are always circles or ellipses The cone s axis coincides with the coordinate z axis, and the cone s vertex coincides with the origin The flare angles, as well as the eccentricity of the crosssectional ellipses, depend on the values of a, b, and c
Surfaces in Three-Space
Figure 17-13
An elliptic cone in Cartesian xyz space, centered at the origin
If we move the minus sign so it s in front of the term containing y2, we get the general equation x2/a2 y2/b2 + z2/c2 = 0 whose graph is an elliptic cone with the axis along the coordinate y axis, and whose center is at the origin If we move the minus sign so that it s in front of the term containing x2, the general equation becomes x2/a2 + y2/b2 + z2/c2 = 0 and the graph becomes an elliptic cone whose axis lies along the coordinate x axis, and whose center is at the origin
An example Consider the object in Cartesian xyz space represented by
36x2 16y2 + 36z2 = 0 We can divide through by 144 to obtain x2/4 y2/9 + z2/4 = 0 This is the equation for an elliptic cone whose vertex is at the origin, and whose axis coincides with the coordinate y axis
Other Surfaces
Another example Consider the object in Cartesian xyz space represented by
x2 + y2 + z2 = 7 When we divide through by 7, we get x2/( 7) + y2/( 7) + z2/( 7) = 1 which simplifies to x2/7 y2/7 z2/7 = 1 This equation describes a hyperboloid of two sheets whose center is at the origin, and whose axis lies along the coordinate x axis
Still another example Consider the object in Cartesian xyz space represented by
15x2 + 10y2 = 6z2 + 30 We can subtract 6z2 from each side, getting 15x2 + 10y2 6z2 = 30 Dividing through by 30 gives us x2/2 + y2/3 z2/5 = 1 This is the equation for a hyperboloid of one sheet whose center is at the origin, and whose axis lies along the coordinate z axis
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