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Lines and Curves in Three-Space
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The z-value at the absolute minimum point is zmin = 2xmin2 + 4xmin + 3 = 2 ( 1)2 + 4 ( 1) + 3 = 2 4 + 3 = 2 + 3 = 1 Now we know that the coordinates of the parabola s vertex are ( 1,5,1) As the basis for our next point, let s choose x = 3 We can plug it directly into the function to get z = 2x2 + 4x + 3 = 2 ( 3)2 + 4 ( 3) + 3 = 18 12 + 3 = 6 + 3 = 9 This gives us ( 3,5,9) as the coordinates of a second point on the curve Finally, let s set x = 1 Plugging it in, we obtain z = 2x2 + 4x + 3 = 2 12 + 4 1 + 3 =2+4+3=9 The third point on our curve is (1,5,9) We now have three points: ( 3,5,9), ( 1,5,1), and (1,5,9) Figure 18-7 shows these points, along with a graph of the parabola passing through them, as seen in the plane where y maintains a constant value of 5
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( 3, 5, 9)
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Each axis increment is 1 unit
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Figure 18-7
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Graph of a parabola in a plane parallel to the xz plane, such that y has a constant value of 5 On both axes, each increment represents 1 unit
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Think about the graphs of higher-degree polynomial functions confined to specific planes in Cartesian xyz space For example, consider the cubic function in the plane where x = 2, such that x=2 y=t z = t3 or the quartic function in the plane where z = 7, such that x = 3t 4 + 6 y=t z = 7 Can you draw graphs of these curves
Circles
In Chap 13, we learned that the equation of a circle centered at the origin in the Cartesian xy plane can be written in the form x2 + y2 = r2 where r is the radius In Chap 16, we learned that the parametric equations for such a circle are x = r cos t and y = r sin t where t is the parameter Let s expand these notions to deal with any circle in xyz space that s centered on, and exists entirely in a plane perpendicular to, one of the three coordinate axes
Hold x constant Consider a plane x = c in Cartesian xyz space, where c is a constant This plane is parallel to the yz plane, and it intersects the x axis at (c,0,0) Imagine a circle of radius r in the plane x = c that s centered on the x axis as shown in Fig 18-8 The variable y follows along with
Lines and Curves in Three-Space
+y Circle in plane x=c
(c, 0, 0)
Radius of circle =r
Figure 18-8
Circle in a plane where x is held to a constant value c The plane is perpendicular to the x axis, and intersects that axis at the point (c,0,0) The circle has radius r and is centered at (c,0,0)
r cos t, while the variable z follows along with r sin t Therefore, we can define our circle with the system of parametric equations x=c y = r cos t z = r sin t For the circle to be fully circumscribed, the parameter t must range continuously over a span of values sufficient to ensure that a moving point makes at least one full revolution around the x axis The smallest such span is any half-open interval that s at least 2p units wide
Hold y constant Now suppose that we restrict ourselves to a plane such that y = c, where c is a constant This plane is parallel to the xz plane, and it intersects the y axis at (0,c,0) Imagine a circle in the plane y = c that s centered on the y axis, as shown in Fig 18-9 In this case, the circle is described by the system
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