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For a moment, suppose that the coefficient in the first equation was 3 rather than 2 In that case, the set of parametric equations would be x = 3 cos t y = 3 sin t z = 3 and we d have a circle in the plane z = 3 Figure 18-10, on page 360 is an approximate graph of this circle if we imagine each coordinate axis division to represent 1 unit However, the coefficient in the first equation is 2, not 3 Therefore, the curve is squashed in the x direction; it s only 2/3 as wide as the above described circle This squashed circle is an ellipse centered on the point (0,0, 3) Figure 18-13 shows how its graph looks in Cartesian xyz space, from a vantage point far from the origin but close to the positive z axis
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+y z = 3 Ellipse in plane z = 3 (0, 0, 3)
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Major semi-axis =3 Minor semi-axis =2
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Graph of an ellipse in the plane z = 3, centered on (0,0, 3) Each axis increment is 1 unit
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When we created the generalized circles and graphed them as shown in Figs 18-8 through 18-10, we held one variable constant and forced the other two variables to follow the parametric equations for a circle in a plane Now imagine that, instead of holding one variable
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constant, we let it change according to a constant multiple of the parameter When we do this, we get a three-dimensional object called a circular helix
Center on x axis Consider a moving plane x = ct in Cartesian xyz-space, where c is a constant and t is the parameter This plane is always perpendicular to the x axis, so it s always parallel to the yz plane It intersects the x axis at a moving point (ct,0,0) Imagine a moving a circle of radius r in the moving plane x = ct that s centered on the x axis On this circle, the value of y tracks along with r cos t, while the value of z tracks along with r sin t The complete set of parametric equations is
x = ct y = r cos t z = r sin t When we graph the path of a point on this moving circle as t varies, we get a circular helix of uniform pitch (that means its coil turns are evenly spaced, like those of a well-designed spring) The pitch depends on c Small values of c produce tightly compressed helixes, while large values of c produce stretched-out helixes The helix axis corresponds to the coordinate x axis, so the helix is centered on the x axis Figure 18-14 is a generic graph of a circular helix oriented in this way
Helix is centered on the x axis
+x Radius of helix =r
Figure 18-14
Circular helix of radius r, centered on the x axis The pitch depends on the constant by which t is multiplied to obtain x
Circular Helixes
Center on y axis Now imagine a moving plane y = ct that s perpendicular to the y axis, parallel to the xz plane, and intersects the y axis at a moving point (0,ct,0) The value of x tracks along with r cos t, while the value of z tracks along with r sin t, so we have the system
x = r cos t y = ct z = r sin t The graph of this set of parametric equations is a circular helix of uniform pitch, centered on the y axis as shown in Fig 18-15
Center on z axis Finally, envision a moving plane z = ct that s perpendicular to the z axis, parallel to the xy plane, and intersects the z axis at a moving point (0,0,ct) The value of x follows r cos t, while y follows r sin t Our parametric equations are therefore
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