x = r cos t y = r sin t z = ct

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Helix is centered on the y axis

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Radius of helix =r

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+y Radius of helix =r

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Helix is centered on the z axis

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In Cartesian xyz space, these equations produce a circular helix of uniform pitch, centered on the z axis Figure 18-16 is a generic graph

An example Consider a circular helix centered on the x axis, described by the parametric equations

x = t /(2p ) y = cos t z = sin t Here are some values of x, y, and z that we can calculate as t varies, causing a point on the helix to complete a single revolution in a plane perpendicular to the x axis: When t = 0, we have x = 0, y = 1, and z = 0 When t = p /2, we have x = 1/4, y = 0, and z = 1 When t = p, we have x = 1/2, y = 1, and z = 0 When t = 3p /2, we have x = 3/4, y = 0, and z = 1 When t = 2p, we have x = 1, y = 1, and z = 0

Circular Helixes

Every time t increases by 2p, our point makes one complete revolution in a moving plane that s always perpendicular to the x axis Also, every time t increases by 2p, our point gets 1 unit farther away from the yz plane The pitch of the helix is therefore equal to 1 linear unit per revolution

Another example Consider a circular helix centered on the y axis, described by the parametric equations

x = 2 cos t y=t z = 2 sin t Here are some values of x, y, and z that we can calculate as t varies, causing a point on the helix to complete a single revolution in a plane perpendicular to the y axis: When t = 0, we have x = 2, y = 0, and z = 0 When t = p /2, we have x = 0, y = p /2, and z = 2 When t = p, we have x = 2, y = p, and z = 0 When t = 3p /2, we have x = 0, y = 3p /2, and z = 2 When t = 2p, we have x = 2, y = 2p, and z = 0

Every time t increases by 2p, our point makes a complete revolution in a moving plane that s always perpendicular to the y axis Also, every time t increases by 2p, our point moves 2p units farther away from the xz plane The pitch of the helix is therefore equal to 2p linear units per revolution

Are you confused

You might ask, When describing a helix with parametric equations, does it make any difference if we multiply t by a positive constant or a negative constant That s an excellent question The answer is yes; it matters a lot! The polarity of the constant affects the sense in which the helix rotates as we move in the positive direction For example, suppose we have a helix described by the parametric equations x = 3t y = 3 cos t z = 3 sin t In this case, the helix turns counterclockwise as we move in the positive x direction If we observe the situation from somewhere on the positive x axis while the value of t increases, a point on the

Lines and Curves in Three-Space

helix will appear to approach us and rotate counterclockwise If the value of t decreases, a point on the helix will appear to retreat from us and rotate clockwise Now suppose that we reverse the sign of the constant in the first equation, so our system becomes x = 3t y = 3 cos t z = 3 sin t If we watch this scene from somewhere on the positive x axis while the value of t increases, a point on the helix will appear to retreat from us and rotate counterclockwise If the value of t decreases, a point on the helix will appear to approach us and rotate clockwise