Match the parameterizations with their graphs in Figure...

Match the parameterizations with their graphs in Figure 17.9.
(a) $x=2 \cos 4 \pi t, y=2 \sin 4 \pi t, z=t$
(b) $x=2 \cos 4 \pi t, y=\sin 4 \pi t, z=t$
(c) $x=0.5 t \cos 4 \pi t, y=0.5 t \sin 4 \pi t, z=t$
(d) $x=2 \cos 4 \pi t, y=2 \sin 4 \pi t, z=0.5 t^{3}$
(FIGURE CAN'T COPY)

Key Concepts

Parametric Equations

Parametric equations describe curves by expressing the coordinates as functions of one or more parameters. They allow for a flexible representation of curves in space, where changing the parameter t traces out a path that can incorporate rotations, translations, scalings, and more complex behaviors.

Helical Curves

A helical curve is characterized by circular motion in the horizontal plane (typically using cosine and sine functions for the x and y coordinates) combined with a steady vertical (z) motion. When the z-component is a linear function of the parameter, the result is a regular helix with constant pitch.

Elliptical vs. Circular Cross Sections

When the coefficients of the cosine and sine functions in the x and y coordinates differ, the resulting projection in the horizontal plane is an ellipse rather than a circle. This leads to an elliptical helix, where the cross-sectional shape of the helix deviates from perfect circular symmetry.

Spiraling with Varying Radius

In some parameterizations, the coefficients multiplying the cosine and sine functions can depend on the parameter (such as being multiplied by t). This causes the radius of the circular component to vary with t, producing a spiral whose distance from the central axis grows (or shrinks) over time.

Effect of Nonlinear Vertical Components

When the z-component of a parametric curve is a nonlinear function of the parameter, such as a cubic function, the vertical spacing of the curve changes non-uniformly. This affects the overall appearance of the helix, leading to variations in pitch and a deformation of the typical helical structure.

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