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Fermat's spiral is the Archimedean spiral with m=2. Both the left-hand and right-hand spirals are shown below.
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The Hyperbolic spiral is the Archimedean spiral with m=(-1).
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The Lituus is the Archimedean spiral with m=(-2). Both the left-hand and right-hand spirals are shown below.
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The Logarithmic spiral is the locus of the equations shown below.
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The Sinusoidal spiral is the locus of the equations shown below.
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The Tractrix is the evolute of the catnary.
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A Curtate cycloid is a cycloid where the locus is drawn from a point closer to the center of the circle than the circle's circumference.
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A Prolate cycloid is a cycloid where the locus is drawn from a point farther from the center of the circle than the circle's circumference.
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