Two typical comets that I know are Donati's and Halley's. So comets also have orbits, but are their orbits on the same plane (such as the ecliptic) as the other planets?
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Those two comets happen to have large orbital inclinations to the Ecliptic: 117$^\circ$ for C/1858 L1 (Donati) and 162$^\circ$ for 1P/Halley. Long-period comet orbits are oriented essentially at random, suggesting that they originate in the Oort cloud, a sphere several thousand AU from the Sun. Many short-period comet orbits are inclined only a few degrees to the Ecliptic, suggesting origins in the scattered disc, a torus several tens to a few hundred AU from the Sun.
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1$\begingroup$ The fact that short-period comets tend to have small inclinations (and orbits that are prograde like planets and asteroids) is taken as evidence that they originate from a flattened, rotating distribution: e.g., the Kuiper Belt. $\endgroup$ – Peter Erwin Sep 23 '16 at 21:44
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$\begingroup$ Noted. The scattered disc seems to be favored over the proper Kuiper belt. $\endgroup$ – Mike G Sep 23 '16 at 23:23
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$\begingroup$ It seems the disk isn't a torus, it's a disk. Instead, the belt is the torus: "The innermost portion of the scattered disc overlaps with a torus-shaped region of orbiting objects traditionally called the Kuiper belt…" 2nd paragraph. $\endgroup$ – uhoh Sep 24 '16 at 2:02
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1$\begingroup$ "...but its outer limits reach much further away from the Sun and further above and below the ecliptic" $\endgroup$ – Mike G Sep 24 '16 at 2:26
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