3189: Conic Sections

Conic Sections

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Title text: They're not generally used for crewed spacecraft because astronauts HATE going around the corners.

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Explanation

This is one of 64 incomplete explanations: This page was created by a section through a scone. Don't remove this notice too soon. If you can fix this issue, edit the page!

A Kepler orbit describes the simplified motion of two celestial objects around each other. Such an orbit will form a conic section. A conic section is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes considered a fourth type. In reality, this model is based only on their basic gravitational forces and ignores any other factors such as the gravity of other objects, atmospheric drag, each object being a non-spherical(/non-point) body and any relativistic effects, but it serves as the basis for most orbital calculations before further refining with the most relevant additional perturbations.

How conic sections emerge from various planar intersections with bidirectional cones, which technically continue beyond the 'top' and 'bottom' of each diagram.

The conic section as depicted in the comic will be familiar to many readers, since models illustrating these curves using a cone with a base are commonplace and anyone who has learned about these curves in a mathematics class is likely to have seen one.

In real conic sections, the cone extends to infinity. In the comic, however, the "conic section" representing the satellite's orbit has been assumed to have a base at a distance that inconveniently coincides with the presumed otherwise eliptical path, resulting in sharp corners where the angled planar section through the cone meets that base. As alluded to in the title text, these corners would be extremely uncomfortable for an astronaut in a crewed spacecraft. Such an extreme and sudden change in direction would require a very large, potentially dangerous, G-force.

Being in a free orbit necessarily means following an ellipse (or very similar, outside of the mathematically strict two-body problem) without any experienced net acceleration, as adding up the forces due to external gravity and your path's continually changing direction will zero out. Being forced off this ellipse to move across the totally imaginary and arbitrary conic-base would force an instantaneous acute change of direction for no other reason than to follow the imperfectly understood mathematical 'model' at two arbitrary points (reaching the baseline and rejoining the true curve again), which would technically require infinite acceleration each time. Alternately, deliberately 'cutting the corner' would require the best effort of the spacecraft to stick to the truncated-orbital path, requiring as much complex and wasteful thrusting as the craft could provide (which any occupants must endure), including the less uncomfortable but still forceful passage along the 'straight' bit of the orbit through the curved gravitational field of the parent body.

Transcript

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[A view of the Earth, focused on Asia and the Indian Ocean with East Africa at left and the Western Pacific and Australia at right. A satellite is shown in an unusual orbit around the planet.]

[Caption below the panel:]

All Keplerian orbits are conic sections. For example, this one uses the base of the cone.

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Discussion

Isn't the base of a cone, just a circle? How would this have "corners"? SDSpivey (talk) 01:41, 3 January 2026 (UTC)

The cone upon which a conic section exists doesn't actually have a base, it's just arbitrarily large (possibly infinitely so) in order for the section to only ever lay along the 'curve' of the cone part.

But, here, the base is wwhere you give up on plotting how far 'down the cone' you go, of the sufficiently large ellipse (or possibly parabolic/hyperbolic curve), which is indeed round but has an sharp (i.e. acute) angle between its flat (and incidentally circular) plane-section and the 'wrapped' pseudo-euclidean plane of the conic-section it intersects with. 92.23.2.208 01:50, 3 January 2026 (UTC)

Bring a jacket and spoon for orbits that go through the ice cream.Lord Pishky (talk) 01:43, 3 January 2026 (UTC)

I'm pretty sure this is the shape of the flat bottom of a cake cone. 71.212.56.254 03:02, 3 January 2026 (UTC)

They REALLY hate the flat-bottom cone orbits and the waffle cones make for a bumpy ride.Lord Pishky (talk) 18:57, 3 January 2026 (UTC)

It appears to be a cut-off section of an ellipse, so basically a regular orbit with a sharp line. (Desmos) Tanner07 (talk) 04:29, 3 January 2026 (UTC)

https://media.licdn.com/dms/image/v2/D5622AQH3CYoPXy1cqg/feedshare-shrink_2048_1536/feedshare-shrink_2048_1536/0/1727242249609?e=1769040000&v=beta&t=UdAX9TH3joo-vpvj4pRWXoCQyF6JVUPVmyONWghcj5E --PRR (talk) 05:06, 3 January 2026 (UTC)

I feel like there needs to some explicit acknowledgement that the cone in question is an ice cream cone.99.239.23.54 00:11, 4 January 2026 (UTC)

But it's likely not. It's just one of the variations of the conic section. (Example 3 in the illustration on the right).71.94.164.106 20:21, 5 January 2026 (UTC)

But ice-cream cones have the 'flat bit' (actually the opening; give or take the scoop of ice-cream, which is a ball, or else the soft-served 'twirly-dollop', which another more convoluted form of inverted cone) at the top. Which just really doesn't fit with anything the comic says about the conic. Unless you see some obscure connection that I'm just not getting out of it. (Beyond that both are considered 'cones', which is as tenuous as if I suggested traffic cones was the ultimate reference, for example.)

But if you can give any better referencing connection, you look like you should know how to edit things to enlighten those of us who are missing it. Explain away, as that's the point of this site... 82.132.236.68 01:39, 4 January 2026 (UTC)

It's not obvious to me why anybody would think an ice cream cone is implied. Ice cream cones do not have flat bases - there's a hole to put the ice cream in. Furthermore, they are significantly smaller than the Earth. Jeremyp (talk) 14:18, 5 January 2026 (UTC)

I think it's just not drawn that well. We're seeing like a cross section of an ice cream cone with a scoop of ice cream in it, with the line between them going from lower left to upper right. The near side looks tangent when it should have a knee. If you were looking edge on, that knee would appear to open up, but that would apply to both the near and far sides. So we're getting a weird perspective here.163.116.145.34 19:44, 5 January 2026 (UTC)

You're weird. 92.23.2.208 21:35, 5 January 2026 (UTC)

I'm not the one insulting people on a webcomic wiki, though. 163.116.145.34 17:35, 6 January 2026 (UTC)

It's clearly a playing piece from Cones of Dunshire. 82.13.184.33 10:36, 6 January 2026 (UTC)

Shouldn't the people from the title text also be following the same orbit? Cobl703 (talk) 18:35, 4 January 2026 (UTC)

Might depends on if they share the same precise centre of gravity (the Explanation goes into some detail about that sort of thing).

Or if the effective orbit obeys the idential 'cone-based' rules. At any given time (depending on where you last positioned yourself), you might effectively be floating in a very similar elliptical orbit (could be the same period, same semi-major, same semi-minor, same periapsis, same periapsis, inclination, etc, but in a very slightly rotated orientation), so hit the change to the 'conic-baseline' section at a different time.

That's if the orbit equation defines the location of the transition into the conic-base (e.g. effectively when hitting the "semi-parameter" 'width', but on the non-focuse side of the original ellipse), or there's always some particular definite absolute (or proportional?) distance between the hypothetical cone's tip and when the normal orbital effect 'runs out'.

Too many little questions need to be asked about what is forcing the orbit to be off-elliptical. And if it's not a mere function of reality, but a deliberate manoeuvre by the craft, then of course the occupants will feel the sudden change in motion that the accompanying thruster-kick invokes. 92.23.2.208 21:03, 4 January 2026 (UTC)

How far the cone extends and where these effect occur depends on the units of distance used and the number of digits & format used to represent the length of the cone on the computer. Larger units avoid cone-end effects but make for a bumpier ride, especially when the exponent changes.Lord Pishky (talk) 05:58, 5 January 2026 (UTC)

Kinda reminds me of SCP-1778. --DollarStoreBa'al^Converse 14:08, 5 January 2026 (UTC)

Very nice, handy etc figure. might be good to label which conic section is which, even though it may be clear-ish, not everyone knows what a hyperbola is for example (although this is xkcd, lol) R128 (talk) 15:41, 5 January 2026 (UTC)

Did my best. Tried to include the salient features. Too much, though? Or did I still leave too much informative stuff out while trying to not make it too long? Should it have merited a 'main text' summary (would have let me use "#" markup!), or even shoved it into Trivia or its whole new section? Could an angled-hyperbollic diagram have been better than an axially-parallel one, for number 4 (or 4a, with that one as 4b, or 1 and 2 as 1a and 1b)? Questions for the ages..! 82.132.237.110 16:25, 5 January 2026 (UTC)
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