3189: Conic Sections

Conic Sections

|< < Prev #3189 (January 2, 2026) Next > >|

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

|< < Prev Random Explanation Next > >|

Explanation

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

A Kepler orbit describes the simplified motion of one celestial object relative to another. Such an orbit will form a conic section — 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 is sometimes considered a fourth type, while intersections of the plane with the point of the cone (just that point, a straight line through that point or else four converging lines that all meet at the point) are possible constructions that are usually excluded. (In reality, this model is based only on the most simple modeling of two point masses, and ignores any other factors such as the gravity of other objects, non-gravitational forces (e.g. atmospheric drag), each object being a non-spherical(/non-point) body of non-uniform density and any relativistic effects, but it serves as a good basis for most orbital calculations before needing further refinements to cover the most relevant additional perturbations for a given scenario.)

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

---

1) Plane intersects perpendicular to conic axis, results in a circular line (often counted as an ellipse of zero eccentricity) around one cone.
2) Plane intersects at a small angle away from the perpendicular, results in an elliptic line around one cone.
3) Plane intersects exactly at the angle of the (opposite) slope of the cone, results in an open-ended line that continues parabolically to infinity at an ever-increasing width (by decreasing degree) but at constant offset from the parallel slope of that cone.
4) Plane intersects at an angle closer to the axis than the cone slope (including being exactly parallel to the axis, as here), resulting in two open-ended hyperbolic lines to infinity (eventually tending to diverge at the rate of the conic slope itself), one upon each cone.

---

In this comic, the orbital shape is similar to the one in figure 3 (a parabolic trajectory that does not technically 'orbit' the focal mass) with the 'end' of the lower cone included. Or, given the implication of this being based upon a mostly standard non-circular orbit, it might be a version of figure 2 but with the angled plane being lower so that the ellipse is cut off by the nominal 'bottom' of the diagrammatic cone.

In real conic sections, the cone effectively extends to infinity (whether or not the useful section of the intersecting curve does). In the comic, however, the "conic section" representing the satellite's orbit (with its unseen point pointing generally to the left of the image) has been assumed to be through a cone which has a flat circular base (presumed to be somewhere close to vertical, towards the right of the image) set at a distance that inconveniently crosses the indicated orbital path (that might be assumed to be fully elliptical, otherwise), resulting in sharp corners where the angled planar intersection through the cone meets that base.

The comic does not indicate why or how this orbit involves the 'base' of the cone. Being in a free orbit normally means following an ellipse (or very similar, outside of the mathematically strict two-body problem) in which there is net zero acceleration, combining the pull of gravity and the forces that would be felt due to the continually changing direction alone.

A sudden change in absolute direction could be due to some alteration in the fabric of space, but even very similar orbits rarely trace the exact same conic sections. Though there are at least two imaginary cones that could intersect the orbital plane exactly along any given orbital ellipse, the dimensions and directions of different orbital cones will be unlikely to have coincident 'bases' (i.e. to be parallel, even discounting the question of what their distances must be from their respectively chosen conic points). If the point of orbital discontinuity was different for every individual orbit that was taken, then any component not firmly connected to the satellite (and not positioned exactly at its centre-of-gravity) would be required to experience (at the very least) a slightly different moment at which it is suddenly expected to start moving in a different direction.

If the change in direction is instead due to a commanded manoeuvre, the applied thrust necessary to change orbit (and, for a time, maintain a straight trajectory even through the curved gravity well) is both wasteful of resources (compared to the normally completed orbit) and requires a rather sudden and obvious change of momentum to the whole craft.

Whatever the reason behind the diversion, the result would be extremely uncomfortable for an astronaut in a crewed spacecraft. The transition from experiencing freefall/microgravity to suddenly being out-of-synch with the ship's momentum (whether just momentarily, twice each orbit, or for extended periods as continual corrections are made) would be disruptive. Such an extreme and sudden change of direction would involve a very large G-force, to a degree that may be not merely uncomfortable, but potentially dangerous.

We also aren't given any indication of how the 'radial' velocity of the craft might be intended to change during the 'flat' phase, such as if it obeys a suitably modified version of the constant 'area sweeping' rule, as for the elliptic part of the path, or instead perhaps attempts to maintain a constant relative velocity to take the same time to cross the new path as it otherwise would. The consequences of any of these might add further difficulties to the operability of a satellite and/or discomfort to any occupants.

Transcript

This is one of 35 incomplete transcripts: Don't remove this notice too soon. If you can fix this issue, edit the page!

[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. This orbit is similar in shape to an ellipse, except it has two corners and a straight edge on one side, giving it a hill-like appearance.]

[Caption below the panel:]

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

  Add comment        Create topic (use sparingly)       Refresh 

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)
        Add comment