3189: Conic Sections

2026-01-21T15:49:30Z

2620:72:0:60C:C46F:9E8E:9110:51CA:

{{comic
| number = 3189
| date = January 2, 2026
| title = Conic Sections
| image = conic_sections_2x.png
| imagesize = 288x322px
| noexpand = true
| titletext = They're not generally used for crewed spacecraft because astronauts HATE going around the corners.
}}

{{incomplete|This explanation is full of jargon from both orbital mechanics and geometry, making the explanation unnecessarily complicated. It buries the punchline in the middle of a discussion about the two-body problem, moment inertia, and general relativity, none of which are relevant to this comic.}}
==Explanation==
A {{w|Kepler orbit}} describes the simplified motion of one celestial object relative to another. Such an orbit will form a {{w|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 {{w|Relativistic angular momentum#Orbital 3d angular momentum|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.)

[[File:TypesOfConicSections.jpg|thumb|alt=Example conic sections|How conic sections emerge from various planar intersections with bidirectional cones, which technically continue beyond the 'top' and 'bottom' of each diagram.
<hr/>1) Plane intersects perpendicular to conic axis, results in a circular line (often counted as an ellipse of zero eccentricity) around one cone.
<br/>2) Plane intersects at a small angle away from the perpendicular, results in an elliptic line around one cone.
<br/>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.
<br/>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.
<hr/>In this comic, the orbital shape is similar to the one in figure 3 (a {{w|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 {{w|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 {{w|Delta-v|applied thrust}} necessary to change orbit (and, for a time, maintain a straight trajectory even through the curved {{w|gravitational field|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 {{w|Automan#Features|sudden change of direction}} would involve a very large G-force, to a degree that may be not merely uncomfortable, but potentially dangerous.{{Citation needed}}

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 {{w|Kepler's laws of planetary motion#Second law|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.

{{clear}}

==Transcript==
:[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.
{{comic discussion}}<noinclude>

[[Category:Space]]
[[Category:Geometry]]

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3189: Conic Sections