So on a sphere, the shortest distance between two points is a portion of a great circle. A great circle is a circle which has its center at the center of the sphere (right?). If I’m picturing things correctly, on the Earth (which I’m taking to be perfectly spherical - just roll with it - like a sphere), at any point, if you travel due north or due south, you’re on a great circle at that instant. And if you start at the equator, and you travel in any direction, you’re on a great circle at that instant. But off the equator, some directions are not on a great circle (right?): If you’re not on the equator and you travel due east or west, that can’t be a great circle path.
My headache is that a sphere is symmetrical - so any direction can be defined as north (or am I missing something).
But if you go due east starting at a point on the Tropic of Cancer, you end up not on a great circle (because the Tropic of Cancer isn’t a great circle). That’s what’s confusing me.
Yeah, if you follow the tropic line, you aren’t on a great circle. There is a great circle line that starts off due east, but it traces a path that encircles the earth’s center so it deviates southward as it progresses. Half way around, on the other side of the world, it would be pointed due east along the Tropic of Capricorn briefly.
(So, at those two points – the starting point and halfway around – it would be a due-east path. Otherwise, it’s either got some south-ness or north-ness as well.)
Thanks. I guess it’s the rate of change of direction that matters (you can start in any direction as you say, but if you don’t change heading in the right way, you will leave the great circle path)
You can stand on the Tropic of Cancer, and start going due east along a great circle. But if you do that, you won’t stay on the Tropic of Cancer.
The easiest way to help think about this is to literally get a globe and some string, and stretch the string along the globe. Or tape; that works for a geodesic, too, including on non-convex surfaces: Just lay the tape out as smoothly as you can, and let the smoothness determine the direction the tape goes.
Ah. A circle on a sphere is the intersection between a sphere and a plane. A great circle is the intersection of a plane and the sphere such that the center of the sphere is on the plane. A non-great circle and a great circle can touch at only one point. So, “going east” from any point on the Tropic of Cancer (sticking with the non-great circle called the Tropic of Cancer) deviates from the great circle that also goes east from that point, immediately
In geodesy this is known as the orthodrome, and the arc length describe between two points as the ‘orthodromic distance’. This often confuses people when they are first learning geodesic navigation because in looking at a standard Mercator projection people are inclined to chart courses using straight lines which gives the ‘loxodromic distance’ as projected on a flat map. For most coastal cruising or sailing within a small region like the Mediterranean, or flying a light aircraft the deviation doesn’t really matter but when sailing or flying transcontinental distances the orthodromic line cuts out a lot of unnecessary distance assuming you don’t have any weather conditions or other obstacles to navigate around.
If you need really accurate orthrodomes (for, say, a ballistic missile trajectory or to hit a really precise set of osculating elements for an orbit) that can be quite complicated because the Earth isn’t actually a sphere (is approximated as an oblate spheroid with mass concentrations that create further deviations in the gravity field) so the ‘exact’ orthodromic path is route dependent and not necessarily a ‘straight’ geodesic line. Of course, you also don’t want to waste energy constantly shifting azimuth and contributing to uncertainty so in order to define the ‘best’ (lowest energy and least cumulative deviation) trajectory you need to quadratic (or sometimes higher order) regression.
The reason going north (or south) from a point is always an on an orthodrome is because all of the longitude lines converge at the poles. Of course, this is purely a convention; although we place the poles on the axis of rotation for convenience, they could actually be defined as a line in any orientation running through the geodetic centroid of the Earth, and the datum could be defined relative to the plane of rotation. This would make a lot of math really complicated but that is why William Rowan Hamilton made quaternions.
Very simple - if you go “due east” and maintain that compass heading, you are travelling on a line of latitude - i.e. on the Tropic of Cancer. If you travel due north-east, you actually spiral in to the pole, which you can see if you try to plot your course on a Mercator projection (Straight line is a constant compass heading) then try to transfer those points of travel onto a globe.
The flaw in the argument is if you make your current position the “north pole” then every direction is “due south” along a great circle. The resulting paths have no relation to what path you follow if you maintain an easterly compass direction from Miami or the Jersey Shore using the real north pole…
Basically, the issue is that we define north and south differently than east and west. We consider ourselves going north if we are moving towards a fixed point: the north pole. We do the same for south. Because of this single point, going north or south is going along a great circle.
But, for east and west, there is no poles. We just move in a direction that it orthogonal to both pole, maintaining an equal distance from both. This results in us traveling in smaller circles, and not ever reaching any fixed points. The circles are parallel, and thus only one (the one at the equator) is actually a great circle.
Actually, that’s a very helpful note. The deviation is very small - because the circles are tangent, having the same first derivative, but different second derivatives. So while moving along the non-great-circle, there’s a horizontal acceleration - which is the horizontal centrifugal effect of moving on a non-geodesic path. So I’ve accidentally come up with a general relativity fact (not even an analogy, really)
