If the two points are on the same parallel of latitude, then I can't imagine a great circle through them. Is it right?
I believe you are having trouble imagining it because you are allowing the smaller circles formed by the lines of latitude themselves to constrain your thinking. Consider two points at roughly 45 degrees North latitude - One point is at 0 degrees longitude, and the other is at 180 degrees longitude.
Now how would you connect them?
You could connect them by following the line along 45 degrees North, but that would not follow the great circle. The great circle route would take you up and over the North pole.
If it helps, take your mental picture of the common latitude “slice”, use the two points as a fixed hinge, then in your mind imagine stretching this ring, and rotating it downwards until the imaginary plane slices through a point at the very center of the earth. These three points, (the center of the earth plus your two on the surface) define a plane that forms the largest (great) circle that can be made which intersects your points.
If your imagination is insufficient, get a bag of oranges, a pen, and a knife. Make two dots any where on the orange and then slice through them in a way that creates two equal sized halves. Keep going until you have freed your thinking.
ADDENDUM: Federico’s excellent answer provides food for though on the non-spherical nature of our planet. I was curious, so I ran some numbers for a comparison to put things in perspective:
If you circumnavigated the globe around the equator vs a polar route at 420 MPH, after almost two and a half full days continuously airborne the equatorial route would be a mere 6 minutes longer. That's a 0.1 difference in your logbook after almost 60 hours of flight time.
That's not much. Probably less than the average time spent taking vectors for landing opposite direction vs a straight-in. Just an FYI for anyone curious about the practical effect of the equatorial bulge from a pilot's point of view.
A great circle drawn along a sphere will, with the exception of the one drawn exactly along the equator, intersect all parallels of latitude it passes by exactly twice.
To make it intersect any two points on a specific parallel of latitude is simply a question of alignment of the great circle:
The earth, however, is not a perfect sphere so this answer does not perfectly apply to the case of the earth. See Federico's answer for more detail
The shortest path between any two point on a sphere follows a great circle. There is exactly one great circle for any pair of two non-identical points on the surface of a sphere, unless they are antipodal, where you have infinitely many great circles (think of the meridians, they go through both poles, and they are all the same). The notion of great circles is most useful when talking about spheres.
Earth has an irregular shape. A spherical approximation is useful and sufficient in very many cases. A better approximation that is indeed used and required for accurate calculations in aviation (like GPS processing) is an (oblate) ellipsoid shape. The shortest path between two arbitrary points on an ellipsoid does not follow a great circle.
Geodesics are the shortest path on a curved surface (e.g. sphere or ellipsoid) - like a straight line on a flat plane. A great circle is the shortest path on a sphere, but not on an ellipsoid (as long as both axes are not of equal length). So every great circle is a geodesic, but not every geodesic must be a great circle (on an ellipsoid for example).
This means that the shortest path between two points on the surface of the Earth is neither a piece of a circle nor even a piece of a closed loop, with the exception of
- two points that lay on the Equator (leading to an actual circle)
- one of the two points is one of the poles (leading to an ellipse)
- two points that differ in Longitude by exactly $180^\circ$ (again leading to an ellipse)
For all other cases, no matter their latitude, the shortest line connecting them will not be a closed line returning to its starting point after only one loop.
Can great circle be drawn between any two points on earth?
Technically speaking, no. Not with a strict definition of "Great Circle", and not with an accurate model of the Earth.
Consider geometric planes
Yes, and the simplest evidence for that reduces to the notion that you can draw a unique geometric plane between any 3 points not on the same line, so there exists a specific plane for any 2 points on surface of a sphere and its center.
By definition, the intersection of that plane and the sphere is a great circle, and it passes through these 2 points.
The question appears to be mathematical but also navigational implying a practical side to it.
Lines of latitude on a sphere are technically called "small circles" and it must be appreciated that two points on a sphere may lie on an infinite number of smaller but on only one great circle simultaneously (except antipodal points, of course). That's the mathematical answer.
Since your post is tagged "navigation", the shape of the Earth is not in actuality a sphere; there are several 3D models of the Earth's shape in use (the World Geodetic System defines them) and these shapes preclude a "great circle" as such. You can see the WGS quoted on ECDIS* charts used on ships. This shape is a parameter in the calculations used to plot a course to a destination. In such a system, a "great circle" must be redefined. On a practical note, however, if a ship should arrive with a nautical mile or so of the entrance to a harbour, the crew "done good" so there are limits to how much accuracy you need in calculation and Great Circle Navigation (where the Earth is treated as a sphere) is adequate. Get that: adequate! An autonomous ship would need tighter precision. A crewed ship would not. The crew spies the entrance and manually sails it into the harbour with a pilot on board. On aircraft, these types of electronic charts save navigators the sometimes mind numbing calculations required for great circle navigation (navigation where the Earth is assumed to be a sphere) which is often okay enough. In range of the airport an aircraft picks up the local radio navigation aids to land. The trouble with such systems is that crews should still KNOW how to calculate these routes; the equipment tends to take the effort out of it, which is fine. Until the electronics give out at which time the pilot/sea captain is on his own! Thus celestial navigation is STILL mandatory for Master's Licenses for sea. And crews should still know how to calculate them. Yachtsmen still practice them but GPS has been taking all the skilled effort of calculation out of even this culture.
Keep looking at these things, defining words and asking questions and you will find yourself knowledgeable beyond belief in the subject. And you'll never get lost and if by chance you do, you'll find out where you are soon enough. Good luck in your studies.
*Electronic Chart Data Information System.
Imagine taking a rope connecting the two points, and pulling it taut. If the Earth were a perfect sphere and there were nothing for the rope to catch on, the rope would eventually take the shape of a chord of a great circle.
I assume that you agree that given any point other than the North pole, there is a great circle route from the North pole to that point. This gives two more proofs that, given two points A and B, there is a great circle route between them:
Rotate the globe so that A is where the North pole was. By spherical symmetry, there is a great circle route between where A now is and where B now is. We can rotate the globe back to where it was before, and this route will rotate along with it.
Take a great circle route between the North pole and A. Now rotate that circle about the point A. As you rotate the circle, it will cover the whole globe, including B. That is, as you rotate the circle, it will hit B, and when it does, you have a great circle route between A and B.
I think you are getting confused using the lines of lat and long and cross referencing them against great circle routes. On a Sphere it does not matter where you are if you mark 2 spots on a Ball X and Y .There will be two arcing paths to each of these points both equidistant. In other words. The shortest distance between two points on a plane is a straight line on a sphere its two arcing lines that are the same length and equidistant to each other (Mirror Image).