Why is the intersection of the Earth’s equatorial plane and the ecliptic important in astronomy?
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The ecliptic is the path in space that we see traced out as the earth rotates around the sun.  The angle between this plane and the equatorial plane is what gives us the seasons.
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Well I'm not a geometer ... so I may err slightly here as i only had one non-euclidean geometry class and going over some links has only refreshed some of it.\n\nRiemannian geometry is otherwise known as elliptic geometry :\n"It can be visualized as the surface of a sphere on which "lines" are taken as great circles. In elliptic geometry, the sum of angles of a triangle is > 180 degrees"  [this last part taken from mathworld]. It is a "non-euclidean" geometry "with positive curvature" which replaces the "parallel postulate" with *through any point in the plane, there exist no lines parallel to a given line.*\n\ntaken from wikipedia :\nSpherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. This is of great importance for calculations in astronomy and earth-surface and orbital and space navigation.\n\nSo one could say ... paraphrase the above defn of spherical trigonometry mentioning how it is a study of the relations of polygons, angles etc on objects under (or in) a riemannian geometry system\n\nincluded below are some of the links so you can formulate a better answer than mine :) gain a better understanding and even give examples that you find in those links and related links :)\n\nenjoy :),\nTom
337086
The planets are actually trying to move in a straight line at a high speed due to the intertia from some force long ago.  When two objects are moving by each other, their gravity interacts and pulls them toward each other, bending the path that would otherwise be a straight line.  If they are too massive and too close, the gravity pulls them so much that they collide.  If things are balanced just right, the path is bent into a complete ellipse.  It's an ellipse instead of a circle because the sun is actually also being pulled a tiny bit toward the earth, as well.\n\nYou could consider gravity to be sort of a "perpendicular inertia", and the planets would revolve around the sun FOREVER if there were no other forces involved.
385100
The volume of an ellipse is a calculus problem. The fact that it's eggshaped isn't enough. There are hundreds of varying ellipses, all egg shaped.
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It is due to the slight orbital eccentricity of Earth rotation around the Sun. Since the orbit is not perfectly circular the distance between the two bodies varies.
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Ellipse is for positive curvature and hyperbola for negative curvature.
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A is correct, the moon muct be full, meaning the earth is between the sun and moon.\n\nThe other condition necessary is that the moon must be on the ecliptic.  The orbital path of the moon is tilted five degrees from the ecliptic, which is the plane that holds the orbit of the earth about the sun.  If the moon is above or beow the ecliptic, then the earth's shadow will miss it.  In the summer and the winter, there are two times when the moon crosses this ecliptic.  If there's a full moon in that time, then ther's a good clance of a lunar eclipse.  If there's a new moon in that time, there's a good chance of a solar eclipse.
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because the earth rotates on an axis
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if you mean an ellipse....\n\narea is equal to pi * minor radius * major radius
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It is the imaginary sphere on which all celestial objects are plotted. It is basically a map of the sky as seen from the Earth. Something like a globe, but seen from the inside. Knowing the coordinates of a star or planet on the celestial sphere tells you which way to look to see it but does not tell you how far away it is.
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Well you can't really have a center of the diameter line of a sphere.  But I guess the common dividing point would be where the prime merdian (zero degrees longitude) passes through the equator.  The intersection point is in the eastern Atlantic ocean, south of Ghana in Africa.
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the second proposition is a formulation of Euclid's postulate of the paralleles.\nIt's precisely a postulate, as the same Euclid posed it; rinascimental mathematicians tried to prove it as a theorem, but they failed: in fact Euclid knew well what he was doing.\nThe postulate of the paralleles defines the Euclidean geometry, which is "platonic": it needs abstraction, because you can't apply it on the spherical surface of the Earth.\n\nSo the answer to your first question is no: if you assume spherical geometry, and thus parallels as maximum circles on the sphere, the angles postulate doesn't work anymore.\n\n(Notice that ancient Greeks were able to measure the Earth's radius by astronomycal calculations: so they knew very well how the Earth was not planar ... )
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If you imagine a solar system that includes the sun and the Earth,and no other bodies, the Earth would travel about the sun in an ellipse (strictly speaking, the Earth and the sun would travel around their common center of gravity in ellipses).  An ellipse has two foci, which are points that define the ellipse in the same way that a point defines the center location of a circle.  Because the sun is so large and the orbit is so nearly circular, both of the foci lie near the center of the sun  So it is not correct to say that one of the foci is the sun -- both foci are in the sun, but they're really just mathematical points that define the shape of the orbit.\n\nWhen you add more planets to the mix, things get much more complicated, but one can think of each approximately elliptical orbit separately, and then there is a different set of foci for each pair of bodies if one neglects the influence of the other bodies.
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The path of the apparent motion of the sun is called ecliptic.\n\nThe lunar orbit is at angle of 5 degree 8 minutes to the ecliptic.\n\nThe two points of intersection of the lunar orbit and ecliptic are called the nodes.\n\nThe sun comes to the same node every 346.6 days.\n\nHence it is clear that the sun goes to north and south directions  with respect to the moon also.
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the tilt of the earth and when it revolves around the sun
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Hi. A good one might be Regulus because it lies pretty close to the ecliptic and is very bright. But you can use any star. For much better accuracy scientists use quasars due to their vast distance. This eliminates parallax error.
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The axis of earth is tilted 23.45 degree from the ecliptic; and the orbit of the earth around the sun is not a perfect circle (it is an ellipse, and thus the distance from earth to the sun changes over the course of a year)