the arc one is wrong as that penetrates the surface |
I suggest that anyone who wants to debate question 1 should read these:
http://en.wikipedia.org/wiki/Great_circle To prove that the minor arc of great circle is the shortest path connecting two points on the surface of a sphere, one has to apply calculus of variations to it. http://mathworld.wolfram.com/GreatCircle.html The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. http://en.wikipedia.org/wiki/Great-circle_distance Through any two points on a sphere which are not directly opposite each other, there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is the Riemannian circle. Or for those wishing to concentrate on the definition of an arc http://en.wikipedia.org/wiki/Arc In geometry, an arc is a closed segment of a differentiablecurve in the two-dimensional plane; for example, a circular arc is a segment of the circumference of a circle. If the arc is part of a great circle (or great ellipse), it is called a great arc. |
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