Polar triangle

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A spherical triangle has a polar triangle emerging from it , the corners of which are the so-called poles of the triangle points. Various relationships can be derived from it, such as B. the angle cosine law from the (usual) side cosine law .

A pole of a great circle means the two points of intersection of the circular axis with the carrier sphere. The pole of a triangle point is the pole of the great circle of the other two triangle points that is on the same side.

The name "polar triangle" is based on the name given on the globe : the two poles of the equator - which corresponds to a great circle in a special position - are the north and south poles .

A polar arc can be assigned to the intersection angle α of two great circles by connecting the “outer” poles of the two great circles with a great circle arc a * . If this is carried out for all 3 angles, then each original triangle angle β corresponds to a polar b * , and conversely, each side of the triangle b corresponds to a polar angle β , with which the polar triangle can be defined: It consists of the three polar arcs a *, b * and c * the three angles α, β and γ .

With a formula, one obtains relationships of the form a + a * = p and then a spherical theorem (e.g. the cosine law of the 3 sides a, b, c and an angle) can be converted into its equivalent (the cosine law of the 3 angles and one page).

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