Spherical 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.

Hence, a sphericaltriangle is specified as usual by its corner angles and its sides, but the sides are given not by their length, but by their arcangle.

The identity may be derived by considering the triangles formed by the tangent lines to the sphericaltriangle subtending angle C and using the plane law of cosines.

A capital letter is customarily used to designate a vertex of a triangle, the angle at that vertex, or the measure of the angle in angular units; the corresponding lower case letter designates the side opposite the angle or its length in linear units.

Therefore, the three angles of a scalene triangle are of different sizes; the base angles of an isosceles triangle are equal; and the three angles of an equilateral triangle are equal (an equilateral triangle is also equiangular).

For example, the sum of the angles of a sphericaltriangle is between 180° and 540° and varies with the size and shape of the triangle.

Spherical trigonometry, which is used principally in navigation and astronomy, is concerned with sphericaltriangles, that is, figures that are arcs of great circles (see Navigation) on the surface of a sphere.

In the right-angled or quadrantal triangle, however, as in the case of the right-angled planetriangle, only two elements are needed to determine all of the remaining parts.

Thus, given c, A in the right-angled triangle, ABC, with C = 90°, the remaining parts are given by the formula as sin a = sin c sin A; tan b = tan c cos A; cot B = cos c tan A.

Spherical 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.

The sides of these polygons are most conveniently specified not by their length but by the angle under which its endpoints appear when looked at from the sphere's center.

Hence, a sphericaltriangle is specified as usual by its corner angles and its sides, but the sides are not given by their length, but by their arcangle.

The capital letters (A, B, C) denote the angles between the great circle arcs of the triangle as measured on the surface of the sphere.

A sphericaltriangle, differs from a planetriangle in that the sum of the angles is more than 180 degrees.

The Cosine Rule allows the length of one of the arcs of a sphericaltriangle to be evaluated if the other two arcs and the angle opposite the arc are known.

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Determine the arc *distances* d1, d2, d3 along the three sides of the "standard" sphericaltriangle which uniquely identifies the location of the point P2.

Because each of the 20 sphericaltriangles are the same shape and size, we only need to understnad how to transform an arbitrary point within a single triangle from (longitude, latitude) to (x, y).

So, the first step of the algorithm is to determine and remember which of the 20 sphericaltriangles the point is within.

Thus the area of a sphericaltriangle on a unit sphere is equal to the spherical exess, which is the sum of vertex angles in excess of π radians.

This relationship for the area of a sphericaltriangle generalizes to convex sphericalpolygons with the spherical excess being the sum of the angles - (n-2)π, where n is the number of sides of the polygon.

A side of a sphericaltriangle is the intersection of a plane passing through the center of a sphere with the surface of the sphere.

The area of the spherical cap is the integral, from zero to alpha, of the circumference, with respect to alpha.

Then the volume of the spherical cap is the difference of the volumes; that is, the volume of the cone and the spherical cap less the volume of the cone.

The radius of the circumscribing circle of the triangle is rho sin(alpha).

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SPHERICAL TO PLANAR TRANSFORMATION(Site not responding. Last check: )

To illustrate the model, the icosehedral sphericaltriangle and the equilateral planar triangle are each divided into six similar triangles by projecting the three perpendicular bisectors.

To transform a point P on the Icosahedral sphericaltriangle, multiply the distance of the point P from the centroid, (=Dg from Rotation 3, Step 5), by the Amplitude factor and add the Angle factor to the bearing of the point P, (=B"' from Rotation 3, Step 7), from the perpendicular bisector.

This paper explained a scheme using spherical coordinate notation to transform the spherical facets of a regular icosahedron to planar facets suitable for plotting on an X-Y plotter.

Spherical Trigonometry(Site not responding. Last check: )

The sphericaltriangle in the diagram above is the same as the trihedral, except that the sides a, b and c are great circle paths on the surface of the sphere centred at V rather than straight lines (as in the trihedral).

vik dhillon: phy105 - spherical geometry - the geometry of the sphere

Hence, in Figure 2, triangle PAB is not a sphericaltriangle (as the side AB is an arc of a small circle), but triangle PCD is a sphericaltriangle (as the side CD is an arc of a great circle).

which shows that the length of a side of a sphericaltriangle is equal to the angle (in radians) it subtends at the centre of the sphere.

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The projection of these triangles onto the surface of the sphere is known as spherical tessellation.

R) projecting from the common centre of the cube and the sphere to the vertices of the sphericaltriangle.

The surface area of the triangle being projected onto the sphere is one eight of the total surface area of the top face which is broken into eight equal triangles each having a vertex at the centre of the face.

Thus, the maximum number of equilateral triangles comprising a spherical surface is 20 which is related to the icosahedron (formed by connecting the vertices of each equilateral triangle with a straight line).

An examination of the sides of the smaller triangles in the least common denominator (LCD) triangle indicates that there are 22 different triangles for the frequency of 10 and each triangle in the LCD is different.

Thus, the number of different triangles is proportional to the square of the frequency and is approximately the frequency squared divided by six.

The sides of a sphericalpolygon are measured in degrees in the same way as a sphericalangle, because the sides of a sphericalpolygon are always arcs of great circles--all great circles on the same sphere have the same size and length.

In the nine-fold subdivision method, the three edges of a triangle are trisected, and the trisection points as well as an inner center point are connected with one another by small circles, producing nine sub-triangles within the triangle.

Denote the vertex angles of the sphericaltriangle by ÐA, ÐD and ÐF.

In the nine-fold method, all three edges of a triangle are trisected and the trisection points, as well as a central inner point, are connected by small circles, producing nine sub-triangles within the parent triangle.

Upon connection of sides of the spherical regular triangles and spherical regular pentagons, six great circles are formed, and since one of the great circles is overlapped with a parting line of a split metallic mold, dimples cannot be arranged on the great circle.

When one of the spherical squares is taken up for consideration, the dimples D to be disposed therein should be arranged to provide a good symmetrical characteristic as far as possible, and no dimples can be arranged on the sides of the spherical square.

In the case of the sphericaltriangle, the number of dimples to be arranged therein becomes 3n (n is a natural number) or 3n+1 in the similar manner as in the case of the sphericaltriangle of the icosahedron-dodecahedron arrangement referred to earlier with reference to FIGS.

There seems to be a real need for a short yet rigorous text in spherical trigonometry which shall contain little more theory than is needed for the solution of sphericaltriangles, with a few applications to add interest to the subject.

A knowledge of spherical geometry is not presupposed, but a brief presentation of the principal concepts and theorems is given in Chapter I. Most of the theoretical examples bear directly on the development of the theory in the text.

The time required for the discussion and solution of the general sphericaltriangle may be reduced by half by considering only Cases 1, 2, and 3 in Chapter III, explaining how the other cases may be solved by means of the polar triangle.

Similarly, to perform calculations with sphericaltriangles it is necessary to use the formulae of spherical trigonometry, which are given below.

a sphericaltriangle with arcs of length (a,b,c) and vertex angles of (A,B,C).

Note that the angle between two sides of a sphericaltriangle is defined as the angle between the tangents to the two great circle arcs, as shown in Figure 4 for vertex angle B.

The pair of triangles with an apex at one of the centers and bounded by the cord are congruent by SAS.

This is an isosceles triangle, with the vertex at the center of the polygon.

For each law, we give the spherical (a, b, c are the sides; A, B, C are the angles), its dual for the polar triangle (A. C are the angles; a, b, c are the sides), and the plane (a, b, c are the norms of the sides; A, B, C are the angles) version.

Spherical geometry is defined as "the study of figures on the surface of a sphere" (MathWorld), and is the three-dimensional, spherical analogue of Euclidean or planar geometry.

The sphericalangle formed by two intersecting arcs of great circles is equal to the angle between the tangent lines formed when the great circle planes touch the circle at their common point (an antipode of the sphere since two great circles intersect each other in a line passing through the sphere's centre).

Given two sides of a sphericaltriangle and the angle between these sides, the solution for a sphericaltriangle yields the length of the third side.