Spherical astronomy: The Triangles – I

We discussed the great and small circle in our post about the circles in spherical astronomy. A brief summary for the previous post is as follows,

1. The great circle is one of the most fundamental concepts.
2. When a plane intersects a sphere at its mid-point, it forms a great circle.
3. One, and only one great circle can uniquely intersect two points on the surface of a sphere that are not antipodal.
4. There are an infinite number of great circles that can intersect antipodal points.
5. The great circle arc between two points is the shortest distance on the sphere between these points.
6. A small circle is formed if the plane does not intersect the sphere at its center.

Spherical triangles

A spherical triangle is a figure formed, on the surface of a sphere, by three intersecting great circles. The great circles are circles on the surface of a sphere that have the same center as the sphere. The internal angles of a spherical triangle are less than $180^\circ$, while the sum of the angles in a triangle on a sphere is always greater than $180^\circ$ due to the curved surface of the sphere.

What are they used for?

Spherical triangles have a variety of uses in mathematics and physics. Some examples include:

1. Navigation: Spherical triangles are used to determine the position of a ship or aircraft relative to a known point.
2. Astronomy: Spherical triangles are used in astronomy to calculate the position of celestial bodies relative to the observer.
3. Geodesy: Spherical triangles are used in geodesy to measure and map the Earth’s surface. Geodesy is the study of the Earth’s shape and gravity field.
4. Geography: Spherical triangles are used in geography to measure and map the Earth’s surface.
5. Physics: Spherical triangles are used in physics to calculate the properties of waves propagating through a spherical medium.
6. Robotics: Spherical triangles are used in robotics for kinematic and dynamic analysis of manipulator arms.
7. Computer Graphics: Spherical triangles are used to render 3D models of spheres. They are also used for simulating the movement of objects in a virtual 3D space.
8. Geometry and Trigonometry: Spherical triangles are also used in geometry and trigonometry to understand and solve problems involving curved surfaces, and complex trigonometric identities.

Basic properties

1. Sum of Interior Angles: The sum of the interior angles of a spherical triangle is always greater than $180^\circ$. The exact amount by which the sum exceeds $180^\circ$ depends on the size and shape of the triangle.
2. Triangle Side Inequality: The sides of a spherical triangle must satisfy the triangle inequality. It states that the sum of the lengths of any two sides must be greater than the length of the third side.
3. Congruent Spherical Triangles: Two spherical triangles are congruent if they have the same size and shape, meaning that their corresponding sides and angles are equal.
4. Law of Cosines: The law of cosines can be used to find the lengths of the sides of a spherical triangle given the angles between them. This is the spherical equivalent of the familiar plane law of cosines.
5. Law of Sines: The law of sines can be used to find the angles of a spherical triangle given the lengths of its sides. This is the spherical equivalent of the familiar plane law of sines.
6. Area of a Spherical Triangle: The area of a spherical triangle can be calculated using the spherical excess formula. It takes into account the sum of the interior angles and the size of the triangle on the sphere.