Spherical Geometry I

In this chapter, we show that geodesic curves on the sphere are great circular arcs, and we verify that the cylindrical projection preserves the areas of figures. Girard’s formula for the area of a spherical triangle is also proved. From the area formula for spherical polygons obtained by applying Girard’s formula, Legendre’s proof of Euler’s polyhedral formula is derived. The theorem on inscribed angles is presented, and the notion of polar set is introduced.