How can Spherical Geometry be Applied to Solve Real World Problems?IB Math SLCandidate Number:Logan SorrelsCheck turnitin.comAlthough it had not been fully explored in the classroom, for my math exploration, I chose spherical geometry as a topic for its interesting properties and applications in the real world. Architecture has always been a fascinating subject to me … alternative purposes of spherical geometry in architecture aside from the aestheticsDianglesA great circle lies on a plane that passes through a sphere’s center and divides the sphere in half. Two different great circles may intersect on the surface of a sphere at two antipodal positions which create a diangle, otherwise known as a lune. Antipodal is defined as diametrically opposite.
It is called a diangle as the lune contains only two angles within itself.Figure 1Figure 2Girard’s TheoremSpherical triangles formed by the intersection of three great circlesThe most common method of solving for the area of a spherical triangle may be seen using Girard’s formula where T is the triangle, r is the radius of the sphere, and A, B, and C represent the interior angles.T=r2(A+B+C?12?)Figure 3Figure 4Show limitations Aside from the Gauss-Bonnet theorem, which can be seen below, another alternative proof exists.Figure 5If we label points A,B,C on a spherical triangle as the vertices of a unit sphere with center D, it is then possible to consider the tetrahedron DABC. Letting ?A be the dihedral angle between the planes from face DAB and DAC, this coincides with edge DA.
Similar to ?A, let us assign ?B and ?C to their same edges DB and DC. By summing ? to the amplitude of the trihedral solid angle (a trihedral angle is one solid angle that is bounded by three plane angles) corresponding to a vertex, one property of tetrahedra is that it is possible to get the sum of the three dihedral angles which originated from that specific vertex. In this relation below, ?i is the four trihedral angles and ?i represent all six dihedral angles.
Figure 6We may then apply this to our tetrahedron DABC where the three dihedral angles are measured in radians and ? is measured in steradians.?+?=?A+?B+?CIt is now possible to show that angle ?BAC of spherical triangle ABC may be defined by the two planes passing through the center D and containing the sides AB as well as AC, such that it is equal to the dihedral angle between the tetrahedral faces DAB and DAC. Thus, we have ?BAC=?A?BAC=?A. Similarly, we get ?ABC=?B?ABC=?B and ?ACB=?C?ACB=?C. We can then rewrite the equation above as?=?BAC+?ABC+?ACB???=?BAC+?ABC+?ACB??where the RHS corresponds to the spherical excess of the Girard’s formula.
Reminding that the amplitude in steradians of a solid trihedral angle in the unit sphere is by definition given by the portion of the spherical surface subtended by the angle itself, we directly obtain that ?? is equal to the area AA of the triangle ABCABC, and soA=?BAC+?ABC+?ACB??Don’t put “reflection” section at the end of the paper. Reflect throughoutApplicationsWhere real world applications are concerned, spherical geometry is able to assist in multiple areas. Considering that the Earth is almost a sphere (The Earth is, in actuality, an oblate spheroid), spherical geometry is used in navigation constantly. With both planes and ships, spherical geometry is able to calculate the most efficient route to the destination.
The reason for applying spherical geometry to assist with navigation instead of Euclidean geometry is the result of Euclidean geometry making it impossible to join two points on a sphere together without creating a line that would travel into or under the globe itself. Large triangles, such as those that span city lengths apart, have angles much greater than 180 degrees which makes it possible to locate the optimal travel path through finding the great circle(s) which lie between those points.Figure 9Spherical Geometry has also been a revolutionizing factor in terms of architectural design and its expandable possibilities into the realm of form versus function. Coming in at 180 meters tall with forty floors, the 30 St.
Mary Axe located in London is a prominent example as the engineers managed to apply non-Euclidean geometry in order to reduce wind pressure and drafts that occured at street levels with skyscrapers. The tower also manages to use an advanced double glass technology on the exterior which cools in the summer and insulates in the winter. Alternative uses of spherical geometry include manufacturing specific shapes in computational origami with computer-aided design or CAD machines, nanotechnology in chemistry through the use of hexagons and pentagons to produce spherical shapes in order to calculate bond angles between molecules.
In the future, the applications of spherical geometry may only expand into an even broader spectrum of fields such as space exploration and foldable satellite wings.