Navigation Course in Mathematical Examples – Utilization of the Spherical Geometry

DOI 10.17818/NM/2017/3.9 UDK 656.052.1:514 Professional paper / Stručni rad Rukopis primljen / Paper accepted: 27. 6. 2017. Lenka Juklová Department of Algebra and Geometry Palacky University at Olomouc e-mail: lenka.juklova@upol.cz Dana Smetanová Department of Informatics and Natural Sciences Institute of technology and Business České Budějovice e-mail: smetanova@mail.vstecb.cz Marie Chodorová Department of Algebra and Geometry Palacky University at Olomouc e-mail: marie.chodorova@upol.cz


INTRODUCTION / Uvod
Euclid described the five basic axioms of geometry in his work Elements around the year 300 BC.Today, they are called the axioms of Euclidean geometry.You could say that Euclidean geometry describes the flat world.The description of the properties of Euclidean space gave rise to analytical geometry which facilitates approximations of the description of local properties of space and their applications using the Cartesian coordinate system.Such a description can be widely applied in solving various technical and physical issues, travel itineraries, etc.
Considerations over the validity of Euclid´s fifth axiom (the parallel postulate) changed significantly the view of the geometric perception of the world and influenced the modern development of geometry.Its disproof allowed the description of a world which is not flat (the so-called non-Euclidean geometry).The origin of non-Euclidean geometries falls into the first half of the 19 th century and is connected with the names Bolyai and Lobachevsky.The non-Euclidean description of the world is dealt with by e.g.spherical and differential geometry.The development of knowledge of non-Euclidean geometry space introduced a number of applications.One of them is the description of curved space (Earth, the Universe …).
Spherical geometry is applied in the area of planning of long journeys on the Earth and mostly in naval and air navigation (c.f.[3], [4], [8], [9], [11] and [12]).Differential geometry studies curvature, transmission of properties by their display, geodetic curves and other properties of space (c.f.[2], [10]).For example, geodetic curves are curves which have zero geodetic curvature at each point.In other words, they are the curves along which movement is the "most economical" (time, distance or energy).Geodetic curves on a sphere are used in naval navigation as orthodromes (c.f.[9]).The methods of differential geometry allow to study the movements in long distances and are applied in air and naval navigation, in descriptions of the Universe (c.f.[2], [9], [10]).One of the best known methods of application of differential geometry is Einstein´s General Theory of Relativity.
The paper presents exercises (instructions and solutions) from naval navigation which applies the methods of spherical geometry.Let us note that the method of solution is presented in a manner comprehensible for secondary school graduates.The exercises may be used to explain the basic principles of naval navigation.

SPHERICAL GEOMETRY AND ITS APPLICATION TO NAVIGATION COURSE / Sferna geometrija i njezina primjena na navigacijski kurs
Earth's surface can be considered as spherical surface with the radius r = 6371km.The coordinates of points on the sphere are determined in the usual manner.Each point on the sphere (with the exception of the poles) has two geographic coordinates -the latitude and longitude.The latitude is denoted φ and longitude is denoted λ, evidently ; the coordinates of the point A (on the sphere) are written in the form A = (φ, λ).The system of latitudes and longitudes generates an orthogonal coordinate network of parallels and meridians on the globe.Every main circle is the geodetic curve on the sphere.Finding the shortest route from the place to the place B is a basic task of spherical geometry.This route (i.e. the shortest route from A to B in the sentence above) is the length of the arc (≤ π) of the main circle which intersects points A, B.
The voyage time t is computed from the known form , where v is average speed.Therefore The angles α, β are computed from equations (6).The side c of the spherical triangle ABC has the smallest deviation from the right angle, therefore only the angle γ belongs to a quadrant other than it's opposite side.We obtain sinα 0,4463 and α 0,466rad 27°46´26'', sinβ 0,55 and β 0,5823rad 33°21´51'' .Navigation course on arrival is π -β 146°38´09´´.
b) Now let's consider the spherical triangle AMC, spherical distance of points M, C is is denoted as the inner angle at vertex C and x as the side opposite to angle (Fig. 2).The equations ( 4 It means that the longitude of the point M is λ 3 = λ 1 + = -0,5508 -31°33´31´´, i. e. 31°33´31´´ of the west longitude. Result: The ship crosses the equator at the point M whose -longitude is 31°33´31´´W and its distance from the point A is 2840km (1533,35NM).
Example 2. 3 The ship still sails on the main circle and it crosses the equator at the angle α = 50° and the parallel l of latitude φ = 15° at an angle ψ.

Compute the angle
Solution: A is denoted as the point in which the ship crosses the equator e, B as the point in which the ship crosses the parallel l, m as the meridian of the point B. We can consider the right spherical triangle ABC, where (Fig. 3).Clearly α = φ and φ = 15° 0,2618 rad, α = 50° 0,8727rad are given.

Definition 2 . 1 1 ) 2 . 1 Remark 2 . 2 Theorem 2 . 3
Let A, B be points on the sphere and k the main circle passing through points A, B. The size of the central angle (≤ π) relevant to arc AB of the main circle k is called the spherical distance of points A, B. Definition 2.2 Let A, B, C be three different points on the sphere.The triangle ABC consisting of arcs of main circles relevant to central angels less than or equal to π is called the spherical triangle.The spherical distances a, b, c of points B, C; A, C; A, B, respectively, are called the sides of the spherical triangle ABC.The angles between arcs in the point A, , in the point B, , in the point C are denoted α, β, γ, respectively and they are called the inner angles of the spherical triangle.The spherical triangle with at least one inner right angle is called the right spherical triangle.Definition 2.3 Let's consider the point A on the sphere and the rotation, which turns the north point of horizon to the east.The navigation course is the angle φ of this rotation.The rotated point indicates the direction of movement in a given navigation course.The angle φ measures from 0° to 360°.Definition 2.4 The number |π -φ| is called deviation of angle φ (or deviation of angle φ from right angle).The following theorems hold for spherical triangles: Theorem 2.1 (Spherical sinus theorem) Let ABC be the spherical triangle, a, b, c be the spherical distances of points B, C; A, C; A, B and α, β, γ be the inner angles of spherical triangle ABC, respectively.Then the following forms are valid (1) (Remark If the spherical triangle ABC is the right triangle (where ) then equations (1) have the form (2) Theorem 2.2 (Spherical cosinus theorem) Let ABC be the spherical triangle, a, b, c be the spherical distances of points B, C; A, C; A, B and α, β, γ be the inner angles of spherical triangle ABC, respectively.Then the following forms are If the spherical triangle ABC is the right triangle (where ) then equations (3) have form aLet ABC be the spherical triangle, a, b, c be the spherical distances of the points B, C; A, C; A, B and α, β, γ be the inner angles of the spherical triangle ABC, respectively.Only one of the inner angles at most belongs to a quadrant other than the opposite side.It is the angle, which is opposite the side with the smallest deviation.

Example 2 . 1
Let A, B be two places on the globe given by the latitude and longitude, A = (φ 1 , λ 1 ), B = (φ 2 , λ 2 ).Determine the shortest route d from A to B. Solution: The spherical distance of these points has to be found and the arc length d is computed -the shortest route from A to B. They are three trivial cases -one of the points is the pole , both points have the same longitude or the absolute value of the difference of longitudes is π.In these cases, the points A, B lie on the same meridian or on the opposite meridians.Than the spherical distance of points A, B is equal to |φ 1 -φ 2 | and d = r .|φ 1 -φ 2 |.Now, suppose points A, B are not in the above-mentioned special positions.Let's consider the spherical triangle ABC, where C is the North Pole.Denote α the spherical distance of points B, C, b the spherical distance of points A, C and c the spherical distance of points A, B. Clearly (and with respect to Definition 2.2 0 < a < π, 0 < b < π).The angle γ = |λ 1 -λ 2 | if 0 < |λ 1 -λ 2 | < π or γ = 2 π -|λ 1 -λ 2 | if π < |λ 1 -λ 2 | < 2 π (Fig. 1).It means that 0 < γ < π and cos γ = cos|λ 1 -λ 2 | = cos(λ 1 -λ 2 ).