GEOMETRIE SFERICA

Frunza Paul Adrian

4 iulie 2012

I Este o forma de geometrie non-Euclidiana mai intuitiva decatgeometria hiperbolica in privinta figurilor si mai putin intuitivain privinta teoremelor. Marele matematician german BernhardRiemann este creditat cu observarea ca geometria sfericapoate fi o alternativa la geometria euclidiana.

I In geometria plana conceptele de baza sunt punctul sidreapta. Pe o sfera punctele sunt definite ın sensul uzual.Echivalentele liniilor nu sunt definite ın sensul uzual de liniidrepte, ci ın sensul celor mai mici drumuri dintre doua puncte,numite geodezice. Pe o sfera geodezicele sunt cercuri mari;alte concepte geometrice sunt definite ca ın geometria plana,dar cu liniile dreapte ınlocuite prin cercurile mari.

I Astfel, ın geometria sferica unghiurile sunt definite ıntredouacercuri mari, rezultand ca ın trigonometria sfericaunghiurile difera de cele din trigonometria plana ın multeprivinte; de exemplu, suma unghiurilor interioare aletriunghiurilor sferice este mai mare de 1800.

I Geometria sferica este cea mai simpla forma de geometrieeliptica, ın care o linie nu are paralele fata de un punct dat,contrastand cu geometria euclidiana, ın care o linie are oparalela fata de un punct dat si geometria hiperbolica, ın careo linie are dou paralele si un numar infinit de ultraparalele fatade un punct dat.

I O importanta geometrie legata de cea sferica este aceea aplanului proiectiv real, fiind obtinut prin identificarea punctelordiametral opuse pe o sfera. Acesta este un alt tip degeometrie eliptica. Local, planul proiectiv are toateproprietatile geometriei sferice, dar are proprietati globalediferite. In particular, este neorientabil sau cu o singurasuprafata, gen inelul lui Mbius. Conceptele geometriei sfericepot fi aplicate si sferelor alungite, cu toate ca trebuiesc facutemodificari minore anumitor formule.

I Geometria sferica a fost studiata din antichitate dematematicienii greci precum Menelaus din Alexandria, care ascris o carte de trigonometrie sferica numita Sphaericadezvoltand teorema lui Menelaus.

FLATLAND

If you object to calling these great circles ”straight lines” becauseyour 3-dimensional insight allows you to connect points by shorterpaths that are not confined to the sphere, then imagine yourself asa tiny being that lives on the sphere and does not project outsidethe sphere at all. Before trying to imagine this, you might thinkabout Flatland, a 2-dimensional world described in the classicbook, Flatland, a Romance of Many Dimensions, written by EdwinAbbott Abbott in 1884. Abbott was one of the leading scholarsand theologians of the Victorian era, and is best known for hisShakespearian Grammar and his biography of Francis Bacon. Theworld of Flatland is a plane inhabited by figures such as squares,triangles, and circles, whose motion and sight is completelyrestricted to this plane.

POSTULATES

It is quite difficult to set up this spherical geometry as a formalaxiom system. Of Euclid’s five postulates, two (numbers 2 and 5)must be changed drastically, two (the first and third) must bechanged slightly, and one (the fourth) remains the same. Thesubstitute for 1 is: ”Two points lie on a unique straight line unlessthose points are antipodal, in which case they lie on many straightlines.” Here we have introduced the word antipodal for points on asphere that are directly opposite from one another. Each point hasa single antipodal point.Spherelanders would not have the perspective to see points asdirectly opposite, and so they might just note that for every pointthere seems to be exactly one other point to which it is connectedby more than one straight line, and give that point a name such as”antipodal.”

In Figure 2.50a, points P and P’ are antipodal, and we

Figure : .

have pictured three straight lines passing through them; one is theequator, one is the circle that defines our drawing, and one is thegreat circle passing through Q and R. Only one straight line passesthrough Q and R, the indicated great circle. If we want to talkabout line segments between Q and R, we should consider theshort portion of the great circle that connects them, not the longportion that goes around the back of the sphere. This is one of themessy parts of axiomatizing.

PROPOSITIONS

We have seen that some of the propositions of Euclidean geometry,such as 1800 in triangles, and the existence of lines that areeverywhere equidistant, are almost true in spherical geometry whenviewed on a small enough scale. Some other propositions ofEuclidean geometry are actually true for small figures but false forlarge ones. For example, Proposition 1 should be replaced bysomething

Figure : .

like: ”For lines less than a certain length, there is an equilateraltriangle with that line as one side, but this is not true for lineslonger than that length.”

In Figure 2.53a, we are given a line connecting points A and B.Draw circles (not great circles) with one of the points A or B ascenter and passing through the other, and let C be a point ofintersection of these circles. The distance from A to C and from Bto C equals the distance from A to B. The equilateral triangle ABCis pictured in Figure 2.53b, slightly enlarged, using arcs of greatcircles (the straight lines in this geometry) to connect the points.If we try the same construction with a big line AB, as in Figure2.54a, the circles centered at A and B will not meet, and so there isno point C whose distances from both A and B equal the distancefrom A to B. Thus there is no equilateral triangle with AB as side.

Figure : .

AREA AND EXCESS

The most quantitative result measuring the deviation of sphericalgeometry from Euclidean is the following. THEOREM: On asphere of radius R, if a triangle has area A and angle sum S, then

A = π180 ·R

2·(S−180)This formula appeared in a book by Albert Girard in 1629, whichmakes it somewhat surprising that the prediscoverers of hyperbolicgeometry found the analogous results so strange. The amount S -180 by which the angle sum of a triangle exceeds 180 is called theexcess of the triangle.This formula makes it clear how the ratio is the same for alltriangles on a sphere, but the ratio can have any value, dependingon the size of the sphere.Note that a consequence of Theorem isthat triangles have the same angle sum if and only if they have thesame area.

In particular,there are no ”similar triangles,” triangles that have thesame angles but the sides of one are a multiple (6= −1)oftheother .Theorem can be used by Spherelanders to determine the radius oftheir sphere (provided they know that it is a sphere). If theymeasure a triangle to have area 10 square miles and angle sum180.10, they can find the radius of their sphere by solving theequation

10 = π180 ·R

2·(180.1-180)

for R, yielding

R =√

10 .1·180π=75.7miles