Euclidean verses Non Euclidean Geometries. Euclidean Geometry

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1 Eulidean verses Non Eulidean Geometries Eulidean Geometry Eulid of Alexandria was born around 35 BC. Most believe that he was a student of Plato. Eulid introdued the idea of an axiomati geometry when he presented his 13 hapter book titled The Elements of Geometry. The Elements he introdued were simply fundamental geometri priniples alled axioms and postulates. The most notable are Eulid s five postulates whih are stated in the next passage. 1) Any two points an determine a straight line. ) Any finite straight line an be extended in a straight line. 3) A irle an be determined from any enter and any radius. 4) All right angles are equal. 5) If two straight lines in a plane are rossed by a transversal, and sum the interior angle of the same side of the transversal is less than two right angles, then the two lines extended will interset. Aording to Eulid, the rest of geometry ould be dedued from these five postulates. Eulid s fifth postulate, often referred to as the Parallel Postulate, is the basis for what are alled Eulidean Geometries or geometries where parallel lines exist. There is an alternate version to Eulid fifth postulate whih is usually stated as Given a line and a point not on the line, there is one and only one line that passed through the given point that is parallel to the given line. This is a short version of the Parallel Postulate alled Fairplay s Axiom whih is named after the British math teaher who proposed to replae the axiom in all of the shools textbooks. Some individuals have tried to prove the parallel postulate, but after more than two thousand years it still remains unproven. For many enturies, these postulates have assumed to be true. However, some mathematis believed that the Eulid Fifth Postulate was suspet or inomplete. As a result, mathematiians have written alternate postulates to the Parallel Postulate. These postulates have led the way to new geometries usually alled Non-Eulidean Geometries. Non-Eulidean Geometries In later 18 th entury Carl Friedrih Gauss beame interested in proving Eulid Fifth Postulate as a young teenage student. After attempt at proving the postulate, young Gauss deided to try a different route. Gauss then wrote what is alled Gauss s Alternate to the Parallel Postulate.

2 Gauss s Alternate to the Parallel Postulate Through a given point not on a line, there are at least two lines parallel to the given line through the given point. This alternate postulate was proposed to Russian mathematiian Nikolai Lobahevsky in 186. Lobahevsky beame very interested in the problem and provided a detailed investigation into the problem of the Parallel Postulate. He onluded that Gauss s Postulate was an independent postulate and it ould be hanged to produe a new geometry. Ironially, Lobahevsky alled the geometry an imaginary geometry beause he ould not omprehend the model for this type of geometry. Instead, he proposed that was possible to onstrut suh a geometry. For this reason, this type was referred to a Lobahevskian Geometry. Today, Lobahevskian geometries are often referred to as hyperboli geometries. Bernard Riemann Born in 196, Riemann was a student Gauss further studied Gauss s work on non- Eulidean geometries. Gauss was able to develop his own alternate to the Parallel Postulate. Reimann s Alternate to the Parallel Postulate Through a given point not on a given line, there exist no lines parallel to the line through the given point. Gauss s Alternate to the Parallel Postulate reated the idea of geometries where parallel lines are non-existent. The non-eulidean geometry developed by Gauss ould be model on a sphere where as Lobahevskian s geometry had no physial model. For this reason, Riemannian geometries are also referred to as a spherial geometry or elliptial geometry. Riemann also made several ontributions in areas of alulus and physis before he died of tuberulosis at age 39.

3 Spherial Triangles In Riemannian geometry, geometri shapes suh as triangles have a different appearane than what they would in Eulidean geometry. A spherial triangle may have sides that urvature whih would allow these triangles to have some speial properties. For example a spherial triangle an have more one right angle and a sum of interior angles that is more than 180 degrees. A spherial triangle with three right angles A B C A spherial triangle with two right angles C A B The Area of Spherial Triangle The area S of the spherial triangle ABC on sphere with radius r is given by S π ( m A + m B + m C 180 ) r where eah angle is measured in degrees.

4 Example 1 Find the area of spherial triangle with three right angles and a radius of feet. Find both the approximate and exat values for the area to the nearest hundredth. m A m B m C r feet S S S S ( m A + m B + m C 180 ) ( ) ( ft) ( ) ( ft) π ( ) ( 4 ft ) S π ft 6.8 ft π π r π Example Find the area of spherial triangle with two right angles and a third angle that measures 75 degrees and a radius that measures 3 feet. Find both the approximate and exat values for the area to the nearest hundredth. m A : m B : m C 75 : r 3 feet S π π π ( ) ( 3 ft) ( ) ( 3 ft) ( 75 ) ( 9 ft ) 135 π ft ft

5 Contemporary Non-Eulidean Geometries The City Distane Formula Suppose that you a ity that is laid out on in grid where the street either run north and south or east and west. If you need to travel from one point in the ity to another point in the ity that is not on the same street, then you wound not be able to travel in a straight line. Instead, you would walk along the streets until you would reah your destination. For example, suppose you need to walk from the City Library loated on the orner of First Street and Main Street to the Post Offie loated on the orner of Lake Street and Fourth Street. In the ity map below point A represents the library and point B represents the post offie. Using the map above, two paths, the red and blue, are outlined from point A to point B or from the library to the post offie. If you were to walk along either path, it turns out you would walk 5 bloks. The distane of eah little path that is traveled along the ity bloks an be represented by the ity distane formula. The City Distane Formula If P ( x, y 1 1) and Q ( x, ) y are two points in a ity, then the ity distane between the point P and Q is given by the following formula. d( P, x x + y y 1 1

6 Example 3 Using the ity grid above with point A as the library and point B as the post offie, let point A have the oordinates of (1,4) and point B have the oordinates of (3,1) Use the ity distane formula to find the distane between points A and B. Solution: d( P, x x1 + y y ity bloks Example 4 Use the ity distane formula to find the distane between the points P(,3) and Q(7,8). Solution: d P, x x + y y ity bloks ( 1 1 The Eulidean Distane Formula Now let s we refer bak to the example 3 with the Library, point A, loated at point (1,1) and the post offie, point B, loated at point (3,4). What if we wanted find shortest distane between the points A and B. The easiest way to aomplish this is to make a right triangle and find the length of the hypotenuse as shown in the next diagram.

7 If you ount the bloks in the diagram, the two legs of the triangle turn out to be bloks and 3 bloks. Next, use the Pythagorean Theorem to find the hypotenuse. a b bloks The Eulidean distane formula basially find the distanes between two points as shown above but use the atually oordinates instead of ounting the blok in the diagram. Eulidean Distane Formula If P ( x, y 1 1) and Q ( x, ) y are two points in a ity, then the Eulidean distane between the point P and Q is given by the following formula. ( x x ) + ( y ) d( P, y 1 1

8 Now, let s use the Eulidean distane formula to find the distane between point A(1,4) and B(3,1). ( x x ) + ( y y ) ( 3 1) + ( 4 1) bloks d( P, Using the Eulidean distane formula is essentially the same using the Pythagorean Theorem to find the distane between two points. Example 5 Find the ity distane and Eulidean distane between the points (,3) and (10,1). (Round answers to the nearest tenth of a blok) Part 1: Find the ity distane between the points (,3) and (10,1) d P, x x + y y bloks ( 1 1 Part : Find the Eulidean distane between the points (,3) and (10,1) ( x x ) + ( y y ) ( 10 ) + ( 1 3) bloks d( P, 0 1 1