Test #1 Geometry MAT 4263

Transcription

1 Test #1 Geometry MAT 4263 I. Consider some of the differences and similarities among the three geometries we looked at with respect to each of the following: 1) geodesics (line segments) 2) triangles (appearance, angle sum, area, etc.) Geodesics (segments) Euclidean Spherical Hyperbolic Modeled in flat Euclidean plane Continuous finite portion of a straight line Modeled on the surface of a sphere minor arc of a great circle on sphere Can be modeled in Poincare disk arc of circle orthogonal to P.disk joining points in Pdisk measure is unbounded measure is bounded measure is unbounded Any two points determine a unique segment Only non-antipodal points determine a unique segment any two points determine unique segment triangles Measurement of segments is usually via the Euclidean metric Any three non-collinear pts determine a triangle Measure = r* where r = radius of sphere and = central angle Any three non-collinear pts. determine a triangle measured by logarithmic cross-ratio Any three non-collinear pts. determine a triangle Example: Example: Example: Angle sum (a.s.) = 180 degrees = radians 180 < a.s < 540 degrees < a.s. < 3 radians 0 < a.s. < 180 degrees 0 < a.s. < radians Area is unbounded Area < 2 Area < Area formula = ½ bh Area = a.s. - Area = - a.s.

2 II. Solve each of the following spherical triangles (or show its impossibility). 1) Spherical right triangle with angles = = /3 (unit sphere). Using the basic formula cos (c) = cot( )cot( ) = (1/ 3) 2 = 1/3, leads to c = cos -1 (1/3) = Using this value of c, the fact that a = b, and the formula cos(c)= cos(a)cos(b), we have a = b = cos -1 { cos(c)} = ) General spherical triangle with =20 b a=2 =60 c Use one of the variants of the Law of Cosines to find : So,

3 NowuUse the Law of Sines to find b and c: III. Derive one of the following trig identities for spherical right triangle ABC (with c the hypotenuse and 1) From the accompanying picture we note 2). Working with the right hand side

4 B M a c O a c b L C P b A IV. 1) Describe the Poincare Disk model for Hyperbolic Geometry. In Hyperbolic Geometry one of the models that can be used is the Poincaré Disk. The entire hyperbolic plane is considered to be the interior of a circle (generally taken to have radius = 1). The points on the boundary of the disk (the circle) are not part of the hyperbolic universe. Lines in this model are the open circular arcs contained in the disk that derive from circles orthogonal to the disk (see illustration) or open diameters of the disk. Segments in this model are closed arcs on a hyperbolic line. The length of a segment is determined by a logarithmic cross ratio (see notes).

5 Poincare disk hyperbolic lines circle orthogonal to disk 2) Describe the Saccheri Quadrilateral (SQ) and prove in Neutral Geometry that the summit angles of the SQ are congruent. (see notes on-line) 3) Outline Saccheri's plan for "proving" Euclid's 5th axiom and the motivations behind it. Saccheri s motivation was that of many mathematicians since the time of Euclid to prove that Euclid s 5 th postulate was unnecessary (i.e., it was really a theorem, dependent on the first 4 postulates and their consequences). In order to do this, Saccheri knew that proving the existence of a rectangle was logically equivalent to the 5 th postulate. So, he constructed a special quadrilateral (see problem 2) and set out to show that it was, in fact, a rectangle. After being unable to prove directly that it was a rectangle, he proceeded to attempt an indirect proof. He was able to show that the summit angles were congruent, from which one concludes that they are either both acute, both obtuse, or both right which he referred to as the Hypothesis of the Acute Angle (or Obtuse or Right, respectively). If the first two possibilities could be eliminated then the only possibility was that the angles were right angles and that the quadrilateral was a rectangle. He was easily able to dispense with the Hypothesis of the Obtuse Angle. After much work he thought he had found a contradiction eliminating the Hypothesis of the Acute Angle. He hadn t. In fact, in his attack on this Hypothesis, he developed much of what would become Hyperbolic Geometry. 4) Define/describe each of the following terms. Sketch an example. lune spherical line Law of Sines (spherical) hyperbolic sine

6 (see notes on-line) Basic trigonometric relations for right triangles cos c = cos a cos b = cot cot sin = sin a sin c ; sin = sin b sin c cos = tan b tan c ; cos = tan a tan c tan = tan a sin b ; tan = tan b sin a sin = cos cos b ; sin = cos cos a