ࡱ > B D A '` bjbj .( _ % z z z 8 $ $ # " ( ( ( ( 0 X d $ h Y ^ ! l ( ( l l ( ( \ \ \ l ( ( \ l \ \ \ ( /! z p \ 0 # \ d \ \ * l l l \ l l l l l X l l l # l l l l I n c l a s s a s s i g n m e n t T o p i c 1 G e o d e s i c o n a c u b e N a m e : F r o m W i k i p e d i a , w e f i n d : I n H Y P E R L I N K " h t t p : / / e n . w i k i p e d i a . o r g / w i k i / M a t h e m a t i c s " \ o " M a t h e m a t i c s " m a t h e m a t i c s , a g e o d e s i c / d i Yd [s jk , - d i s jk / [ j e e - u h - d e s - i k , - d e e - s i k ] i s a g e n e r a l i z a t ion of the notion of a " HYPERLINK "http://en.wikipedia.org/wiki/Line_%28mathematics%29" \o "Line (mathematics)" straight line" to " HYPERLINK "http://en.wikipedia.org/wiki/Manifold" \o "Manifold" curved spaces". In presence of a HYPERLINK "http://en.wikipedia.org/wiki/Metric_%28mathematics%29" \o "Metric (mathematics)" metric, geodesics are defined to be ( HYPERLINK "http://en.wikipedia.org/wiki/Locally" \o "Locally" locally) the shortest path between points on the space. The term "geodesic" comes from HYPERLINK "http://en.wikipedia.org/wiki/Geodesy" \o "Geodesy" geodesy, the science of measuring the size and shape of HYPERLINK "http://en.wikipedia.org/wiki/Earth" \o "Earth" Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's HYPERLINK "http://en.wikipedia.org/wiki/Surface" \o "Surface" surface, namely, a HYPERLINK "http://en.wikipedia.org/wiki/Line_segment" \o "Line segment" segment of a HYPERLINK "http://en.wikipedia.org/wiki/Great_circle" \o "Great circle" great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in HYPERLINK "http://en.wikipedia.org/wiki/Graph_theory" \o "Graph theory" graph theory, one might consider a geodesic between two vertices/nodes of a graph. Geodesics are of particular importance in HYPERLINK "http://en.wikipedia.org/wiki/General_relativity" \o "General relativity" general relativity, as they describe the motion of inertial test particles. HYPERLINK "http://en.wikipedia.org/wiki/Image:Spherical_triangle.svg" \o "A geodesic triangle on the sphere. The geodesics are great circle arcs." HYPERLINK "http://en.wikipedia.org/wiki/Image:Spherical_triangle.svg" \o "Enlarge" Exploration: A Common Cube Conundrum Suppose one desired to move along the surface of the cube from the top far corner (point A) to the near bottom corner (point B). Would the dark line path marked in the picture be the shortest route from point A to point B? A1. If you agree, justify your answer; use the new vocabulary in your answer. A2. If you do not agree, describe the geodesic and justify your answer in a mathematical way. Compare the suggested geodesic with your path length and argue mathematically which is the shorter route. In either case, find both the exact answer using algebra and an approximation from your calculator. How would you demonstrate the answer using graph paper for the faces of the cube? Sometimes a low-tech approach is very convincing! (graph paper is attached). Can you build a cube using origami and test out what you think with strips of paper? B. How would you teach that the exact answer, the graph paper approach, and the approximation are the same? Answers: A: The shortest route is: B: Some ideas for teaching this are: PAGE PAGE 1 L N d f n ~ r s J K Q R q r 乩 h6} 0J OJ QJ ^J mH sH h6} 0J OJ QJ ^J mH sH h6} 0J OJ QJ ^J mH sH h6} 0J mH sH h6} 5\mH sH h6} 0J mH sH j h6} UmH sH h6} mH sH hkU mH sH hl-0 h mH sH 2 j l n p | ~ < ! # gd4 gd4 ^gd4 gd4 gd6} &d P gdl-0 gdl-0 gd6} _ K L S T % & e f m n z { * + f g ! " ĺĺĺĺĺĺвЮ j hqV h4 56Uh^X h4 0J 6_H h4 h mH sH h6} 0J mH sH j h6} UmH sH h6} mH sH h6} 0J 6]mH sH h6} 6]mH sH j h6} 6U]mH sH 8" # ) P R T W X g K Y ( $ 6 7 : ] ^ _ ` b c e f h i ·ƨƠƨƑ h~ j h~ Uhl-0 hww hww mH sH h2 hl-0 hc hl-0 j hc hc Uh/ hww h/ h! hc h4 har hKb h^ hkU h/ hkU 5hqV h4 h4 567 S T , - % gdww gd4 % & ' ( ) * + , - . / 0 1 2 3 4 5 6 8 ^ _ a b d e g h j k gdww i k l r s t u v w } ~ hww h/ 0J mH nH uh/ h/ 0J j h/ 0J Uh~ k t u v gdww h]hgd/ &`#$gd~ , 1h/ =!"#$% D d Zeh > C A b :\GՕ D n :\GՕPNG IHDR QT sRGB pHYs od `IDATx^ݡoTM$BRd I 4%h[II[R %ӄUC8Lqh0(}}yw۽sΜsw9s7nxO$' B8 #"k1"$p 8.c B8 #"k1"$PǏ#j'PVZF?B?}4巸!HP5:~]'/5FYK@ϱOT+"ZL%cSA3 838J Ǧf2qfp3 Mmd&f*L Ǚ1T864 3cplj;h&3g0S vLf8apL%cSA3 838J Ǧf2qfp3 Mmd&qQ? #3yK `({X?e2PE&_~ݼyԩS˻s.ATB{x߿D "xO>R1:P( uѣGO1S-8@<;;+1K WSqܟcˁ؟cKPؙcˁؓcKP؍cˁ؇cKPcˁ\u &WbӎA,NrEvX\ͺ:8[tb jr5- 6r$*Al1%մؐcˁl &WbA,NAX\M+;8ftb jr5-#r$*G\1%մcˁcKPcˁcKPcˁcKPcˁcKPcˁcKPcˁ qy &W3A,Nrą˚ Wf$)bḃ'Tgcc#}Ч''ON8a(VWW6'Z'oBf2n-m²߿Hp㽽UW@?ﴛ@1l%&(i)z ^;C[qe.-@1Iݯc({YPvq⎡X57ׂ8?Nw三2Пc1
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