Starting from a familiar curve

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1 In[]:= NoebookDirecory Ou[]= C:\Dropbox\Work\myweb\Courses\Mah_pages\Mah_5\ You can evaluae he enire noebook by using he keyboard shorcu Al+v o, or he menu iem Evaluaion Evaluae Noebook. Saring from a familiar curve In[]:= Mahemaica commens In he nex subsubsecion here is a simple picure in which I presen only one poin. This is o demonsrae how o plo geomeric objecs in Mahemaica. For ha we use Graphics[] command. One can ge help on Mahemaica commands by placing? before he command name.? Graphics Graphics primiives, opions represens a wo dimensional graphical image. In[3]:= Ou[3]= In[4]:= Ou[4]= In[5]:= In he command below here is only one primiive: PoinSize. Blue, Poin PoinSize. RGBColor, Poin and several opions, he firs opion being Frame True Frame True The example given in Mahemaica help is Thick, Green, Recangle Red, Disk Blue, Circle Yellow, Polygon 4, 4, Purple, Arrowheads Large Arrow 4, 3 3 Black, Dashed, Line 4, Ou[5]= This graphics command has six primiives and no opions. I don like how hey wrie his command. In my opinion i is much nicer if we pu each primiive in a separae lis and all primiives we pu in one lis. Below is a nicer way of wriing he above example

2 Walking_v8.nb In[6]:= he lis of primiives sars here Thick, Green, Recangle he firs primiive Red, Disk he second primiive Thick, Blue, Circle he hird primiive Yellow, Polygon 4, 4, he fourh primiive Thick, Purple, Arrowheads Large Arrow 4, 3 3 he fifh primiive Thick, Black, Dashed, Line 4, he sixh primiive he lis of primiives ends here Ou[6]= The only disadvanage is ha we have o repea he graphics direcive Thick hree imes. You can experimen by adding opions o he above command. Ploing poins This is how o plo one poin.

3 Walking_v8.nb 3 In[7]:= Graphics command sars here he lis of primiives sars here PoinSize. Blue, Poin he lis of primiives ends here, he opions follow Frame True, his opion pus a frame around he graph PloRange his opion deermine he range of he plo AspecRaio Auomaic, horzonal uni verical uni GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range his opion draws he grid lines command ends here Ou[7]= Nex I wan o show a family of poins. I do i in several seps. Firs I inroduce a variable, and I give his variable a specific value.5. Then I plo one poin wih coordinaes {Cos[],Sin[]}.

4 4 Walking_v8.nb In[8]:=.5; PoinSize. Blue, Poin Cos Sin Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Ou[9]= Nex I use command Manipulae[] o show many poins wih coordinaes {Cos[],Sin[]}, as a varies. Noice ha he Graphics[] command from he previous cell is wrapped ino Manipulae and he variable is given range from o. To emphasize he change in I show he value of as PloLabel.

5 Walking_v8.nb 5 In[]:= Clear ; Manipulae Manipulae sars here PoinSize. Blue, Poin Cos Sin PloLabel N Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range ends here., Pi his ells Manipulae o use in his range Manipulae ends here. Ou[]= In he nex command I ell Mahemaica o remember he poins ha have been ploed previously, so ha we can see which curve is being ploed.

6 6 Walking_v8.nb In[]:= Clear ; Manipulae PoinSize. Blue, Table Poin Cos v Sin v v,,, Pi 64 PloLabel N Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range, Pi, Pi 64. Ou[3]= In he nex several plos I show variaions on a uni circle. The only hing ha I change is ha I make he radius o be a funcion of. I call ha funcion fr[]

7 Walking_v8.nb 7 In[4]:= Clear ; fr _ : Cos 3 ; Manipulae PoinSize. Blue, Table Poin fr v Cos v Sin v v,,, Pi 64 Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Pi, Pi, Pi 64 Ou[6]=

8 8 Walking_v8.nb In[7]:= Clear ; fr _ : Cos ; Manipulae PoinSize. Blue, Table Poin fr v Cos v Sin v v,,, Pi 64 Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Pi, Pi, Pi 64 Ou[9]=

9 Walking_v8.nb 9 In[]:= Clear ; fr3 _ : Cos ; Manipulae PoinSize. Blue, Table Poin fr3 v Cos v Sin v v,,, Pi 64 Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Pi, Pi, Pi 64 Ou[]= The las radius funcion is more complicaed. As a reward, he resuling graph is any regular n-gon. Jus change 4 o any of 3,4,5,6,7,... in fr4[v,4] in he Graphics[] command below.

10 Walking_v8.nb In[3]:= Clear ; fr4 _, n_ : Cos Pi n Cos Mod n Pi n ; Manipulae PoinSize. Blue, Table Poin fr4 v, 4 Cos v Sin v v,,, Pi 64 Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Pi, Pi, Pi 64 Ou[5]= In he nex few examples we demonsrae curves in hree-space. We sar wih a helix above he uni circle and which climes one uni for each complee uni circle.

11 Walking_v8.nb In[6]:= Clear ; Manipulae Graphics3D PoinSize. Blue, Table Poin Cos v Sin v PloLabel N PloRange 3 Axes True 3 Pi, 6 Pi, Pi 3 v Pi v,,, Pi Ou[7]= The nex example shows a helix like curve ha climes on a cone

12 Walking_v8.nb In[8]:= Clear fx, fy, fz ; fx _ : Cos 4 ; fy _ : Pi Pi Sin 4 ; fz _ : Pi ; Manipulae Graphics3D PoinSize. Blue, Table Poin fx v fy v fz v v, Pi,, PloLabel N PloRange Axes True, AxesEdge BoxRaios,.5 Pi Pi, Pi, Pi 8 Pi Ou[3]= The nex plo is he same helix shown as line, no jus a collecion of poins.

13 Walking_v8.nb 3 In[3]:= Clear fx, fy, fz ; fx _ : Cos 4 ; fy _ : Pi Pi Sin 4 ; fz _ : Pi ; Manipulae Graphics3D Thickness.5 Blue, Line Table fx v fy v fz v v, Pi,, PloLabel N PloRange Axes True, AxesEdge BoxRaios,.5 Pi Pi, Pi, Pi 8 Pi Ou[33]= The nex helix is on he same cone, bu winds more ofen hen he previous one.

14 4 Walking_v8.nb In[34]:= Clear fx, fy, fz ; fx _ : Cos 8 ; fy _ : Pi Pi Sin 8 ; fz _ : Pi ; Manipulae Graphics3D Thickness.5 Blue, Line Table fx v fy v fz v v, Pi,, PloLabel N PloRange Axes True, AxesEdge BoxRaios,.5 Pi Pi, Pi, Pi 8 Pi Ou[36]=

15 Walking_v8.nb 5 Lines Poin and a vecor Given a poin say P and a direcion given by a vecor, say v, how do does a poin walk saring from P in he direcion specified by he vecor v? In[37]:= pp ; vv ; PoinSize. Blue, Poin pp Green, Arrow vv Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Ou[38]= Afer one second, he poin will be a he green poin whose posiion vecor is OP v

16 6 Walking_v8.nb In[39]:= pp ; vv ; ; PoinSize. Green, Poin pp vv PoinSize. Blue, Poin pp Green, Arrow vv Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Ou[4]= Afer / second, he poin will be a he green poin whose posiion vecor is OP v

17 Walking_v8.nb 7 In[4]:= pp ; vv ; ; PoinSize. Green, Poin pp vv PoinSize. Blue, Poin pp Green, Arrow vv Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Ou[44]= Now we are ready o illusrae he moion of he poin wih he Manipulaion[] command

18 8 Walking_v8.nb In[45]:= pp ; vv ; Clear ; Manipulae PoinSize. Green, Poin pp vv PoinSize. Blue, Poin pp Green, Arrow vv Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range, 4 Ou[47]= The same illusraion wih poin s posiions remembered.

19 Walking_v8.nb 9 In[48]:= pp ; vv ; Clear ; Manipulae PoinSize. Green, Table Poin pp s vv s,,,. PoinSize. Blue, Poin pp Green, Arrow vv Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range, 4 Ou[5]= Two poins In his subsecion I illusrae how o find he line deermined by wo poins.

20 Walking_v8.nb In[5]:= pp ; pq, ; PoinSize. Blue, Poin pp PoinSize. Cyan, Poin pq Tex P, pp, Tex Q, pq, Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range P Ou[5]= Q

21 Walking_v8.nb In[53]:= pp ; pq, ; PoinSize. Blue, Poin pp PoinSize. Cyan, Poin pq Cyan, Arrow pq pp Tex P, pp, Tex Q, pq, Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range P Ou[54]= Q

22 Walking_v8.nb In[55]:= pp ; pq, ; Manipulae PoinSize. Green, Table Poin pp s pq pp s,,,. PoinSize. Blue, Poin pp PoinSize. Cyan, Poin pq Cyan, Arrow pq pp Tex P, pp, Tex Q, pq, PloLabel N Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range,.5, 4.5 P Ou[56]= Q The same logic applies in hree dimensions:

23 Walking_v8.nb 3 In[57]:= pp 3,, 3 ; pq, 3, ; Manipulae Graphics3D PoinSize. Green, Table Poin pp s pq pp s,,,. PoinSize. Blue, Poin pp PoinSize. Cyan, Poin pq Cyan, Arrow, pq pp Tex P, pp, Tex Q, pq, PloLabel N Boxed True, Axes True, PloRange AxesLabel x, y, z, 3 4, 4.75 y P Ou[58]= z Q x

24 4 Walking_v8.nb Two poins and he uni sphere In[59]:= pp 3,, 3 ; pq 4, 3, 3 ; Manipulae Graphics3D PoinSize. Green, Table Poin pp s pq pp s,,,. PoinSize. Blue, Poin pp PoinSize. Cyan, Poin pq Opaciy.75 Sphere, Tex P, pp, Tex Q, pq, PloLabel N Boxed True, Axes True, PloRange AxesLabel x, y, z, 4, 4 Ou[6]= An relevan quesion for he above graph would be: Does a person locaed a he poin P sees a person locaed a he poin Q? To answer his quesion we need o calculae wheher he line joining P and Q inersecs he uni sphere. I will do his in Mahemaica.

25 Walking_v8.nb 5 In[6]:= pp 3,, 3 ; pq 4, 3, 3 ; In[6]:= The equaion of he line joining hese wo poins is pp pq pp Ou[6]= 3 5 4, 5, 3 3 Now we calculae if here are poins on his line which are a he disance from he origin In[63]:= Solve , Ou[63]= Or, look for a numerical soluion In[64]:= NSolve 3 5 Ou[64]= , Yes, here are wo poins on he line joining P and Q which are on he uni sphere. Therefore a person locaed a he poin P canno see he person locaed a he poin Q. This changes if we change he posiion of Q In[65]:= pp 3,, 3 ; pq, 3, 3 ; In[66]:= The equaion of he line joining hese wo poins is pp pq pp Ou[66]= 3, 5, 3 3 Now we calculae if here are poins on his line which are a he disance from he origin In[67]:= Solve , Ou[67]= There are no real soluions. Therefore here are on poins on he line joining P and his new Q which are on he uni sphere. Here we can calculae he closes poin on his line o he uni sphere. Firs plo

26 6 Walking_v8.nb In[68]:= Plo ,, PloRange Ou[68]= Now calculae derivaive In[69]:= Simplify 3 Ou[69]= In[7]:= Solve D , Ou[7]= Thus, he closes poin o he uni sphere is In[7]:= pp 34 pq pp 65 Ou[7]= 7 3, 4 3, 9 3 Is disance from he origin is In[7]:= Ou[7]= 37 3 approximaed by In[73]:= N 37 3 Ou[73]=.657 Thus his poin is really close o he uni sphere. Finally see i in hree-space

27 Walking_v8.nb 7 In[74]:= Manipulae Graphics3D PoinSize. Green, Table Poin pp s pq pp s,,,. PoinSize. Blue, Poin pp PoinSize. Cyan, Poin pq Opaciy.75 Sphere, Tex P, pp, Tex Q, pq, PloLabel N Boxed True, Axes True, PloRange AxesLabel x, y, z, 4, 4 Ou[74]= In[75]:= We need a differen ViewPoin o see wha is happening. VP `, `, ` Ou[75]=.54354,.533,.87655

28 8 Walking_v8.nb In[76]:= Manipulae Graphics3D PoinSize. Green, Table Poin pp s pq pp s,,,. PoinSize. Blue, Poin pp PoinSize. Cyan, Poin pq Opaciy.75 Sphere, Tex P, pp, Tex Q, pq, PloLabel N Boxed True, Axes True, PloRange AxesLabel x, y, z ViewPoin VP, 4, 4 Ou[76]= Now i is clear ha his line ges very close o he uni sphere, bu does no ouch i. Two lines Two pairs of poins deermine wo lines.

29 Walking_v8.nb 9 In[77]:= pp, 3 ; pq,, ; pp 3,, 3 ; pq,, ; Manipulae Graphics3D PoinSize. Green, Table Poin pp s pq pp s,,,. PoinSize. Magena, Table Poin pp s pq pp s,,,. PoinSize. Blue, Poin pp, Poin pp PoinSize. Cyan, Poin pq Poin pq Tex P, pp, Tex Q, pq, PloLabel N N Boxed True, Axes True, PloRange,.75, 4.5, 4.75,.5 P Ou[79]= Q Do hese lines inersec? Here is he algebraic answer. The parameric equaions of hese lines are

30 3 Walking_v8.nb In[8]:= pp pq pp Ou[8]= In[8]:= Ou[8]= In[8]:= Ou[8]=, 3 pp s pq pp, s,, 3 5 s Do hey have a common poin? Solve 3 s, 3, s s, No soluions, so hese wo lines do no inersec. Miscellaneous An egg This parameric equaion of a cross secion of an egg I found on he Inerne.

31 Walking_v8.nb 3 In[83]:= Line Table.78 Cos Sin Cos Pi, Pi, Pi 4 8 Frame True, PloRange.,.5,.5 AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range, 4 GrayLevel.5 Dashing.,. & Range, 4..5 Ou[83]= And you can draw an egg using Manipulae[]

32 3 Walking_v8.nb In[84]:= Manipulae Line Table.78 Cos Sin Cos Pi,, Pi 4 8 Frame True, PloRange.,.5,.5 AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range, 4 GrayLevel.5 Dashing.,. & Range, 4 Pi 4 Pi, Pi, Pi 8..5 Ou[84]= Velociy

33 Walking_v8.nb 33 In[85]:= Each parameric curve sudied above can be inerpreed as a moving paricle which leaves a race: he parameric curve. For each curve we will name is parameric equaion, find he velociy vecor and illusrae on he graph of he curve. The uni circle Clear r ; In[86]:= In[87]:= Ou[87]= In[88]:= r _ : Cos Sin D r Sin Cos Clear v ; v _ : Sin Cos

34 34 Walking_v8.nb In[89]:= Manipulae Thickness. Blue, Line Table r v v,, Pi, Pi 64 Thickness.7 Blue, Line Table r v v,,, Pi 64 Thickness.35 Magena, Table Arrow r v r v v v v,,, Pi Thickness.7 Magena, Arrow r r v PoinSize.5 Blue, Poin r PloLabel N Frame True, PloRange AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range, Pi, Pi Ou[89]= In[9]:= Clover Clear r ; In[9]:= r _ : Cos 3 Cos Sin

35 Walking_v8.nb 35 In[9]:= D r Ou[9]= In[93]:= Cos 3 Sin 3 Cos Sin 3 Cos Cos 3 3 Sin Sin 3 For esheic reasons, in he picure below I will uniformly shoren each velociy vecor o half of is magniude. Clear v ; v _ : Cos 3 Sin 3 Cos Sin 3 Cos Cos 3 3 Sin Sin 3

36 36 Walking_v8.nb In[94]:= Manipulae Thickness. Blue, Line Table r v v,, Pi, Pi 64 Thickness.7 Blue, Line Table r v v,,, Pi 64 Thickness.35 Magena, Table Arrow r v r v v v v,,, Thickness.7 Magena, Arrow r r v PoinSize.5 Blue, Poin r PloLabel N Frame True, PloRange 3, 3 3, 3 AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Pi 3, Pi, Pi 64 Pi Ou[94]= In[95]:= Cardioid Clear r3 ; r3 _ : Cos Cos Sin

37 Walking_v8.nb 37 In[96]:= D r3 Ou[96]= In[97]:= In[98]:= Cos Sin Cos Sin Cos Cos Sin Clear v3 ; v3 _ : Cos Sin Cos Sin Cos Cos Sin Manipulae Thickness. Blue, Line Table r3 v v,, Pi, Pi 64 Thickness.7 Blue, Line Table r3 v v,,, Pi 64 Thickness.35 Magena, Table Arrow r3 v r3 v v3 v v,,, Thickness.7 Magena, Arrow r3 r3 v3 PoinSize.5 Blue, Poin r3 PloLabel N Frame True, PloRange 3.5,.5 AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Pi 3, Pi, Pi 64 Pi 6.47 Ou[98]= 3

38 38 Walking_v8.nb Unnamed curve In[99]:= In[]:= Ou[]= In[]:= Clear r4 ; r4 _ : Cos Cos Sin D r4 Cos Sin 4 Cos Cos Sin Cos Cos 4 Cos Sin Sin Clear v4 ; v4 _ : Cos Sin 4 Cos Cos Sin Cos Cos 4 Cos Sin Sin

39 Walking_v8.nb 39 In[]:= Manipulae Thickness. Blue, Line Table r4 v v,, Pi, Pi 64 Thickness.7 Blue, Line Table r4 v v,,, Pi 64 Thickness.35 Magena, Table Arrow r4 v r4 v v4 v v,,, Thickness.7 Magena, Arrow r4 r4 v4 PoinSize.5 Blue, Poin r4 PloLabel N Frame True, PloRange 3, 3 3, 3 AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Pi 3, Pi, Pi 64 Pi Ou[]= Egg In[3]:= Clear r5 ; r5 _ :.78 Cos Sin Cos 4

40 4 Walking_v8.nb In[4]:= D r5 Ou[4]=.78 Cos 4 Cos.95 Sin Sin Sin 4 In[5]:= Clear v5 ; v5 _ :.78` Cos 4 Cos.95` Sin Sin Sin 4

41 Walking_v8.nb 4 In[6]:= Manipulae Thickness. Blue, Line Table r5 v v, Pi, Pi, Pi 64 Thickness.7 Blue, Line Table r5 v v, Pi,, Pi 64 Thickness.35 Magena, Table Arrow r5 v r5 v v5 v v, Pi,, Thickness.7 Magena, Arrow r5 r5 v5 PoinSize.5 Blue, Poin r5 PloLabel N Frame True, PloRange.5,.5 AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range, 4 GrayLevel.5 Dashing.,. & Range, 4 Pi 4 Pi, Pi, Pi 64 Pi Ou[6]=

42 4 Walking_v8.nb In[7]:= In[8]:= Helix Clear r6 ; r6 _ : Sin Cos D r6 Ou[8]= Cos Sin Pi In[9]:= Clear v6 ; v6 _ : Cos Sin

43 Walking_v8.nb 43 In[]:= Manipulae Graphics3D Thickness. Blue, Line Table r6 v v,, 8 Pi, Pi 64 Thickness.7 Blue, Line Table r6 v v,,, Pi 64 Thickness.35 Magena, Table Arrow r6 v r6 v v6 v v,,, Pi Thickness.7 Magena, Arrow r6 r6 v6 PoinSize.5 Blue, Poin r6 PloLabel N Boxed True, Axes True, PloRange 4 BoxRaios, 3 Pi, 8 Pi, Pi Ou[]= 3 In[]:= Conical helix

44 44 Walking_v8.nb In[]:= Clear r7 ; r7 _ : Sin 8 Cos 8 Pi In[3]:= D r7 Ou[3]= In[4]:= 8 Cos 8 Sin 8 Cos 8 8 Sin 8,, Clear v7 ; v7 _ : 8 8 Cos 8 Sin 8 Cos 8 8 Sin 8,,

45 Walking_v8.nb 45 In[5]:= Manipulae Graphics3D Thickness. Blue, Line Table r7 v v, Pi, Pi, Thickness.7 Blue, Line Table r7 v v, Pi,, Pi 8 Pi 8 Thickness.35 Magena, Table Arrow r7 v r7 v v7 v v, Pi,, Pi Thickness.7 Magena, Arrow r7 r7 v7 PoinSize.5 Blue, Poin r7 PloLabel N Boxed True, Axes True, PloRange 3, 3 3, 3.5 BoxRaios, Pi Pi, Pi, Pi Ou[5]=

46 46 Walking_v8.nb Lengh Smile Wha is a smile mahemaically? I could be defined as a graph of he square funcion near he origin; for example for x beween -/ and /. In[6]:= Thickness.5 Circle 3 Blue, PoinSize.7 Poin 3 8, 9 8 Poin 3 8, 9 8 Thickness. Line Table x, x x,,,. Frame True.5. Ou[6]=

47 Walking_v8.nb 47 In[7]:= Thickness.5 Circle 3 Blue, PoinSize.7 Poin 3 8, 9 8 Poin 3 8, 9 8 Thickness. Line Table x, x x,,,. Frame True, GridLines GrayLevel.5 Dashing.,. & Range, 4 GrayLevel.5 Dashing.,. & Range, 4.5. Ou[7]= The parameric equaion of a smile is In[8]:= In[9]:= Ou[9]= rs _ : D rs Then magniude of his vecor is In[]:=. Ou[]= 4 The lengh of his smile is In[]:= Inegrae 4,, Ou[]= ArcSinh

48 48 Walking_v8.nb In[]:= Cardioid Clear rc ; rc _ : Cos Cos Sin ; Thick, Blue, Line Table rc v v,, Pi, Pi 8 Frame True, PloRange.5,.5.5,.5 AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range.5..5 Ou[4]=..5. In[5]:= rc Ou[5]= In[6]:= Ou[6]= In[7]:= Ou[7]= Cos Cos Cos Sin FullSimplify D rc Cos Sin Cos Cos FullSimplify Cos Sin Cos Cos. Cos Sin Cos Cos Cos The lengh of he cardioid is In[8]:= Inegrae Cos,, Pi Ou[8]= 8

49 Walking_v8.nb 49 In[9]:= Inegrae Cos, Ou[9]= Cos Tan Egg In[3]:= Line Table.78 Cos Sin Cos Pi, Pi, Pi 4 8 Frame True, PloRange.,.5,.5 AspecRaio Auomaic, GridLines GrayLevel.8 & Range, 4 GrayLevel.8 & Range, 4..5 Ou[3]= I will modify his egg o

50 5 Walking_v8.nb In[3]:= Line Table 3 4 Cos Sin Cos Pi, Pi, Pi 4 8 Frame True, PloRange.,.5,.5 AspecRaio Auomaic, GridLines GrayLevel.8 & Range, 4 GrayLevel.8 & Range, 4..5 Ou[3]= In[3]:= D 3 4 Cos Sin Cos 4 Ou[3]= 3 4 Cos 4 Cos 3 6 Sin Sin Sin 4 In[33]:= Ou[33]= FullSimplify 3 4 Cos 4 Cos 3 6 Sin 4 Sin Sin. 3 4 Cos 4 Cos 3 6 Sin Sin Sin Cos Cos Sin The inegral below is a difficul inegral, i akes oo long o evaluae. In[34]:= Inegrae 49 3 Cos Cos 4 4 Sin Pi,Pi So, find a numerical approximaion

51 Walking_v8.nb 5 In[35]:= NInegrae Cos 3 Ou[35]= In[36]:= Ellipse Clear a, b, rel ; rel _, a_, b_ : a Cos b Sin ; 4 5 Cos 5 4 Sin, Pi, Pi Thick, Blue, Line Table rel v, 3, v,, Pi, Pi 8 Frame True, PloRange 3.5, 3.5.5,.5 AspecRaio Auomaic, GridLines GrayLevel.5 Dashing.,. & Range GrayLevel.5 Dashing.,. & Range Ou[38]= In[39]:= Ou[39]= In[4]:= 3 3 D rel a, b a Sin b Cos a Sin b Cos. a Sin b Cos Ou[4]= b Cos a Sin Thus, he lengh of he specific ellipse ha we ploed above is In[4]:= Inegrae Cos 3 Sin,, Pi Assumpions And a, b Ou[4]= 8 EllipicE 5 4 This shows ha his inegral is no calculable using he funcions ha we learn in Pre-calculus. A numerical approximaion is

52 5 Walking_v8.nb In[4]:= N 8 EllipicE 5 4 Ou[4]= We can expec ha he general case will involve EllipicE funcion. However, o calculae he general inegral one needs o use an opion for he Inegral[]. Calculaing he general inegral akes 48 seconds In[43]:= Timing Inegrae b Cos a Sin,PiAssumpions And ab I is a lile easier o calculae In[44]:= Timing Inegrae Cos a Sin,, Pi Assumpions And a Ou[44]= 7.75, 4 EllipicE a Then he general inegral equals In[45]:= 4 b EllipicE a b Ou[45]= 4 b EllipicE a b since b Cos a Sin b Cos a b Sin I is clear ha exchanging he role of a and b does no change he lengh of an ellipse. Therefore 4 b EllipicE a b 4 a EllipicE b a. I is ineresing ha Mahemaica does no know ha he preceding expressions are equal In[46]:= Ou[46]= FullSimplify b EllipicE a b EllipicE a b a EllipicE b a The above expression should simplify o. b a EllipicE And a, b b a

53 Walking_v8.nb 53 In[47]:= Plo3D a EllipicE b a b EllipicE a,, b,, a b Ou[47]= Now explore he funcion for he lengh of an ellipse as a funcion of a and b. In[48]:= Plo3D 4 b EllipicE a a,, b,, b Ou[48]=

54 54 Walking_v8.nb In[49]:= Show ConourPlo 4 b EllipicE a a,, b,, b Conours Pi Range, ConourLabels All PloRangePadding. 4 Ou[49]=

55 Walking_v8.nb 55 In[5]:= Show ConourPlo 4 b EllipicE a a,, b b,, Conours Pi Range, ConourLabels 4 Tex Framed 3 3 Pi, 3 Background Whie & PloRangePadding. Pi Ou[5]=

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