Spiral Symmetry on the TI-92 Paul Beem Indiana University South Bend South Bend, IN

Size: px

Start display at page:

Download "Spiral Symmetry on the TI-92 Paul Beem Indiana University South Bend South Bend, IN"

Marjory Price
6 years ago
Views:

Proceeings of he Thir Inernaional DERIVE/TI-92 Conference Spiral Symmery on he TI-92 Paul Beem Iniana Universiy Souh Ben Souh Ben, IN pbeem@iusb.eu We all hink an reason in conex.

1 Proceeings of he Thir Inernaional DERIVE/TI-92 Conference Spiral Symmery on he TI-92 Paul Beem Iniana Universiy Souh Ben Souh Ben, IN We all hink an reason in conex. In a linear algebra class, we ry o use linear equaions an homomorphisms an, when in a geomery class, lines an circles. The same suen who easily fins he exponenial funcion as he answer o a raioacive ecay problem in a calculus class won' necessarily fin i as he soluion o a similariy problem in a geomery class. This alk is abou a uni I each on spiral symmery in my Topics In Geomery course for seconary eucaion majors. Spiral symmery explois he exponenial funcion o analyze he geomeric opic of similariy. The TI-92 is use hroughou. We sar wih he noion of a ilaive roaion. This is a ilaion similariy (a sreching or shrinking cenere a a poin P followe by a roaion aroun he same poin. The amoun of sreching (or shrinking is calle he ilaion number an may be negaive. (Two ilaive roaions are consiere he same if heir ilaion numbers are opposies an heir angles iffer by 18 egrees. Suppose ABC has a righ angle a C an a perpenicular is roppe from C o he hypoenuse AB meeing AB a D. There is a ilaive roaion cenere a D ha maps ACD o CBD. The angle of he ransformaion is +9 egrees an he ilaion number is he raio BC/AC. Dilaive roaions are someimes calle spiral symmeries an we will now see why. Suppose we sar wih a cerain riangle an perform he same ilaive roaion on i an is ieraes a number of imes. We ge, of course, a series of riangles, each roae by he same angle from he previous one an expane (or conrace by he same raio. (In he figure, he raio is 1.2 an he angle is 6 egrees. Suppose we keep rack of a paricular verex of his riangle an recor is various posiions as i roaes aroun he cener of he ilaive roaion. I is clear ha hese poins lie on a spiral of some sor. We will use he TI-92 o make his precise. Suppose we pick he verex a he larger acue angle, on he original riangle, an consruc he coorinaes of i an is ieraes. We arrange hese coorinaes so ha hey separae sufficienly o be selece easily. Then we pu

Proceeings of he Thir Inernaional DERIVE/TI-92 Conference hese aa ino he aa base variable SYSDATA.

Now plo his aa an we see he plo of poins we collece in he previous sep. To make sense of his aa, i is convenien o conver i ino polar coorinaes.

If we were o use he buil-in inverse angen funcion, we woul ge values in he range -π/2 o π/2, which is no wha we wan.

Nex, we plo he raius agains he angle. Make sure ha he calculaor is in FUNCTION moe an ha angles are measure in raians, no egrees.

2 Proceeings of he Thir Inernaional DERIVE/TI-92 Conference hese aa ino he aa base variable SYSDATA. Because of he way he selecion proceure works, i is imporan o selec he x-coorinae firs an o eselec boh coorinaes before selecing he nex pair. Now plo his aa an we see he plo of poins we collece in he previous sep. To make sense of his aa, i is convenien o conver i ino polar coorinaes. Reurning o SYSDATA, we label he firs wo columns, XCOORD an YCOORD. We label he nex wo columns RADIUS an ANGLE an efine C3 o be SQRT(C1^2+C2^2. If we were o use he buil-in inverse angen funcion, we woul ge values in he range -π/2 o π/2, which is no wha we wan. Bu i is easy o efine a cusom Arcan funcion using he buil-in ANGLE funcion. Using his funcion an aing 2π, "by han", o accoun for he wining naure of he aa, we ge i in usable form. Nex, we plo he raius agains he angle. Make sure ha he calculaor is in FUNCTION moe an ha angles are measure in raians, no egrees. Because of he way he poins were consruce, we know ha heir secon coorinaes are increasing by a consan raio for every regular change in he angle. (In he example, he moulus of he poin is increasing by 1.2 for every 6 change in he angle. This ype of funcion is usually calle exponenial growh. We can check our guess ha his is an exponenial funcion by oing an exponenial regression on he aa. We can view he resuling exponenial funcion (he regression Beem: Spiral Symmery on he TI-92 Page 2

3 Proceeings of he Thir Inernaional DERIVE/TI-92 Conference funcion in boh FUNCTION moe, in orer o see he usual graph, or in POLAR form, o see he resuling spiral fi o he original aa. We can erive he equaion of he spiral easily. Firs noe ha, if an enoe he consan angle an ilaion number, respecively, hen he coorinae (r, ransforms, via he ilaive roaion, o (r, +. Since he ilaive roaion is a symmery of he curve, r( + = r(. Seing =,, 2,..., we ge r( n = n r(. Le a enoe r( an = n. Then r ( = a. The simples assumpion o make now is o assume ha can ake on any value. This hen becomes he equaion of our "self similar" curve. (In he example, 3 π r ( =.5(1.2. Le D(ρ,ξ enoe he ilaive roaion cenere a he origin wih ilaion number ρ an roaion angle ξ. Theorem: The spiral ( a has D(ρ,ξ as a symmery if an only if r = Proof: D(ρ,ξ is a symmery of he curve + ξ This is equivalen o a = ρ( a logarihms yiels he resul. ln( ln( ρ =. ξ ( a if an only if r( + ξ = ρ r(. r =. Diviing boh sies by a an aking Corollary: The above spiral can also be wrien as r ( = a, where enoes he ilaion number require o say on he spiral hrough a roaion of one raian. Proof: Jus ake ξ = 1, in he heorem. The number is calle he naural ilaion number of he spiral. If > 1, he spiral wins ou couner-clockwise an if < < 1, hen he spiral wins ou clockwise. Beem: Spiral Symmery on he TI-92 Page 3

4 Proceeings of he Thir Inernaional DERIVE/TI-92 Conference We can also express he spiral in erms of he base e: ln( has a geomeric inerpreaion. ln( r ( = ae. The quaniy Theorem: The equaion of he spiral can be wrien in he form ( co( φ r = ae, where 1 φ = co (ln( is he angle beween he raius vecor o a poin on he spiral an he angen vecor o he spiral a he same poin. Noe, in paricular, ha he angle φ is a consan, an is no epenen on. Logarihmic spirals are also calle equiangular spirals because of his propery. Noe also ha we can assume ha φ is a firs or secon quaran angle, epening on wheher he spiral wins ou couner-clockwise or clockwise. φ Proof: Le O enoe he origin an P a poin on he spiral r ( = a wih (polar coorinaes (r,. Incremen he angle by a small amoun δ an le Q be he inersecion of he angen line a P an he augmene raius vecor (see figure. Le R be he foo of he perpenicular from P o he line OQ. Then OP = r, m QPS = φ, RQ δr, an m OQP = φ - δ. Le η = m OQP. Then RP δr = r sin(δ. Since sin(δ δ, when δ is small, co(η = RQ/RP, when δ is rδ small. Beem: Spiral Symmery on he TI-92 Page 4

5 Proceeings of he Thir Inernaional DERIVE/TI-92 Conference Taking he limi on boh sies of co(φ - δ 1 co( φ r = ln( =. r δr rδ, as δ goes o zero, yiels Nex, we will use he calculaor o suy he erivaive spiral. We sar by efining r 1 ( = (.5(.8 an graphing i, in polar moe, using a square winow. Now, on he home screen, efine x 1( = r1( cos( an y 1( = r1( sin(. Graph hese equaions in parameric moe, again wih a square screen. Now reurn o he home screen an efine x 2( = ( x1( an y 2( = ( y1(. Now graph boh he spiral an is erivaive. (The graphing will go faser if you obain explici formulas for he righ han sies of hese expressions before efining he lef han sies. If you graph hese in simulaneous moe wih heir syles se o PATH, you will see an ineresing relaionship beween hese wo curves. Now reurn o he home screen an simplify he expressions for he erivaives. If you facor an hen Collec each expression, you shoul ge ( x1( = (.8 sin( Beem: Spiral Symmery on he TI-92 Page 5

6 Proceeings of he Thir Inernaional DERIVE/TI-92 Conference ( y1( = (.8 sin( Who are hese srange numbers an wha o hey wan? Firs of all, noe ha π + ( = , so ha he equaions can be wrien 2 ( x1( = (.8 cos( ( y1( = (.8 sin( which is a lile beer. Nex noe ha co( = ln(.8, so ha is he consan angle φ. To ge beer conrol over his, we reurn o some heory. Since he spiral has he equaions: x( = a cos( he erivaives will have he form: y( = a sin( x ( = a(ln( cos( sin( = ln( x( y( y = a(cos( + ln( sin( = x( + ln( y(. ( In marix form, hese equaions become: x ( ln( = y ( 1 1 x(. ln( y( Facoring ou he square roo of he eerminan (why no? yiels: ln( x ( = D D y ( 1 D 1 D x(, where D = 1+ ln 2 (. ln( y( D The marix is now in he form of a roaion marix, he roaion being by an angle, 1 ln( where sin( = an cos( =. I follows ha co( = ln(, so ha D D = φ, he consan angle for he spiral. Noe ha, in his form, is a firs or secon quaran angle epening on wheher 1 or 1. Beem: Spiral Symmery on he TI-92 Page 6

7 Proceeings of he Thir Inernaional DERIVE/TI-92 Conference So, we have shown ha he erivaive curve is a ilaive roaion of he original spiral. 2 The ilaion number is D = + ln ( = csc( φ an he angle is φ, where φ enoes 1 he consan angle for he original spiral. We can check his agains our example above. In our case, r 1 ( = (.5(.8, so ha is.8. Hence, ln( = an D = Hence is a secon quaran angle an is given by 1 π sin 1 ( = This checks wih he above. The prouc of D an a is D (.5(1.245 =.5122, which also checks. I is no har o see ha he curve ( x (, y ( is also a logarihmic spiral, since: x ( = Da(cos( cos( sin( sin( = Dr( cos( + = D r( + cos( + y ( = Da(cos( sin( + sin( cos( = Dr( sin( + = D r( + sin( + Hence, he poin ( x (, y ( lies on he curve r( = ( D a, a spiral parallel (i.e., having he same naural base o he original spiral. We have seen, in he above, ha he erivaive of a logarihmic spiral is anoher logrihmic spiral an is a ilaion of he original spiral. I is no har o show ha any ilaion of a logarihmic spiral is anoher logarihmic spiral. In fac, he same is rue of any roae logrihmic spiral. Theorem: Le C be he graph of he logarihmic spiral r ( = a. Then boh D(ν,(C an D(1,(C are graphs of logarihmic spirals an hey are he same graph if ν =. Proof: Suppose (r, maps o he poin (r','' via he ilaion D(ν,. Then 1 1 D (,( r, = ( r, = ( a,. Hence, r = a. Since = ', he ilae spiral has ν ν equaion r ( = ( aν, which is a spiral parallel o he original spiral. Suppose, on he oher han ha (r, maps o (r','' via he roaion D(1,. Then r' = r an ' = +. Since r ( = a on C, r = a = (a. Tha is, he image of D(1,(C of C has polar equaion he original spiral. r ( = ( a, which again is a spiral parallel o Equaing he wo expressions for he raius vecor, yiels he resul ν =. Beem: Spiral Symmery on he TI-92 Page 7

Proceeings of he Thir Inernaional DERIVE/TI-92 Conference In paricular hen, he above heorem says ha ilaing he spiral r ( = a by a facor has he same effec as roaing he spiral by.

8 Proceeings of he Thir Inernaional DERIVE/TI-92 Conference In paricular hen, he above heorem says ha ilaing he spiral r ( = a by a facor has he same effec as roaing he spiral by. This can be illusrae on he calculaor. Change o polar graphing moe an se a small square winow wih a symmeric range. Then ype in he following program: rospirl( Prgm Local i For i,1,6,1 min-(i-1*π/3 min max-(i-1 *π/3 max.5*(.8^( +(I-1 *π/3 r1( DispG SoPic #("spir"&sring(i EnFor EnPrgm This program creaes an sores six picures (.PIC files each of which requires 397 byes for a oal of byes of memory. If you on' have ha available, eiher elee some unneccessary files or increase he incremen angle (which is π/3 in he program an make fewer picures. The picures will be sore as SPIR1.PIC, SPIR2.PIC,, SPIR6.PIC. You can view he picures using he VAR-LINK viewing funcion or by opening hem in he graph screen. The following picure shows all six spirals: To see even more vivily he equivalence beween ilaions an roaions of his spiral, wrie he comman CYCLEPIC "SPIR", 6,.1, 1, -1. The resuling animaion seems o be of a spiral ilaing in an ou, even hough i was wrien as a roaing spiral. Beem: Spiral Symmery on he TI-92 Page 8

Two Coupled Oscillators / Normal Modes

Two Coupled Oscillators / Normal Modes Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own