G%Welcome *x0200.05y0800.2o y x2x 3x9Welcome to Calculon-TCalculon is a next-generation mathematical "Xvisualisation tool; a medium for educational,Ncommunication; and an amazing piece of 'Tsoftware whch truly shows off the power of*your PDA. r^In Calculon's innovative interface, equations,  $Ptext notes and sketches are arranged on (Bfreeform worksheets (like this!).!Equations are mathematicallytypeset:VThe plots of all equations on the worksheet+Nappear instantly on the View Page (see 'RView tab, above). Calculon can be used to `visualise points, lines and surfaces in 2, 3, 4 0 or n-dimensions. :You can sketch on worksheets:hRWorksheets are collected into a "Library")Rof packages (see the Lib tab), which can Hbe exchanged or downloaded from the $internet. LSuper-fast custom math routines and 3D&Pengine provide calculation and rendering(Tspeeds unprecedented in a handheld device.*JAmaze your colleagues by showing them%Psome of the included examples (under the(Lib tab). LFor detailed instructions see the Help&PPackage (included in this installation).(>For support or comments, email:*calculon@vaagmaer.com@@Thanks for purchasing Calculon ! VKeep an eye onH http://www.vaagmaer.com/calculon Rfor updates and free packages to download)  CalculonPNG  IHDR+HIDATxKh@QKA/ɲ/p0\e%y Y{RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4Jn=_mvt2Թ̶iKF#ЏgYpL 穳w-#v.yYg67uV9:ws9@1 W[ t{(ϗ㩩mtq_k5\ {N0Kj+s|UF'n@H=#wzFޝ4fzmp/_|]MZ14ˡ^Mo,ht˩L:ޓ6=Lo?3F8n"ө/LЁF'xr˓'@c_'"Qf>˺Ssqu,F_M}7Fg<]ћ]w,ì~<]%iM<'Ť]\qd]Fx^@џR0w&Ԅ!ZFF'ЦC?oӈL Ѭ4@Q*и5z@U&с@⸍ t+بm9> }oLb]>?=8On_oFGQ8Do`>54 vyңoWL-LӦm7ٓMc.e۞wYwfsK.Qs'esoo{;ggRQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4JF(uNi:RQ4J[TqqIENDB`$Quick Instructions = $Quick InstructionsLUsing Calculon to do a quick plot of a Xsingle function in two or more dimensions is,very easy: JT1. Select Work Page (Work tab at the top).aD2. Select and clear the Temporary ": Worksheet with this icon:<3. Enter equation editing mode* with this icon: P4. Tap on the worksheet to start the new( equation.  R5. Edit the equation using the pickbox or)B the keyboard and PDA buttons.! T6. Select the View Page to see the plotted* function.  VThe most likely cause of dissatisfaction at+Vthis point is that the range of values over+Twhich the function has been plotted is not*\ideal, in which case edit their limits or step.Rsize in the Value Table. The Value Table )Tis available both on the View Page and the*Work Page. oXTip: If you regularly use Calculon for quick Hplots then disable "Auto-Open First $HWorksheet" and "Load All Packages At$VStartup". Calculon will start up faster and  Pyou will be automatically placed in the (Ptemporary worksheet in equation editing ( mode.ZSee the full Instructions in the Help Package-Zfor more details. Peruse the examples in this-Ppackage and the Help Package for greater(Binsight into how to use Calculon. XIf you don't have the Help Package it can be, downloaded from:@http://www.vaagmaer.com/calculon eneweqnoff2 Dimensional.Trigonometric Functionsqx-440.2y-440.2x-440.2y-440.2x-440.2y-440.2= yxT yxk yx.Trigonometric FunctionsXThe classic functions. Note the use of three,Rvalue spaces to plot the three functions.)&Simple User ControlU>a-10100.2x-330.2y-10100.05:sy x3 ax2(User Control ExampleLThis example demonstrates the use of a&Z"user control". The value a in the expression Tbelow is designated type user in the Value TTable. As such its value can be controlled*Rinteractively on the View Page using the ):slider under the control tab.3 Dimensional Eggbox:x-660.5y-110.05z-660.5CwyxzRA 3-dimensional plot of two trigonmetric )Xfunctions. x and y are are set to role axis    Vin the Value Table, but as they are not the+Rsubject of any equations they also act as)parameters.  Eggbox Torusyr-110.05x-110.05y-110.05z-110.05-3.143.150.2-3.143.150.2ETr2D~ yEx rFz r TorusHA simple 3D parametric plot with two$parameters. 5Vr is an intermediate value which represents *Pthe radius from the centre of the torus,(i>x, y and z are the axes values,   Another Torusyr-110.05x-110.05y-110.05z-110.05-3.153.150.2-3.153.150.2r2p ynz rrux rAnother Torus PThis is exactly the same as the previous(NTorus example but shows some different ',visualisation options.GZNote in particular one of the surface colours-Lhas been set to "Transparent" (on the &VOptions Page), so the interior can be seen.+Lines in 3DL{r-110.05x-110.05y-110.05z-110.05-3.153.150.02-3.153.150.01r3 2 12 y 2!x r5z rLines in 3D RCalculon can draw both lines and surfaces !Rin any number of dimensions. Surfaces are)Pdefined by equations with two parameters(J(which may or may not also be axes), %Jwhereas lines are defined by a single%parameter. nRThis example is derived from the previous)Ttoroidal surface example, but in this case*Pthere is only one independent parameter.(ZThe result is a line which spirals around the-&surface of a torus. <x, y and z are the axes values   $Surfaces and Lines r-110.05x-110.05y-110.05z-110.05-3.153.150.02-3.153.150.01r-330.2x-330.2y-330.2z-330.2-3.153.150.2-3.153.150.2tr3 2psy 2pz rpx r 12 r3l ylz rlx r$Surfaces and LinesNAny number of surfaces and lines can be'Pcombined in a single n-dimensional plot.(:PThis example combines the equations from(Hthe Torus and Lines in 3D examples. $^NThese equations define the coiled line:' F... and these equations define the #"torodial surface:"Intersecting Torir-110.05x-110.05y-110.05z-110.05-3.143.150.3-3.23.20.5g-10160.2r-330.2x-330.2y-330.2z-330.2-3.23.20.2-3.23.20.2~Hr3Kr3` yb xzy rzx rz r z rg"Intersecting Tori >Check out the User Control on g 5one torus: 5the other: PNG  IHDRҕIDATxݱNAEi.]a.(RYݹ!Tj7cSrz?"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%E -8$.];J9Or專r]糒t9EcNkyVnXʍYrc]R'yYnJrkHa."]R>t0"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)C"]R{y\I"]%3]R˯-e;K|gH_g ]e9.?\KXN˫DX΢ˌ\XN |74.?ߎ尸w:9wݍ./8Y^Ng4r^^ |HK%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RK~fZb(cIENDB`4 Dimensional4D Torusr-110.05w-330.2x-110.05y-110.05z-110.05-3.23.20.2-3.23.20.2-330.2r2^ ytz rHx rw 224D TorusRCalculon can handle visualisations in any !Jnumber of dimensions. When viewing an%Nn-dimensional plot, dragging the stylus'>on the view rotates the plot in@3-dimensional space (as normal). TRotations in dimensions above three can be*Jperformed using the sliders under the%Rotate tab. 4Dx, y, z and w are the axes values,  \There is one slider per dimension above three..\Each slider rotates in a plane defined by the .Ntwo axes listed adjacent to the slider.'RThis example is derived from the previous)T3-dimensional torus example. In this case *Zthe values include an extra axis, w, and the" Xequations give the torus a kind of 'saddle' ,Jshape in the 4th dimension. Note the %\projection of all four axes into 3D space (or .2really 2D on the screen). VNote how rotating the plot in the w-z plane+T(i.e. through the 4th dimension) turns the*Vshape inside out. This can be understood by+Nthinking about rotating a 2-dimensional'Pobject in 3-dimensional space. Imagine a(^glass disc with writing on it, lying on a piece/\of paper. If the disc is lifted up and turned .Zover, the writing is reversed. This can only -Xhappen by turning the disc through the third,^dimension. If the disc is confined to the plane/Tof the paper (two dimensions) it cannot be*Tbe turned so that the writing is reversed.*LHSimilarly, a 3-dimensional shape in $H3-dimensional space cannot be turned$Zinside out by rotating it, but it can if it's-@rotated through a 4th dimension! SeashellsNatalina>A-330.2R-330.2a-330.2b-330.2e-330.2s-3.23.20.3t-330.2u-330.2v-330.2x-330.2y-330.2z-330.2-330.2-330.2-330.2 -1460.3-330.2-330.2-330.2 a12 b165R 1 a2 s2 b2 s2fY 240360} 255360s% 210360zB 230360 A25:t A Rs;u A Rs--v A Rsk 280360>x t Rs e  >y u Rs e  z v e  8Digital Seashells - NatalinaDThe shape of many seashells can be"Vdescribed by a single set of equations with+Lvariable parameters. This example uses&Rthese equations with the parameter values)@for Natalina, a kind of snail.  hTThe aperture of the shell is defined as an*Lellipse with major and minor diameters&>given by a and b. For Natalina,    TThe aperture radius R is then defined as a Nfunction of a and b and the parameter s    Vwhich is the aperture-wise direction of the+^surface. Essentially, this defines the ellipse:/ />The following angles govern the&shape of the shell, VThis is the distance of the aperture centre+Bfrom the shell tip when theta = 0 RThree intermediate values, simply because)Nthe equations are too long to fit here,'FTThis is the equianglar angle of spiral for*Natalina `Finally it is all pulled together in the 3D axis0>variable definitions x, y and z   VTo see the effect of varying the parameters+Vwhich govern the shape of the shell see the+<other examples in this folder.*PDownload the full Seashell Package from .the Calculon website -  2www.vaagmaer.com/calculonYReferences lHCortie MB (1993). Digital Seashells.$>Computers & Graphics 17, 79-84. RCortie MB (1991). The Form, Function, and)VSynthesis of the Molluscan Shell. In Spiral% TSymmetry (eds. I Hargittai & CA Pickover). !XWorld Scientific Publishing, Singapore, 369-,387.PNG  IHDR..4IDATxvH@QZ'>zJK%*AϿ@l ]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH钢P_[ Ͽ=_~{[o݊|3׫&6ϒ:߿E:Nݻh_1?L/IMnZˁq t$; m6^FO6K\ǿ\ڙeǗ?Cש_:ԛܣvi{DZmub>cCZ|r{RQ/DșBC2!v,xwOlbeHѴvn޿!n7-Mnbxe*t.Gip[{OuW~(?H v5ݗk|{- lGz#kG^QE^"t4X~'=r?__|:RпtH1;9I?<핏m O^|sm|+?} +{Iq. aiݸ1n9y.3= K6ͣ/lD!p}_VC'}aǗkcGʧ{ ~~!^lh\9 _~,c-g0+VrAAX|_[aٜl4Jqz{&"G6{E!>^>WVUD9.1^hǗ7#s^in꣼D.?}2krANh=>*_"s3_(䛎u)V?ǧ[Z# N2aԯ:~_QCޝCuy5*inh]^wJ:Z7ܵcv9d< G1 o ܮaV !uvߊ_t~l|ViAj2=$~q|9jG{gxJYan=s3MY+]/1C:Ox9|8x-);g[}7/xhx9vEg|_κ9y|n/N)%E/{}]Qˉ?ޡ^'[{0 <>"]~(]>76>R0t@\of.g6iO"]Rd%EKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH"]RKtI.)%EH?\>qHo=~8)+܊Sdȡ|wx9(f̋v>]kU(^Q g/WPrr,&2v.gy|i쯿~m["x9?,!m؞= ,4N8Bu;k됿=63n4r5| ||~=k~pkti]\4;wח(=:,1Q{t.l~ۯůxָnb{caY!#;[J>/_Z_ȃYfۧԝ:>=Ik9>;7?}wO+\9MK<~p =.aKgyuv;_^2@`vG>9]Բ]fxMߐGA>/F5tş3xLX>c%r5k\9RkZ:M|d7a\٪]t~Ndeux t Rs e  R>y u Rs e  qz v e  ^ a40s b14rP 219360uk 245360x 21360v 22360H A50  283.9360 LyriaRAnother example using the same equations )Las the Natalina example. This time the Nparameters are set for the conch Lyria.! QNautilus@A-330.2R-330.2a-330.2b-330.2e-330.2s-3.143.150.3t-330.2u-330.2v-330.2x-330.2y-330.2z-330.2-330.2-330.2-330.2 -1060.3-330.2-330.2-330.2$5R 1 a2 s2 b2 s2 $:t A Rs ;;u A Rs R-v A Rs p>x t Rs e  >y u Rs e  z v e   a2 b1.5w 290360 0 0 0 A2 280360 Nautilus NThis example uses the same equations as'Nthe Natalina example but the parameters Rare those which define a shell shape very)Dclose to that of a Nautilus shell. Q XBSee the Natalina worksheet for an Bexplanation of the parameters and!equations Turitellai@A-330.2R-330.2a-330.2b-330.2e-330.2s-3.153.150.5t-330.2u-330.2v-330.2x-330.2y-330.2z-330.2-330.2-330.2-330.2 -8000.5-330.2-330.2-330.2f A20uj 26360{ a2.2t 255360 b2.6v 21360 288360t 223605R 1 a2 s2 b2 s2:t A Rs7;u A RsR-v A Rsp>x t Rs e  >y u Rs e  z v e  Turritella DYet another example using the same"Hequations as the Natalina example.  PThis time the parameter settings give a (^shell shape similar to Turritella, the "Auger"  "or "Screw Shell". Seashell Control|rc8A10400.2R-330.2a4300.2b4300.2e-330.2s-3.143.150.6t-330.2u-330.2v-330.2x-330.2y-330.2z-330.2-0.50.50.21.31.50.201.60.2 -660.6010.201.60.2$5R 1 a2 s2 b2 s2 :t A Rs ;u A Rs -v A Rs >x t Rs e  !>y u Rs e  @z v e  e 06Seashell with User ControlsNThis example uses the same equations as'Jthe Natalina example but most of the  Lparameters are made available as user &controls. QVVTry adjusting the alpha setting to show the:shell structure more clearly.TopologicalMobius Bandft-1.51.5.5x-110.05y-110.05z-110.05-3.143.1460.2 /y t 2 U"x 3 t 2 "z 3 t 2Mobius Band :A surface with only one side.Dini's Surface˝˝|a-440.2b-440.2u0150.2v0.00110.1x-440.2y-440.2z-340.2zx auvy auv,z av v2 bu a1@ b0.2Dini's SurfaceRDini's surface is an elegant form with a )Xconstant negative Gaussian curvature of -1. ,RA negative Gaussian curvature means that )Zthe surface is locally saddle-shaped at that -Ppoint, so this surface is saddle shaped (Xeverywhere! There are various other surfaces,Rthat exhibit constant negative curvature.)Boy Surface&Boy Surface (Apery) qa-440.2s03.1420.05t03.1420.05x-440.2y-440.2z-440.25x a 2s 2t 2st3a s3 32ss 3t5y a 2s 2t 2st-z a3sSRThis parameterisation of Boy's Surface is),based on Apery (1987).Boy Surface PThe first Boy Surface was constructed by(VWerner Boy around 1901. It is a realisation+Xof the real projective plane in 3-dimensions,>that contains no singularities.`ZApery F (1987) Models of the Real Projective  VPlane: Computer Graphics of Steiner and Boy+ PSurfaces. Vieweg, Braunschweig, Germany. OReferences rVNote the use of the intermediate value a to' `reduce much of the repitition in the definitions0^of x, y and z. As a is calculated only once for    Teach point in the plot, the calculation is*Xsubstantially faster than if x, y and z had    Jbeen defined directly in terms of the%parameters. Hhttp:://www.math.smith.edu/~jposson/$ boy_surface.html:http://mathworld.wolfram.com/BoySurface.htmlWebsites.Boy Surface (Apery v.2)^a-440.2s03.1420.05t03.1420.05x-440.2y-440.2z-440.2b-440.2t03.150.05x-440.2y-440.2z-440.2-440.2 5x a 2s 2t 2st b3a s3 32ss 3t 5y a 2s 2t 2st z a3sb 1 33 6t+x b44 3tt1+y b44 3ttGz b6ZAnother plot of the Boy Surface, but this one-Ralso includes a plot of the line of self-)Pintersection as derived by Apery (1987).("Boy Surface (v.2) N@The surface defined as before... Vand the intersection line is defined in the+*second value space...:Boy Surface (Petit & Souriau) a-440.2b-440.2c-440.2d-440.2f-440.2g-440.2h-440.2t06.2840.1u06.2840.3x-440.2y-440.2z-440.2-440.2-440.2-440.2 a bc a2 b2 d a2 b2Sh  3t8$f cd auu =$gd au bu{,x 3.3 ft ght,y 3.3 ft ghtz 4 gh10 q&10 2 6t 3 " 1.98 3t 6:Boy Surface (Petit & Souriau) \This plot is another possible parameterisation.Vof a Boy Surface, which was found by Petit +Zand Souriau in 1981. However, it has not yet -Bbeen proven to be a Boy Surface. !QXThe calculation consists of a long series of,(intermediate values.*Boy <-> Roman Surfacea-440.2s03.1420.1t-1.571.570.1x-440.2y-440.2z-440.2010.25x a 2s 2t 2st8a s3 32ss 3t5y a 2s 2t 2stz a3sVThe Roman surface is also a realisation of +Zthe real projective plane, however unlike the-VBoy Surface it does contain singularities. +FRHere, a parameter, gamma, has been added  Rto the equations for the Boy Surface. As )Ngamma varies from zero to one the plot  "Nsmoothly deforms from the Roman to the 'Boy Surface. *Boy <-> Roman SurfaceG%