GeomSpace User Manual
Contents
1. They may form or may not form existing faces After that press Calculate button to find relevant measure characteristic and value between lineals 12 Configuration and sessions GeomSpace keeps track on what folders were visited last time for reading models from or writing models to If it is just downloaded or properly installed GeomSpace is able to locate localization data and documentation In case the program fails to find them you can point out the folders for localization and for documentation using Settings Dialog available from Tools gt Settings menu Please take into consideration that if GeomSpace failed to find localization due to missing information and you provide such information supposing the localization file for your language exists you have to restart the program in order to make effect Each time before closing GeomSpace saves its configuration to disk When it starts GeomSpace reads or creates its configuration 132. becomes inactive and Lineal Chooser becomes active so you are able to select the transform Figure 6 right While transform matrix is changing the current transform is immediatly applied to selected figures and you can see the resulting geometry While selected figures are transformed you can observe that original points stand on their positions even if they are not associated with a face Same point names are used also in transformed figure So you can watch how the current transform changes their position even if you are opted to move the figure If you want to reset figure to its original position you can press Reset button 11 When you finish transforming press Stop button Once it is pressed face selection is cleared and transformed points get new names in order to keep them unique The Close button in Transform Dialog is disabled until you stop the transform If you close Transform Dialog by pressing key or button from Transform Dialog the transform stops before dialog closes You can start several transform actions simultaneously Sometimes it is usefull The only idea to keep in mind is that Transform Dialog grabs mobile figure parts so they are not available for selection in the second dialog and so on 10 Calculating Bases You can find homogeneous coordinates of some point or orthonormal basis of some lineal in canonic form by pressing Basis button from Calculate group Basis Dialog appears Figure 7 In this dialog you
3. select Open with option and select GeomSpace as program that reads them You can open and save these models by pressing Open and Save buttons from Space group A file browser appears allowing you to select a model Used Projection For dimensions 1 or 2 no projection is performed In this case OX1 axis is oriented left to rigth and OX is bottom up For dimension 3 the perspective projection is performed by OpenGL runtime User is always positioned in the center of model OX3 axis direction is toward the camera For dimensions greater then 3 first parallel projection is programmatically performed to 3 dimensional plane X X 2X3 and then perspective projection is performed by OpenGL runtime Axes OX n gt 3 have no graphic representa tion User Interaction Once you opened a model you can change camera position orientation regarding the space in the following way e Use arrow keys t R and or keys to move yourself through the space for 2 dimensional spaces the last two keys rotate space around the center e Use left mouse drag to move the space along you and right mouse drag to orbit the space around you This behavior will be changed in future e Use left mouse drag to move yourself in the space and Shift right mouse drag to orbit yourself in the space e Use mouse wheel to move forward backward if space dimension is greater then 2 or rotate space 2 dimensiona
4. can choose a lineal in Construct Chooser It may form or may not form an existing face After that press Get info button The following info will be shown lineal name lineal specification specification of space defined by lineal and lineal orthonormal basis You can repeat this procedure with another lineal In case lineal contain one point its homogeneous coordinates will be shown Its specification is void since point is space of dimension zero Ed Basis les Face Point s A12A4A19 Points Points geveenennnenennsennneneneennennnenennnnnnennenennnennennsenenaseenneneneenneenennnnenennennenennnennennensensnnnnennenevneg Lineal A4 A12 A19 Specification 0 1 Basis 1 00 0 00 0 00 1 0 1 00 0 35 0 86 1 00 0 14 0 33 Close Figure 7 Basis Dialog 11 Calculating Distances and Angles Using measure feature you can find the distance between two points or a point and a lineal as well as angle between two lineals in case lineals are intersected or distance between them in case lineals are not intersected Ge omSpace uses unusual Measure Dialog due to the fact different spaces have dif ferent configurations of mutual positions of lineals GeomSpace measure gives only relevant informations and all these informations They include measure characteristic showing what kind of measure it is and measure value direct and complementary It is important to know both direct and complementary values because for s
5. so you can see the face name before using it If you want to remove some point s from face name you can write its name their names in Point s input and press Points button A face can have the number of points equals at most to the space dimension and at most 3 If Point s editbox contain more valid points then that only the first in specified order points will enter in the face 8 Removing Faces You can remove existing faces by using Remove Dialog Figure 5 which opens by pressing Remove button from Construct group All you need is to select the face or face set to remove with Face Chooser presented in Figure 5 in brown border see below how to use it and press Remove selected button Please note that removing faces process is resources consuming It is better to remove once a large amount of faces then to remove them one by one It isn t possible to remove points as they are not part of geometry If you want to remove all faces and all points just construct new space by pressing New button from Space group and accept default values ted Remove Faces to remove 6 AO A3 A6 AO A5 A6 Pattern A6A0 Exact match Pattern subset Pattern superset Pattern intersects Pattern Remove selected Close Z Figure 5 Remove Dialog Choosing Face Set You can choose a set of existing faces using Face Chooser The Pattern input accepts name patterns A name pattern is a set of point names Their order doesn t m
6. Each characteristic may have one of the following values e for elliptic characteristic Euclidean space 0 1 1 Elliptic spce 1 1 1 Hyperbolic space 1 1 1 Minkowskii space 0 1 1 1 m maximal number of general situated points or minimal number of verteces that form a positive volume figure Simplex for timelike geometry or 0 1 1 1 for spacelike geometry 1 dimensional planes are lines 0 dimensional planes are points Are all i dimensional planes congruend Is measure between 2 i dimensional planes bounded Is measure between 2 i dimensional planes scalable This geometry is inconsistent It is possible to use the model but the terminology may differ Figure 3 Algorithm to determine space specification e 0 for parabolic characteristic and e for hyperbolic characteristic If you use Euclidean elliptic or hyperbolic Bolyai Lobachevsky space of dimension n use the following specification ki k2 kn 0 1 1 for Euclidean space 1 1 1 for elliptic space and 1 1 1 for hyperbolic space If you plan to use another space use the algorithm presented in Figure 3 to determine its specification For more information about what space specification is about and how to find it please refer to the theory behind GeomSpace Uniform Theory of Geometric Spaces presented in popular manner or Analytic Geometr
7. Geomspace User Manual Alexandru Popa amp Team email alpopaQgmail com http sourceforge net projects geomspace version 0 12 1 Why GeomSpace GeomSpace is interactive geometry program for Euclidean and non Euclidean spaces It is unique piece of software that e Gives you feeling of space not of space model e Has no restriction on space dimension e Performs equally well for spaces Euclidean Riemann Bolyai Lobachevsky Galilean including curved ones Minkowskii including curved ones and other homogeneous spaces Actually GeomSpace uses the same code for all them e Uses native OpenGL calls not only for Euclidean like motions but for all possible motions Also advantages of GeomSpace over similar interactive geometry programs include e Small size e Tiny and or standard dependence libraries FLTK e Really cross platform compiles without modification of the code for Linux Windows OS X FreeBSD and Solaris including 32 and 64 bit versions Hire is full list of supported platforms with some recomandations linux64 is prepared for modern Linux system installed as amd64 port It should run with most distributions linux32 is prepared either for older computers with Linux system installed or for Linux i886 port installation In many cases this version will also run in Linux amd64 environment with support of 32 bit able graphic driver and libraries windows64 is prepared with suppor
8. atter You can write them toghether or separated by a space or any other character s different from capital letter and digit The Pattern button adds to the current selection listed in upper part of Face Chooser all existing in space faces that match the pattern by selected match criteria The Pattern button removes from the current selection all faces that match the pattern by selected may be different match criteria There are 4 different match criterias Exact match applies only when the face name is the same regardless the point order as the pattern Pattern subset applies to all faces with names that contain the whole pattern Pattern superset applies to all faces with names contained in the pattern Pattern intersects applies to all faces with names that contain at least one point of the pattern In order to select all existing faces choose Pattern subset match criteria and leave Pattern blank after that press Pattern button If you want to clear selected faces list choose Pattern subset match criteria and leave Pattern blank after that press Pattern button 10 9 Transforming Geometry Transform feature is not the same as camera motion achieved by mouse and keyboard interaction Camera motion changes observer s position and orienta tion in space all motions are around observer but it doesn t change figures position Camera motion is done in projected space model so it is limited to at most 3 dimensional
9. ct you can construct it with the current color by pressing Add face button Points Creation with Lineal Chooser Lineal Chooser allows you to select normalized point coordinates column as well as generalized orthogonal motion square matrix and orthonormal basis of a lineal rectangular matrix using homogeneous coordinates In case of points the origin has coordinates 1 0 0 In order to change shown coordinates or matrix you need to select two radio boxes Then Measure spinner becomes active and you can change its value If you select 0 and 2 radio boxes measure means distance of translation along OX axis If you check i and j checkbox measure means angle of rotation in plane X OX around the origin from lower to higher axis index You can modify the measure value by hand but in this case the matrix coordinates will not be refreshed automatically In order to force it to refresh you can select unselect some radio box or click on Points tab title The Reset button resets coordonate matrix to initial value Constructing Faces with Construct Chooser Construct Chooser allows you to select a valid face name In Point s input you can specify point name s which this face will contain Points order doesn t matter You can write them toghether or separated by a space or any other character s different from capital letter and digit After pressing Points button all valid points in correct order will be added to Face output
10. f Ge omSpace in different languages by setting environment variable LANG between launches In main window Figure 1 space specification and not file name is shown in the titlebar or a message about no space selected At the top of window you can find menubar with all functions The right part contains toolbar with common function buttons After choosing some function a corresponding di alog apears The left side of window is reserved for space geometry and user interaction keyboard and mouse You can interact with geometry from the main window when some dialog is open Sometimes it is very usefull Using GeomSpace you can e Create a space by specification e Construct geometric model in space e View constructed model from any position navigating in space with mouse and or keyboard e Edit constructed model by adding removing or moving some its parts e Do different calculus on you model and get the result find out coordi nates basis vectors distances and angles e Exchange created models in gmsp files 5 Getting Started The easiest way to learn GeomSpace is to open an existing model Ge omSpace uses gmsp files that describe a space with gemetry model in it The file however doesn t contain camera position and orientation because it is not the property of space or model You can find several models on project website You can configure your system to open them with GeomSpace by right clicking on a gmsp file
11. l only around the center e Use key to show camera orientation matrix to show camera state matrix orientation orthogonalness and to reset camera orientation to default e Use key to show hide unused points and point names e Use or keys to display this help If you want to see the model from different angles you need to perform a range of translations and rotations in opposite directions as if you are walking around the model So press the left button and move the mouse in one direction then press the right button and move the mouse in the opposite direction Repeat this several times 6 Space Creation New Space j Ed New Space Space color a h i v Dim 3 v Change A B 192 k1 1 Hyperbolic A 96 v k2 1 Elliptic Change k3 1 Elliptic Cancel Cancel ir E Figure 2 Space Dialog You can construct new space by choosing its specification and color When you press New button from Space group the Space Dialog Figure 2 appears Space Dialog allows you to choose the dimension and after pressing Change button space specification as number of characteristics k1 k2 kn where e k is the characteristic of distances e k is the characteristic of plane angles e kz is the characteristic of dihedral angles between 2 dimensional planes eens e k is the characteristic of dihedral angles between n 1 dimensional planes
12. motions the maximum OpenGL dimension At the other hand transform keeps the camera position and orientation and actually changes figure points coordinates all motions are around origin All possible motions are available via transform feature It is very usefull when you play with spaces of dimension more then 3 You can use camera motion and figure motion simultaneously However please note that while camera motion is done in specialized graphics hardware GPU graphic memory figure motion is done in main hardware CPU main memory It is very probable that motion of large figures is less efficient then camera motion Faces to transform A10 A14 A10 A18 A14 A18 A10 A14 A18 Pattern A10A18A14 Exact match Pattern subset Pattern superset Pattern intersects Vosoossoos00000000000000000000000000000000007 Keep original Keep original Start Stop Transform 1 00 0 00 0 00 0 99 0 07 1 00 0 06 1 00 0 07 1 20 0 00 0 00 Measure 1 50 Start Stop 2000 eoooo 0 al 2 3 Lee Reset FT Close Figure 6 Transform Dialog Transform Dialog Figure 6 allows you to copy or move selected piece of geometry First you have to select figures you want to copy move using Face Chooser see Figure 6 left If you plan to copy selected figures check Keep original chechbox If you want to move them leave it unchecked When you are sure figures are selected correctly press Start button Face Chooser
13. ome measure characteristics one of them can be infinite or unmeasurable for details please refer to theory behind GeomSpace Uniform 12 Theory of Geometric Spaces or Analytic Geometry of Homogeneous Spaces available at project cite rd Measure hes Lineal 1 AO A6 Lineal 2 A3 A5 Face Point s ASA3 Points Points Lineal 1 Lineal 2 Measure between lineals Direct 2 000000 Complementary infinity Characteristic 0 Parabolic Close Figure 8 Measure Dialog In order to explane what means relevant and non relevant informations consider two lines in Euclidean plane In case they are intersected the most relevant information is the angle between them elliptic measure characteristic In case they are parallel the most relevant information is distance between them parabolic measure characteristic If these two lines are in elliptic plane there is no difference between distances and angles as all measures have elliptic char acteristic If these two lines are in hyperbolic plane they can be intersected elliptic measure characteristic parallel parabolic measure characteristic or divergent hyperbolic measure characteristic All these cases are properly han dled by measure feature You can open Measure Dialog Figure 8 by pressing Measure button from Calculate group Using Construct Chooser you can fill out Lineal 1 and Lineal 2 static texts by pressing Lineal 1 respective Lineal 2 buttons
14. t left picture green bordered and pressing Add point button see below how to use Lineal Chooser Each point gets the unique name containing capital letters and digits starting with AO Each new point name have the letter part as previous point and the number part increased by 1 So after AO follow A1 A2 and so on If you press New group button the next point name will be formed by the next letter followed by 0 In this case after A2 follows BO point name after Z point group follows AA AB and so on When new space is chosen the point names are reset Point dots and names are visible in space only when some dialog is open Once the dialog is closed dots and names are hidden You can toggle points visibility by pressing key Points don t make part of space geometry and normally aren t shown when all dialogs are closed In order to construct a true geometric point with the Fr Rosses Face A0 A1 Point s AOA1 New group v Also face Add face Add point Close Close Figure 4 Construct Dialog Point construct tab left and face construct tab right current color a one point face should be constructed It is automatically con structed with the point if Also face checkbox is checked Most faces are constructed from Faces tab using Construct Chooser pre sented in Figure 4 at right picture blue bordered see below how to use it When you are sure that face name shown in Face output is corre
15. t of NET runtime and is recomanded for Windows Vista Windows 7 and above windows32 is optimized for Windows XP but is able to run also in Windows Vista and above osx is prepared as fat binary sutable for OS X 32 and 64 bit versions It should run on Mac OS X Leopard and above freebsd64 is compiled for FreeBSD installed as amd64 port on modern com puters It will also run on other FreeBSD based systems e g PC BSD freebsd32 is compiled for older computers or for 1886 port of FreeBSD system solaris Because of Solaris architecture there is no different 32 64 versions of GeomSpace compiled for it One universal binary will run on Or acle Solaris 10 and above OpenSolaris OpenIndiana and Solaris based systems Enjoy using GeomSpace 2 System Requirements and Limitations GeomSpace makes use of 3D accelerated graphics hardware It is the case of many modern computers GeomSpace will also run with software simulated 3D acceleration however it will dramatically drop its performance On all supported platforms GeomSpace should run out of the box The application is really light it consumes under 20 MB of memory in full load with 3D accelerated graphics adapter OS X version starts localized only from the terminal In order to start GeomSpace localized start Terminal app and issue open path to GeomSpace app change path to to actual path where GeomSpace resides Windows version should work out of
16. that understands this format In order to manually uninstall GeomSpace remove all its files Please note that GeomSpace creates resource file in Windows Vista and above LOCALAPPDATA geomspace geomspace re in Windows XP APP DATA N geomspacel geomspace re and geomspace geomspace rc in all other systems for each system user You may want to remove also these folders files Ed No Space ERS Space View Construct Calculate Tools Info Space New Open Save Construct Add Remove Transform Calculate Basis Measure Info Help Figure 1 GeomSpace main window 4 Using GeomSpace You can start GeomSpace in several ways e By clicking on GeomSpace executable GeomSpace starts with no space selected e By clicking on gmsp model the first time you need to associate this file format with the program that reads it GeomSpace starts with cor responding model opened You can also select several models and open them in GeomSpace Each model opens in separate window OS X will open them one after another in the same window e From the command line In this case you can provide as arguments any number of gmsp models In OS X you should open the Terminal applica tion and type path to GeomSpace app Contents Mac0S GeomSpace lt modeli gt gmsp lt model2 gt gmsp You can also pass arguments to FLTK window position geometry or color On unix like systems you can start different instances o
17. the box in most cases For better expe rience of using it please keep executable documentation and localization files in ASCII only folders including whole path The standard location C Program Files GeomSpace is good This limitation doesn t affect model locations 3 Installation and uninstallation GeomSpace has no installation program yet Fresh downloaded and unar chived program will run properly from the folder where it is unarchived If you want to install it by hand use the following paths you may need adminis trator s privileges Linux FreeBSD Solaris UNIX You should be familiar with lt prefir gt folder It may be usr usr local opt local etc Copy GeomSpace to lt prefix gt bin i18n doc and models folders to lt prefix gt share Ge omSpace folder of course you can keep the models where you want Windows Copy GeomSpace exe with i18n and doc folders to C Program Files GeomSpace You may keep models where you want but it is good idea to keep them together with the program OS X Create folder Programs GeomSpace and put here GeomSpace app with doc and models folders feel free to keep the models in different place In order to complete installation add a shortcut to GeomSpace in start menu desktop or taskbar using system provided mechanism Also register gmsp file format by right click on some gmsp file choose Open with item from popup menu and browse to GeomSpace as program
18. y of Homogeneous Spaces presented in rigurous mathematical manner both available at the project website Default values for space specification in Space Dialog is taken from previous space or dimension 0 with no specification if no space was present So you can clear the space just by pressing New button and accepting default values Choosing the Color Beside space dimension and specification you can choose space background color Space Dialog as well as other dialogs uses a simple color chooser for geometry elements presented at Figure 2 at left picture in red border This element acts as toggle button Four spinners for color components open hide on its press You can choose R red G green B blue and A alpha opacity color components in range 0 255 While modifying the color components Color Chooser button changes its color to reflect the current color selection Its central part shows color with opacity and outer part uses solid color in order to figure out the opacity by contrast 7 Constructing Geometry When the space is chosen you can construct different figures in it Con struct Dialog Figure 4 opens by pressing Add button from Construct group This dialog contains two tabs Points for construction of points by their coordinates and Faces for construction of faces by specifying their points Constructing a point is as simple as choosing its coordinates in Lineal Chooser presented in Figure 4 a
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