MP2GL: prototyping 3D objects with я ь тсвь and OpenGL
Contents
1. 3 Denis Girou The pst ob3d package A PSTricks package for three dimen sional basic objects 2002 25 4 5 a S 11 12 13 14 15 16 17 18 Denis Girou The pst gr3d package A PSTricks package for three dimen sional grids 2004 Luis Nobre Gongalves FEATPOST and a Review of 3D METAPOST Pack ages In Proceedings of TUG 2004 TEX XML and Digital Typography volume 3130 of Lecture Notes in Computer Science LNCS pages 112 124 Xanthi Greece 2004 Luis Nobre Gongalves FEATPOST macros 2004 John D Hobby A User s manual for MetaPost Technical report AT amp T Bell Laboratories Murray Hill New Jersey 1992 Computing Science Tech nical Report 162 Donald E Knuth The METAFONTbook Reading MA Addison Wesley 1986 Manuel Luque and Herbert Vo 3D views with pst vue3d 2005 Brook Moses MetaPlot MetaContour and Other Collaborations with METAPOST In Proceedings of Practical TRX 2004 2004 Denis Roegel Creating 3D animations with METAPOST TUGboat 18 4 274 283 1997 Denis Roegel Space geometry with METAPOST TUGboat 22 4 298 314 2001 Denis Roegel The METAOBJ tutorial and reference manual 2001 Denis Roegel METAOBJ Very High Level Objects in METAPOST In Proceedings of TUG 2002 Trivandrum India 2002 Herbert Vo Three dimensional plots with pst 3dplot TUGboat 22 1 319 329 2001 Herbert Vo 3D plots PST 3dplot v1 63 20052. V8 crossproduct W4 W3 V7 V9 Vitwhatever V2 whatever V3 V4 whatever V2 whatever V5 V6t whatever V 7 whatever V8 In the three previous lines it should be noted that the six whatevers repre sent six different values The last point V9 is the center of the sphere The sphere itself can be drawn by inserting the following code in the scene TranslateV V9 wire_sphere norm W1 V9 30 30 where norm W1 V9 is the radius of the sphere and 30 is both the number of parallels and meridians Equations also allow us to set some coordinates separately Xpart A 0 4 or we can define equations on certain components only Xpart Q Ypart U As usual in METAPOST a value can be refreshed with whatever this time with an assignment 17 Xpart A whatever A Point can be refreshed like this A whatever whatever whatever like above the three whatevers are different unknowns 6 Text support 6 1 Basic text Text can be added to a 3D object but we currently have to provide a 3D posi tion The position of the text will depend on the point of view An example is shown in figure 10 TX labels are obtained by generating two files a PostScript file containing only the drawings and a IATFX picture environment with the la bels Labels are obtained via GL2PS and can be positioned in different ways When doing 2D graphics an incremental approach can be used whereby the final drawing is obtained by a number of modificat
3. Building an object amounts to moving to where the object should be set and then calling the appropriate function For instance building a tetrahedron at coordinates 4 1 5 12 3 is done as follows glTranslatef 4 1 5 12 3 glutSolidTetrahedron Building a square face at coordinates 0 0 0 1 0 0 1 1 0 and 0 1 0 is done as follows glBegin GL_POLYGON glNormal3f 0 0 1 glVertex3f 0 0 0 glVertex3f 1 0 0 glVertex3f 1 1 0 glVertex3f 0 1 0 glEnd The glVertex3f calls specify the vertices and the g1Norma13f call specifies a normal to the face which is necessary for correct shading with lights Such a square can be put anywhere in space by the appropriate use of trans formations before glBegin 3It should be stressed that many features are very easy to add since they are a mere interface to OpenGL The main difficulty is to get the file organization right 3 2 Projections The main OpenGL projections are the perspective and orthographic projections In both cases a frustum is defined and all elements outside the frustum are not drawn In the perspective projection a camera has to be defined 4 The MP2GL interface to OpenGL MP2GL can be used either to create 3D objects using the high level OpenGL objects library or it can create objects by a low level approach We distinguish therefore an OpenGL objects library OL and a METAPOST objects library ML In either case objects can be set an
4. Herbert Vo PSTricks Grafik mit PostScript f r TRX und BTRx Lehmanns 2005 2nd edition Mason Woo Jackie Neider Tom Davis and Dave Shreiner OpenGL Pro gramming Guide Fourth Edition Addison Wesley 2004 26
5. etc we would typically need two runs of METAPOST the first for the main fake labels the second for the second layer possibly there could even be obfuscated cases needing more than two runs 18 os SAR yV RY Wr SVAN gt CLES Pg iS we SSeS OZ CA ko we ras Y axis X axis Figure 10 A scene with T X labels Each label is positionned differently with respect to the ends of the segments We didn t draw arrows but a solution for the representation of vectors in space could be to use small cones 19 7 Animations 3 dimensional objects appear differently when seen from different points of view and therefore a 3D scene is deprived of a lot of interest if it cannot be animated in some way The most elementary form of animation is obtained by choosing the point of view at the same time as the scene is constructed When we need different points of view this then requires either a tedious repetition of the processing or some automation taking advantage of the camera path This approach was the one used in our first 3d package where a series of PostScript files could be produced by slight changes of the camera parameters position and orientation A similar approach is also used by the FEATPOST package Although such approaches are most useful in certain cases a more natural approach is also needed where the scene is merely animated by a high level interface such as a mouse or a keyboard MP2GL pro
6. METAPOST is not a 3D engine 3D objects can be defined in the lan guage but doing so is tedious Moreover the representation of a 3 dimensional scene requires an algorithm for hidden faces removal which is time consuming Currently the few existing 3D extensions to METAPOST only handle special cases For instance our own 3d extension was able to handle convex polyhedra but not always when they were overlapping 11 But even if a general hidden faces removal algorithm were implemented it wouldn t be the end of the story METAPOST has known numerical limitations and such an algorithm as well as other features like shading would stretch it to its limits It appears therefore desirable either to extend the core METAPOST and include native 3D support or to use an external processor for the 3D compu tations With the first approach one is led to rebuild all the 3D algorithms which are already available elsewhere It may be a sound approach and it may provide inherent advantages but there is a long way to go With the second approach on the contrary an existing 3D engine is used and combined with METAPOST This may lead to some inefficiencies for instance when the 3D engine doesn t know about internal METAPOST parameters but the advantages seem to far outweigh the limitations at least for the time being The second approach however leads us to a crucial question if we use METAPOST with a 3D engine do we need METAPOST at all In
7. a cube of edge OpenGL units because METAPOST uses PostScript points and 2 54cm corresponds to 72 PostScript points How big OpenGL units are is irrelevant since it all depends on the position of the camera The first example in figure 3 was obtained with the following commands begin_scene disable_lighting setwirecolor 0 0 0 black setlinewidth 0 5 wire_tetrahedron end_scene and the second is a mere call to solid_tetrahedron with no other settings The following objects are based on quadrics and there is only one version of each Figure 3 A regular tetrahedron in wire frame and another in solid mode Figure 4 The standard teapot in wireframe Disk disk name r R slices rings Partial disk partialdisk name r R slices rings startangle sweepangle Sphere sphere name r slices stacks Cylinder cylinder name rbase rtop height slices stacks 4 3 Objects constructed from paths Paths are a very convenient structure in METAPOST Although METAPOST paths are limited in 2D we can readily build 3D objects using them 4 3 1 Simple cyclic paths The first step is to evaluate a path and store it in a way appropriate for OpenGL Currently we merely store the points used to build the path not the control points In other words only three points are used in a path such as z1 z2 2z3 Future versions may offer options to save control points in order to take advan tage of NURBS Possibly the computa
8. a limited z buffer as well as PHIGS syntax Phan has also worked on simulating transparency in METAPOST Sadly there is no documentation available other than the source code 9 4 MetaGraph3D This package is a Java interface for constructing simple 3D scenes interactively The documentation available is very scarce or in a form which is unfortunately not easily searchable 8 9 5 FEATPOST The FEATPOST package by Luis Nobre Gongalves is written in METAPOST and provides 3D functionality 5 6 As the author describes it he wrote it in METAPOST because he wanted to keep the METAPOST machinery Like MP2GL FEATPOST uses the color type for points Several projec tions are provided including a fish eye projection 1 The hidden object removal is done by sorting the objects by distance from the point of view There are therefore also cases where the objects will not be drawn correctly The package comes with a large number of macros for various physical appli cations For instance it has a provision to draw triangular or hexagonal meshes It can also produce non interactive animations by a procedure similar in spirit to the one used in our 3d package METAPOST outputs are transformed in bitmaps with netpbm and these bitmaps are merged into an MPEG file The author of FEATPOST is now contemplating a reimplementation of his macros into 3DLDF see below because of the numerical advantages the latter provides 8In spite of our efforts
9. cinelerra 8 Limitations textures blending and other advanced features A number of METAPOST features are not handled at least not outside META POST For instance there is no support for special pen shapes However 5It should be remarked here that the author of 3DLDF see 9 6 intends to define 3D pens such as sphere and cube balls 20 Figure 11 JPEG output of the scene shown previously The labels are the raw TEX labels shown as Helvetica strings in orthographic projection 21 if MP2GL is extended to the point where a METAPOST run can create an additional layer to a 3D drawing then such METAPOST features could come into play The main intrinsic 2D feature in use by MP2GL is the path type but even in this case we are only making an elementary use of it So far paths are considered as polygons but this could change if we start generating NURBS There is no support for textures in GL2PS and anyway textures may be thought of inherently non vectorial As a consequence there is no support for textures in MP2GL In principle transparency or blending is supported by GL2PS when its out put is PDF This feature however is not yet supported by MP2GL Since shadows are by default not handled in OpenGL they are also not covered by MP2GL They could be handled if implemented in OpenGL A 3 dimensional version of METAPOST should also be able to compute in tersections both real and projected This is o
10. provides functions for the rendering of 3D scenes and these can include lights shading and various other effects The scene is drawn on a screen and adapted to its resolution An OpenGL frame is a bitmap OpenGL uses the z buffer algorithm for hidden parts removal and this is done seamlessly We can therefore envision producing 3D bitmaps but fortunately there are also ways to save a 3D scene not as a bitmap but as a vector graphics which is independent of the screen size This is made possible by the GL2PS library 2 figure 2 1 MetaPost code ps Mpui OpenGL code GL2ES y 3D PostScript code GL 2Ps 2D PostScript code animation JPEG Figure 2 From METAPOST to 3D PostScript 1However GL2PS has several limitations and this will have as a consequence that there are 3D scenes which are constructible with MP2GL but which cannot be saved in PostScript 2 2 Overview and scope of MP2GL MP2GL currently provides a limited interface to OpenGL and tries to be faith ful to the motivations given above Our aim so far has been to show the feasibility of this approach and we have therefore only coded a few representative features It is hoped that this approach will be extended if it proves useful More specifically it occurred to us that most 3D objects that one wants to build are e either geometrically very simple for instance a cube a sphere e or obtain
11. the origin of the cur rent referential We also have functions to compute the usual vector operations and in par ticular the normal of a surface given three points 4 2 Basic objects The current version of MP2GL provides only a few basic objects among them the regular polyhedra the sphere the cylinder etc All these objects either exist in OpenGL or can easily be constructed Although these objects seem to be special cases they can actually be transformed For instance a sphere can be used to obtain an ellipsoid by changing the scales of the axes A number of basic objects are provided in two versions solid and wire frame e Regular tetrahedron solid_tetrahedron wire_tetrahedron in addi tion there is a new_tetrahedron macro building a solid tetrahedron from four points e Cube solid_cube size wire_cube size e Octahedron solid_octahedron wire_octahedron e Dodecahedron solid_dodecahedron wire_dodecahedron e Icosahedron solid_icosahedron wire_icosahedron e Sphere solid_sphere radius slices stacks wire_sphere radius slices stacks e Cone solid_cone radius height slices stacks wire_cone radius height slices stacks e Torus solid_torus r R nsides rings wire_torus 1 R nsides rings e Teapot solid_teapot size wire_teapot size The dimensions will be the METAPOST dimensions interpreted as OpenGL dimensions So if the object solid_cube 1cm is drawn it will actually be
12. we were not able to locate an independent documentation file It is hoped that future version of MetaGraph3D will contain separately that is not hidden within an archive file a PDF file describing the whole system Our 3d package doesn t use that type although it was suggested to us in 1997 11 10In MP2GL such a projection is naturally obtained when a large field of view is chosen 23 9 6 3DLDF 3DLDF is a 3 dimensional drawing software with a METAPOST output created by Laurence D Finston 1 The program is written in C using CWEB As of version 1 1 5 January 2004 the input code had to be written in C and then compiled 11 In 3DLDF there is a Point class and a Point is subject to the usual oper ations translations rotations etc Similarly there is a Transform class for storing 4 x 4 transformation matrices on homogeneous coordinates Transforms can be applied to Points inverted etc 3DLDF has provisions for drawing Points and labelling Points It also has a 3D Path class similar to METAPOST s path but in 3D These paths can be drawn and filled The system provides a number of plane geometric figures polygon rectangle circle etc There are also a number of solid figures cuboids polyhedra It allows the specification of the projection For the perspective projection 3DLDF needs the specification of a camera 3DLDF s hidden surface algorithm currently doesn t work for intersecting surfaces 3DLDF
13. 44 4 aaise bye eee bed 10 A 3 2 ets Of paths sai m ko eee ed eS Se ace E 11 4 3 3 Other path constructed objects 11 4 4 Low level constructions 2 0000000 0s 11 4 5 More on wire frames e a s acsi wi a 44 09944 2 ee pee 4 13 4 6 More complex surfaces NURBS 13 4 7 Lights and Colors eie a joj ab Be a ek a e a a a er 14 5 3D Equations 14 6 Text support 18 6 1 Basic text o 2 ook We ae ak eS Se Ee ee gh 18 6 2 More complex marks 0 0000004 a 18 7 Animations 20 8 Limitations textures blending and other advanced fea tures 20 9 Related work 22 9 1 Our own METAPOST 3d aka mp3d package 22 9 2 PSTricks 3D ii 4 2 eae te ee eee ea a we a 23 93 GOLA na sh eh of Ge eins oop ss SS BB ehh ns ae ae ges GR A A 23 9 4 MetaGraph383D 0 00002 eee ee 23 05 REAIPOST ix 42454 26 e amp dad o dd ol Soe Mislead dh ak S 23 9 6 SWEDE 22464325 an Re bbe dt oe oar ye 2 BAe eee 24 10 Conclusion 25 11 Acknowledgments 25 1 Introduction METAPOST is a language aimed at the description of technical drawings and is adapted from METAFONT 8 7 With METAPOST one describes a two dimensional drawing with primitive objects such as points and paths connect ing these points The language of METAPOST is very rich and a number of extensions have been written in particular for handling graphs or objects 1 1 Limitations of METAPOST However
14. EE and then call this function from within ML by adding def new_pyramid expr name n r h enddef 4 5 More on wire frames As it was shown previously a number of objects have a wireframe variant Other such objects can be created Moreover whenever an object is made of lines we can set the line width and also the stippling properties figure 6 Figure 6 An icosahedron with line stippling 4 6 More complex surfaces NURBS NURBS Non Uniform Rational B Splines provide a way to have 3D Bezier curves or surfaces We haven t implemented an interface to them yet but there isn t any problem doing it 13 4 7 Lights and colors There is currently a very simple support for colors in MP2GL We can set the color when there is no lighting and these colors are used for wire frames When lighting is on we can define the material properties of the next sur faces These material properties follow the OpenGL model and cover the emis sion properties the diffusion properties which account for the main color effect the specular properties the ambient properties and the shininess Similarly light support is very primitive on the METAPOST side A scene is currently lit by two lights one at a fixed position another on the camera There is no support to add lights from within METAPOST but it could easily be added It is also easy to change the lights within the OpenGL code 5 3D Equations METAPOST comes with a very conveni
15. MP2GL prototyping 3D objects with METAPOST and OpenGL Denis Roegel LORIA Nancy France 7 March 2005 Abstract METAPOST was created with 2D graphics in mind and in spite of various extensions added during the last few years it doesn t seem well adapted for 3D technical graphics However there are cases where simple but realistic 3D graphics are needed for instance for inclusion in an arti cle and there are also cases where 3D objects are mere 2D objects with added depth In such cases an approach combining METAPOST with an OpenGL environment proves very useful and allows for interesting appli cations in particular the prototyping of 3D objects for use independently from METAPOST Such an approach is also a smooth way to get intro duced to OpenGL MP2GL is our first attempt towards this direction Contents 1 Introduction 2 1 1 Limitations of METAPOST 0 00 2 12 Motivationsiof this work sa 444 4 6444094 Hee2 hades a 3 2 MP2GL 3 21 OpenGL aan c ic ec44 n Fat hae ee eewevevadaad 4 2 2 Overview and scope of MP2GL 0 4 5 3 Elements of OpenGL 6 Sul ODCCE a a dade Ga ww aha eae ede seem ia de 6 3 2 Projections s s c saa me ee RS Rw we a ea R 7 4 The MP2GL interface to OpenGL 7 4 1 Coordinates and coordinate transformations 7 4 2 Basic objects sae 206 4 idan 408 Dated eee tw aaa ead 8 4 3 Objects constructed from paths 00 10 A 3 1 Simple cyche paths
16. ecause OpenGL is not always correctly implemented GL2PS uses the feedback buffer of OpenGL and this buffer may lack some of the elements it is supposed to contain or because GL2PS is still in development or for other reasons But it does anyway seem to us that we have achieved our goal and that it is or will be very easy to produce a great variety of high quality 3D graphics with MP2GL Moreover there are several possible future directions which could be developped in particular through a collaborative effort e The MP2GL code could be extended on the METAPOST side in order to cover a larger subset of OpenGL e the OpenGL library could be extended with various objects useful from a METAPOST perspective But for many specific applications some of the reviewed tools may both be more adapted more complete and maybe easier to use than MP2GL especially with respect to text handling when all of the control lies within METAPOST Our tool should therefore be seen as a new possibility to produce not only certain 3D graphics but also to prototype objects which can be used beyond an article for instance as part of a complex animation 11 Acknowledgments The author would like to thank Christophe Geuzaine for his kind help during the development of MP2GL References 1 Laurence D Finston 3DLDF User and Reference Manual Manual edition 1 1 5 1 January 2004 2 Christophe Geuzaine GL2PS an OpenGL to PostScript printing library 2004
17. ed simply from 2D objects a cube a prism e or composed of simpler 3D objects In other words we think that the greatest amount of time when building 3D objects will normally be spent on adding depth to 2D shapes The reason behind this observation lies of course in either the manufacturing process of the objects we want to reconstruct or in the way they function For instance almost every moving object is actually moving in translation or around an axis consider a car a steering wheel etc and this usually is reflected in the shape of the object MP2GL is not limited to such objects but it has facilities for handling them In particular MP2GL makes it possible to use paths for specifying shapes used in 3D objects The main features of MP2GL in this experimental version are e METAPOST input language e structures for points and homogeneous coordinates e interface to OpenGL objects for instance polyhedra e ability to build low level objects made of faces e ability to use METAPOST paths as a basis for prisms including non convex prisms with holes e equations can be used for positioning objects in space e C code using the OpenGL library is produced with a minimal animation interface e from a given point of view either a bitmap or a PS output can be produced o TFX labels can be added and adjusted after the PS production e the C output can be edited and extended e the objects created with MP2GL can be us
18. ed without METAPOST e objects can also be created in OpenGL for use by MP2GL MP2GL has only been tested on linux but should be easy to adapt to other platforms provided GL2PS is ported there The animation produced by MP2GL provides a standard set of lights and a standard position for the observer These can easily be changed but not yet from MP2GL in this preliminary version For some of the objects the METAPOST interface is very simple This is for instance the case for the regular polyhedra Other objects require more work The reasons why an object should be coded in METAPOST or in OpenGL have to do with the need to use a METAPOST structure for instance a path value or with a greater familiarity of the user with one of these two languages MP2GL should actually be seen as a gateway from METAPOST to OpenGL both for the code which is translated in OpenGL and for the user who can seamlessly learn about OpenGL and try to make changes to the produced code 3 Elements of OpenGL 3 1 Objects The scene itself is made of objects which are themselves made of faces For instance a cube is made of six faces Objects or faces are built using points in a given referential OpenGL provides means to change the referential The basic transformations in OpenGL are e translation glTranslatef x y z e rotation glRotatef a x y z e scaling glScalef x y z e saving a position glPushMatrix e restoring a position glPopMatrix
19. ene 15 use_object ti use_object t2 use_object t3 use_object t4 end_scene Figure 9 A scene where the tetrahedron shown left has been split in the four tetrahedra at the right and the sphere is going through one vertex of each tetrahedron The tetrahedra are still bound together In order to split them we introduce translations from the center begin_scene PushPosition TranslateV K D use_object ti PopPosition PushPosition TranslateV K A use_object t2 PopPosition PushPosition TranslateV K B use_object t3 PopPosition PushPosition TranslateV K C use_object t4 16 PopPosition end_scene The initial point K has now been split into four points As an interesting 3D geometry exercise we can use these four points in order to find the sphere going through these points and draw it This is done as follows Let Wi W2 W3 and W be the four vertices stemming from K Point W W1i K K A W2 K K B W3 K K C W4 K K D The center of the sphere is obtained by first finding a median plane on which the center lies then a line then the center itself by a total of three plane intersections each being handled with two whatevers each This is summarized below Point V Vi 5 W1 W2 V2 crossproduct W2 W1 W3 W2 V3 crossproduct W1 W2 V2 V4 5 W2 W3 V5 crossproduct W3 W2 V2 v6 5 W3 W4 V7 crossproduct W3 W2 W4 W3
20. ent resolution of linear equations This feature makes it possible to use a declarative approach for a number of drawings For instance in order to compute the intersections of the medians of a triangle figure 7 one can do beginfig 1 pair A B C D E F 1 A origin B 6u 0 C 2u 4u D 5 B C E 5 C A F 5 A B I whatever B E whatever A D draw A B C cycle draw A D draw B E draw C F pickup pencircle scaled 3pt drawdot I endfig Figure 7 A triangle and its medians Imagine now that we want to do the same in space say with a tetrahedron Can we do that 14 The answer is yes And we can use the Point structure for the equations Like above we first do the computations before drawing anything We start with four points the vertices of the tetrahedron D B Figure 8 A tetrahedron and its center Point A B C D vertices Point E F G H I J edge middles Point K tetrahedron middle A 0 0 0 B 1 0 0 C 0 3 1 0 D 0 5 0 5 1 E 5 B C F 5 C D G 5 B D H 5 A D I 5 A B J 5 A C K whatever G J whatever H E Now we have the center of the tetrahedron and we can split it in four smaller tetrahedra A B C K C B D K A C D K and B A D K We can use the predefined function for tetrahedra new_tetrahedron t1 A B C K new_tetrahedron t2 C B D K new_tetrahedron t3 A C D K new_tetrahedron t4 B A D K and later begin_sc
21. eraction so that a 3D scene created with METAPOST can be animated and the best point of view chosen something that is often best done interactively Once the point of view is found a vector snapshot should be produced Since 3D vector graphics can be obtained by other tools than METAPOST the answer to our question on the need of 3D METAPOST at all is therefore that it is only needed if we want to retain an homogeneous framework common with 2D METAPOST and if we want to be able to have a smooth integration with TEX 2 MP2GL MP2GL is the bridge we have developped between METAPOST on one side and OpenGL on the other Such a bridge is interesting not only for producing 3D figures for inclusion in articles but also for 3D objects independent of articles We will see that linking METAPOST to OpenGL isn t that catastrophic in terms of loss of critical information because METAPOST could reuse later outputs Figure 1 A TeX logo produced with MP2GL This scene combines an object built from METAPOST paths the T with the three holes and a predefined OpenGL icosahedron in solid output to which a slightly larger wire frame icosa hedron was superimposed 2 1 OpenGL OpenGL is an API for 3D graphics which has become a standard in the indus try 18 There are OpenGL libraries for many languages and many applications are using such a library for their advanced graphics Various games for instance Quake also use OpenGL OpenGL
22. ernal processing stage Finally one problem in METAPOST is the problem of the numerical val ues This is one reason why the author of 3DLDF created his package That is by working in C the various computations associated to intersections projections etc are easier and less constrained 11The use of an external processor has also recently been applied to 2D plots by Brook Moses 10 12In MP2GL this is not done at the METAPOST level but at the OpenGL level We could do it at the METAPOST level but we assume the user wants to animate the object and find the most convenient location Nothing prevents him her of setting the projection as perspective parallel isometric or axonometric 24 The MP2GL approach on the contrary uses very little computational fea tures of METAPOST apart from a few matrix operations often confined to very small values Moreover objects can always be created at a small size and then scaled within OpenGL In the rare cases where an overflow would occur a spe cial mode could be selected which would turn off the matrix computations on the METAPOST side In most cases these matrix operations are not needed in METAPOST 10 Conclusion Our package is still in its infancy and our purpose has only been to explore its feasability In particular it should be noted that most of the features presented here are still unstable and that function names are likely to change Technical problems can arise either b
23. f course difficult and we haven t made any attempt to solve this problem This doesn t mean that the problem cannot be solved but it remains to be done either by a representation of the objects in OpenGL or in METAPOST Finally there is currently no support for a CSG construction of objects but a CSG OpenGL library could be linked to MP2GL and offer tremendous new possibilities 9 Related work Our work is not isolated and there have been a number of attempts at handling 3D with or around METAPOST over the years One of the concerns of these packages has been to obtain a smooth integration with TeX and in particular to retain the vectorial characteristics of the output A number of commercial products and other products independent of TFX produce vector output but we will only survey the main TX related tools 7 As it will be readily visible these packages all have many features which are currently not present in our system However many of these features could be added to MP2GL 9 1 Our own METAPOST 3d aka mp3d package This package was described in TUGboat 11 and was later extended with fea tures for space geometry 12 The package provided the basic tools for the construction of 3D scenes made of segments and faces Projections could be either pespective or parallel but there was no support for hidden faces removal The 3d package was created with 3D animations in mind and in particular but not exclusively
24. has four different ways to sort objects It can also find the intersection of a few non arbitrary paths such as two polygons a line and a polygon two ellipses etc So far 3DLDF doesn t have support for lights shading etc 3DLDF does not have linear equations solving in the METAPOST style al though there are plans to add equations It also doesn t have macros Current plans are to implement NURBS in 3DLDF but NURBS do exist in OpenGL and could easily be interfaced in our work 3DLDF appears as an interesting approach somewhat symmetrical to ours It does however remind us of our two major previous METAPOST packages d 11 and METAOBS 13 14 3DLDF recodes a number of objects such as polyhedra as we did modestly in our 3d package in 1997 3DLDF also has an object oriented approach not unlike what we did for plane objects in METAOBJ 3DLDF is bound to code a lot of features which are already present in OpenGL but it may also produce code which can be reused in our OpenGL library For instance the author of 3DLDF plans to code a number of exotic polyhedra which should be easy to recode in C Actually if our C part were written in C we could even output 3DLDF code 3DLDF is also a very interesting approach in that its author is trying to extend to the third dimension features that may not be that easy to transfer such as pen shapes In a way 3DLDF is aiming at the most orthodox extension of METAPOST albeit by using an ext
25. ions to the original source This can also be achieved in 3D but it is sometimes necessary or useful to work on the output and therefore to be able to modify labels and their positions within the TX file For instance it can be necessary to replace a label by a shifted label and an arrow The current version of MP2GL does not yet support this feature but a future version may produce an additional TeX METAPOST file which would be superimposed to the PostScript output and the labels perhaps by the use of a simple postprocessor 6 2 More complex marks The usual 2 dimensional drawings have both curves labels but also additional elements such as dots arrows angle marks textured text etc These elements can all be added to a 3 dimensional drawing provided we know the final 2 dimensional positions However this is often not the case GL2PS only provides for labels at specific positions not for more complex ele ments such as arrows connecting two points There are two simple solutions to this problem 1 the point of view can be frozen from within MP2GL in which case MP2GL would know exactly where a point is projected this would require adding support for the projection transformation something that wouldn t be difficult 2 a less constraining solution would be to introduce fake labels which would make it to the PostScript or T X output and these labels could then in turn be used to specify paths possibly in METAPOST arrows
26. library provides a function for multipath prisms In that case paths need to be split in positive paths outside paths and negative paths holes These paths do not need to be convex and this function can therefore be used to draw simple non convex prisms The user creates two arrays of paths and hands them to the storepaths function path pospaths negpaths numeric np nn np 2 nn 3 pospaths0O pospathsi negpaths0 negpathsi negpaths2 storepaths pospaths np negpaths nn Paths_1 Then the multipath prism is obtained with new_multipath_prism Prism2 Paths_1 3cm After that the object is used like any other to compose a scene Again these paths can be drawn in an ordinary METAPOST figure environ ment The TEX logo in figure 1 was obtained by this process First three ordinary METAPOST paths have been defined one for T one for E and one for X These paths were reduced and fed as holes of the non reduced T into the new_multipath_prism definition A solid and a wire icosahedron were drawn over the TEX logo 4 3 3 Other path constructed objects Other objects based on paths can be imagined and it suffices to either add them to the OpenGL library or to the METAPOST library 4 4 Low level constructions Objects can be constructed using low level components For instance a cube can be constructed from its six faces as follows figure 5 build
27. other words why would we want to use METAPOST for 3D drawings We come to the motivations of our work 1 2 Motivations of this work Among the main reasons for using 2D METAPOST in a T X environment are that e it is a natural partner of T X e it can produce high quality and accurate technical drawings e it produces vector graphics e it has a nice declarative approach where equations can be used to specify points e it has nice types especially the path type e and it is fun to use The main reasons for using 3D METAPOST should at least include the previ ous ones In particular we are looking for a way to produce 3D vector graphics while at the same time retaining the declarative approach and the ability to use types such as paths But we also want more With 3D come more needs Consider first a planar drawing The coordinates of the origin of such a drawing are actually irrelevant Whether a square is drawn with its lower left corner at 0 0 or at 1 1 doesn t make a difference as long as the bounding box of the visible part of the drawing is used for insertion With 3D we have a camera or an observer and its location determines the point of view But with 3D there is also the legitimate desire to have motion or animations Such was actually the motivation of the 3d METAPOST package we wrote in 1997 11 With it small GIF animations could be produced But these animations were not interactive Now we want int
28. s a square on 0 X Y def build_cube_face begin_convex_polygon normal 0 0 1 vertex 0 0 0 vertex 0 1 0 vertex 1 1 0 11 vertex 1 0 0 end_convex_polygon enddef beginobject Cube set_diffuse_color 1 0 0 0 0 0 build_cube_face bottom face PushPosition Translate 1 0 0 RotateY 90 set_diffuse_color 0 0 1 0 0 0 build_cube_face Translate 1 0 0 RotateY 90 set_diffuse_color 0 0 0 0 1 0 build_cube_face Translate 1 0 0 RotateY 90 set_diffuse_color 1 0 1 0 0 0 build_cube_face PopPosition PushPosition RotateX 90 Translate 0 1 0 set_diffuse_color 1 0 0 0 1 0 build_cube_face PopPosition PushPosition Translate 0 1 0 RotateX 90 set_diffuse_color 0 0 1 0 1 0 build_cube_face PopPosition endobject Figure 5 The cube entirely built within MP2GL 12 This however creates an object which is not parametric This isn t very problematic for the cube as we can still scale it afterwards However other objects benefit from being parametric There is currently no extra support for such objects as most of these can either be obtained from paths or can be integrated in the OL library with a minimal effort For instance assume we want to add a general pyramid function having a regular polygonal base We can build any instanciation and this can be enough but we could also define the OpenGL function void create_pyramid GLint n GLfloat r GLfloat h E
29. the animation of regular polyhedra The package was used by Denis Barbier and Sami Alex Zaimi for some experiments We have also implemented extensions for handling parametric 5CSG Constructive Solid Geometry refers to booleans operations union intersection etc used for the construction of objects TA review of several tools has already be given recently by the author of FEATPOST 5 We would like to remark that this review oversimplied the features of our 3d package and focused mainly on the application to convex polyhedra 22 curves surfaces and in particular revolution objects These extensions were not released because they were limited by the absence of a hidden parts removal algorithm which we were reluctant to code in METAPOST 9 2 PSTricks 3D PSTricks was written by Timothy van Zandt and its maintenance was stopped around 1993 but over the years many extensions were added in particular by Denis Girou The number of these extensions shows that the PSTricks commu nity is very active and that the PSTricks model appears very fruitful Several extensions handle 3D objects 3 4 15 16 9 17 9 3 m3dplain Anthony Phan made a number of very interesting experiments over the years He borrowed our ideas for the creation of animations but the rest of his package is different and aimed at the manipulation of various mathematical objects such as polyhedra molecules fractals etc In part of his code he tries to implement
30. tion of control points could also be added to the OpenGL library mimicking the METAFONT METAPOST construction of control points path p ieee cycle storepath p Path_P Once the path is stored which saves its points in a C array an object built on path p will use the name Path_P of the path For instance a prism can be built as follows new_prism Prism1i Path_P 3cm The last parameter is the height of the prism MP2GL functions prefixed with new_ create new C functions represent ing the objects Calls to these functions can be produced within METAPOST with use_object For example the scene involving the previous object can be constructed as follows begin_scene use_object Prism1 end_scene The 2D elements involved in the construction can be examined in the same METAPOST run by the help of an ordinary figure environment beginfig 1 draw p endfig In other words the compilation of the METAPOST file will have two or more outputs a C file which will be postprocessed by a C compiler and one or more ordinary PS figures 4The current version of the prism function works only for convex paths because it relies on the polygon construction of OpenGL For more general cyclic paths the multipath prisms should be used 10 4 3 2 Sets of paths Sets of paths can be used to create shapes with holes These shapes will be constructed in OpenGL using a tessellation algorithm The MP2GL
31. vides this kind of interface allowing the user to examine the scene under various angles and choosing the best configuration for a screenshot PS or JPEG Specifically the user can e get closer PgUp or farther PgDn from the scene e change the orientation roll lt gt pitch Up Down yaw Left Right e rotate the whole scene around the center with the mouse e change the field of view and e and save the scene as a PS file s or a JPEG file j The default animation uses a perspective projection but this could easily be changed and even toggled through a key It could also be selected from METAPOST in the future This is of course a minimal interface We could and actually can go further in that the animation would not only be that of the user but an internal animation The objects which have been created can actually interact in various ways For instance we can have a wheel turn and this can produce a piston translation Such animations are not directly supported but it is currently sufficient to create the objects and then to make slight modifications to the OpenGL code in order to obtain the internal animations The OpenGL code can also be easily extended to produce a motion of the camera and at the same time generating a bitmap at regular intervals Various tools make it then possible to produce MPEGs for instance ppm2mpeg and other tools can be used to edit them for instance
32. ywhere in space and the handling of coordinates is done by mimicking the OpenGL operations Although in many cases we do not really need to know where we are but only how we came there we provide means to obtain an exact location from relative motions translations and rotations 4 1 Coordinates and coordinate transformations Coordinates are seen as triples but are actually manipulated as vectors in a 4 dimensional space Nevertheless we use a 3 dimensional structure to store points namely the color structure renamed Point def Point color enddef def Xpart redpart enddef def Ypart greenpart enddef def Zpart bluepart enddef Internal transformations involve 4x 4 matrices which are stored as 2 dimensional arrays of numerics The user need not care about matrices even less in METAPOST than in OpenGL The only transformations needed are the following e ResetPosition return to the origin e ResetLocalPosition return to the local origin e PushPosition save the current position e PopPosition restore the formerly saved position if there was one e Translate x y z move by x y z e TranslateV v move by vz vy Vz e RotateX a rotate a around the X axis e RotateY a rotate a around the Y axis e RotateZ a rotate a around the Z axis e Scale sgr Sy Sz scale coordinate X by sz Y by sy Z by sz a scale with one negative value does a reflection e CurrentPosition returns the absolute position of
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