Special Relativistic (Flight-)Simulator

Transcription

1 Speial Relativisti (Flight-)Simulator Anton Tsoulos and Wolfgang Knopki Abstrat With speial relativisti visualisation it is possible to experiene and simulate relativisti effets whih our when traveling lose to the speed of light. These effets are geometri distortions, like length ontration, as well as optial effets for example the Doppler and the searhlight effet. There are various approahes to implement these physial effets; image-based rendering, polygon rendering or ray traing to name but a few. In this paper we take a loser look at image-based rendering aswell as polygon rendering with the aim to, reate an other method, alled advaned polygon rendering. Advaned polygon rendering is a further development of polygon rendering to provide better results ompared to regular polygon rendering. Our implementation uses an open soure 3D engine to load senes with little effort, as well as GPU shader programming with modern OpenGL to ahieve best performane. Index Terms Visualisation in Physial Sienes and Engineering, speial relativity, advaned polygon rendering, image-based rendering 1 INTRODUCTION Sine its publiation in 1905 Einstein s theory of relativity has been in the enter of modern physis. But unlike traditional Newtonian physis, its effets are hard to observe and experiments aiming at its observation are beoming inreasingly big and ostly. Computational simulation of relativisti effets has thus beome a vital part in modern physis. These effets in partiular ompose of the Doppler effet, being the shift of spetrum of the light emitted by a visible objet. As the speed of the observer hanges, so does the wavelength of the observed light whih may ause a shift of spetrum from the formerly invisible infrared or ultraviolet part into the visible spetrum. For example heat radiation emitted by objets turns now visible. desribed by Weiskopf [12] and Müller [8], first visualisation attempts date as far bak as Visualisation of speial relativisti effets by means of omputation, however, are a relatively new field of study: In 1989 Taylor [11] developed tools for eduational purposes. Fully interative visualisations of relativisti effets are even more reent: In 2005, Kraus and Borhers offered a relativisti bike ride through the anient enter of Tübingen (see [5]), and in 2007, Savage et al. [10] developed a first person simulation of relativisti effets in an image-based approah. Most reently, there have been further publiations in the field of speial relativisti visualisation: In 2010, Müller et al. [8] and Weiskopf [12] used polygon rendering and loal ray-traing tehniques to visualize relativisti effets. Unlike the formerly widely used image-based methods, these do not have the issue of inreasingly poor resolutions with high veloities and therefore high rates of distortion (details are disussed in Setion 3.1.1) Figure 1: The aberration effet explained: Image shows the observer at rest: Light reahes him equally distributed from all sides. Image shows the observer in motion to the right, light previously reahing him from the sides now omes from ahead Other effets that an be observed are aberration effets suh as the distortion of the surrounding. The light that seemingly reahed the observer from a ertain diretion while he is moving at low veloities seems to ome from other diretions (see Figure 1). Objets that are behind the observer do now appear sideways or in front of the observer and the objets in front of the observer seem to be smaller and thus inreasingly distant. 2 RELATED WORK Due to its nature, the observation of relativisti effets relies purely on omputational simulation, sine these effets an not be observed under normal irumstanes found on earth as it is tehnially impossible to reah veloities even far from the speed of light. As Anton Tsoulos Wolfgang Knopki anton.tsoulos@ilw.uni-stuttgart.de 3 OUR APPROACH The goal attained by this projet was to get a disussion base for visualisation of relativisti effets and hereby ompare the primitive image based approah with the approah of advaned polygon rendering. Advaning in polygon-rendering is the amelioration of simple polygon rendering by reating new verties depending on the urvature of the sene elements. A further goal was to adapt an existing game engine to our needs to provide possible future support for relativisti game development. Conerning the engine used, we deided on Irrliht [2], a free engine whose most attrative feature was the ability to easily load a variety of sene formats inluding Quake 3 levels (.bsp) or OGRE meshes (.mesh). As graphis API OpenGL was used for the sake of OS-interoperability, although Irrliht also offers the usage of DiretX. 3.1 Image-Based Part To omplete the task of speial relativisti simulation in an image-based approah we use a shortened version of the lassial graphis pipeline. The sene is at first rendered six times by six ameras that are orthogonal. These ameras have eah a field of view of The images obtained through these ameras are then put together to one ubemap texture that is projeted on a full sreen quad implemented as a sreen-sized retangle. During this projetion, relativisti effets are added by means of fragment shading. To do so, we reverted to the formula of aberration obtained through the Lorentz-transformation as desribed by Müller et al. [9] wolfgang.knopki@ilw.uni-stuttgart.de Let the veloity of light be denoted as. 1

2 the front ube and faes towards (0, 0, 100), the enter of the middle ube. In image 2, the observer appears to move with a speed of v = 0.5 forwards, towards the enter of the image. It is evident, that the ubes in front of the observer appear more distant than in the image before, and the ube sideways is distorted and more of it is visible. In image 2(), the aberration effet beomes even more apparent: The part of the ube previously being behind the observer is now visible and the ube as a whole appears heavily distorted and rounded. In Figure 3 another observation is visible. The big disadvantage of image based rendering of these effets is evident: Due to the implementation of the aberration by means of fragment shading, the simulation suffers from a lak of texture. This lak is due to the finite resolution of the texture:during omputation of the aberration effet, the parts of the texture are piked aording to the formulas mentioned above (see subsetion 3.1) and aordingly shifted. Thus, a small area of the sreen is blown up to over the whole sreen whih leads to these pixelated artifats seen in the piture. Let S be a system in motion with veloity v with respet to S v β=, 1 γ=p, 1 β2 and S a system at rest. Let Θ resp. Θ0 (0, π) and Φ resp. Φ0 ( π, π) be the angles in spherial oordinates for any given point. This yields the following equations (see M uller et al. [9] for details): os Θ0, γ (1 β sin Θ0 os Φ0 ) (1) sin Θ0 os Φ0 β, sin Θ (1 β sin Θ0 os Φ0 ) (2) sin Θ0 os Φ0. γ sin Θ (1 β sin Θ0 os Φ0 ) (3) os Θ = os Φ = and sin Φ = The equations (2) and (3) for Φ are divided and yield tan Φ = sin Θ0 sin Φ0. γ (sin Θ0 os Φ0 β ) (4) With this equation at hand, it is possible to simply program a fragment shader, that adds relativisti aberration effets to the projeted sene: At first, the texture oordinates are piped through the vertex shader and passed to the fragment shader. Here, the important part happens: The texture oordinates of the full sreen quad have to be transformed into three oordinates on the ubemap. This transformation is similar to the transformation from spherial oordinates to artesian: Let x, y [0, 1] be the texture oordinates of the full sreen quad. To get the oordinates on the ubemap texture, the oordinates have first to be saled. These new saled oordinates are x0 [0, 2π] and y0 [0, π] To obtain three dimensional oordinates in the ubemap (denoted as a,b,), they are transformed as follows: a = sin(y0 ) os(x0 ) b = sin(y0 ) os(x0 ) = os(y0 ) To add the aberration effet, this vetor v = (a, b, ) is first transformed to spherial oordinates to obtain angles φ 0 and θ 0 that are transformed aording to equation (4) and (1). Applying the inverse mappings yields angles φ and θ whih are transformed bak into a three dimensional vetor that points to the orresponding pixel on the ubemap texture. Thus, the image shown by the amera is distorted in aordane with speial relativity. The program does not only offer the omputation of the aberration effet, it does so interatively. It is possible to load different senes via ommand-line options. Basially any format Irrliht uses in supported, so far however, only senes in quake 3 map format (.pk3) [1] and senes in ustom Irrliht format (.irr) have been used. It is furthermore possible to sript the initial appearane of the program. To do so, the position, look at diretion and up vetor of the amera an be passed as options in the ommand line, as well as the veloity β Results The results obtained by this method are as follows: As an be seen in Figure 2, the sene is distorted: Image 2 is the referene image with the observer being at rest (v = 0). The sene seen here onsists of several aligned ubes whose edges have a length of 10 Units. 5 of the ubes are aligned equidistantly from ( 40, 0, 55) to (40, 0, 55), a sixth ube is positioned at (10, 0, 40). The observer is positioned at (0, 0, 40) right beside () Figure 2: The relativisti effets as seen by the image based approah: In image 2, the observer is at rest, in image 2, he moves with v = 0.5, in image 2() with v = 0.9 2

3 () Figure 4: Example of a rod under the influenes of speial relativisti effets; shows the rod at rest with β = 0. In the rod moves into diretion of the observer with β = 0.9 with a kind of zooming effet in the enter of the rod. () shows the same rod as from a sideview, whereby the distortion of the rod in form of a hyperbola is learly visible. The observer is thereby in front and in the enter of the rod. jets without urvatures or with strong edged urvatures an be seen, due to the non-linearity of the transformation into the photo-objet and the minor tessellation of the objet itself. By using advaned polygon rendering we try to minimize the effet to reate graphis and senes with higher quality of rendered objets. The possibility to use advaned polygon rendering ame first in the last ouple of years with the development of modern graphi ards and the introdution of programmable graphis pipeline whih an be ontrolled by shaders. Under these onditions, we were using modern OpenGL (OpenGL 4.0 and greater) whih gave us the important opportunity to use tessellation shaders whih we greatly used. With tessellation shader we an divide huge objets, respetively the triangles or other primitives they are assembled of, into smaller ones to inrease the quality of the piture and the granularity of an objet. In our example the primitives of the rod beomes divided into smaller primitives whih an be plaed to approximate a urve. The tessellation shaders are however not the only way for geometry subdivision. The geometry shader an be used as well, but even if it is also apable of this kind of work, the usage of tessellation shader is preferred sine the exlusive use of the geometry shader arries problems with it. The first problem is that the number of primitives, generated by one instane of the geometry shader varies beause some objets require a higher tessellation level than other whih ould lead to problems with the synhronization of data emissions and the order in what output primitives will take plae in the output buffer. Another problem is that the geometry subdivision within a single shader is done iteratively, whih is quite a waste in ase of a highly parallelized proessors like a GPU, whih results in a reasonable amount of delay. Beause of these reasons, the preferred way to reate a geometry subdivision are the tessellation shaders. These preonditions allow us to use the graphi pipeline as follows: Figure 3: Lak of texture aused by limited resolution of the texture: Movement bakwards with v = 0.9 in image 3, no movement in image Advaned Polygon Rendering The main idea of polygon rendering is the transformation of (all) points or objets from their 3D referene frame into the 3D referene frame of an observer. The result of this transformation is a so-alled photo-objet whih is a virtual objet whih shows the position where the light rays must been emitted to reah the observer simultaneously. However we develop Advaned Polygon Rendering (APR) in order to show and improve the rendered effets of the speial relativity and the quality of the image and onsequently the photo-objets at high speed. Due to the speial relativitiy, in high speed, geometrial objets appear distorted, for example an unbowed rod beomes formed as a hyperbola when it is flying towards an observer, as seen in Figure 4. Sine we want to know what an observer sees, we have to mind the time whih the light needs to reah the observer from an event, to reate a photo-objet. However, speial relativity views spae and time not as separate features but as a unity, whih is alled spaetime. A point within the spaetime an be desribed by a 4-dimensional vetor x = (t, x, y, z) ontaining the time and the 3D-oordinates of the event. The effet of the unbowed rod whih beomes bowed on high speed an be then explained by the finite speed of light. Light rays from both ends of the rod must be emitted earlier than from the enter in order to reah the observer simultaneously. Sine the rod is still approahing, the time whih the light needs to reah the observer desends, points loser to the enter of the rod are emitting light rays muh later than from the edges. Considering both, time and spae, all the single emission points taken together form a hyperbola. Inreasing β would even lead to a stronger urved hyperbola, up to a sharp edge. Unfortunately beause of these effets, an image from a mesh with few verties, reated with regular polygon rendering does not show the reality like it is supposed to be. Instead just distorted ob- The vertex shader is only passing the position of the verties multiplied by the world-matrix, to transform the verties withion the 3D-world and not in the 2D-image and the texture oordinates to appropriate map the textures later, to the next shader. The tessellation ontrol shader determines the tessellation levels and passes the verties to the evaluation shader. The tessellation evaluation shader is evaluating the points and transforms the additional verties aordingly. 3

4 The fragment shader itself does nothing more than to sample the textures and writing the olor Implementation of APR We assume at least three different referene frames; The referene frame of the observer and one or more referene referene frames in whih our objets are, as shown in Figure 5. These referene frames are in a global frame whih ontain all other. To desribe an event, whih ourred in its referene frame, in the referene frame of the observer we need the Poinaré-transformation. Thus it is needed to alulate the required Lorentz-matries, see equitation (5) and their inverse for eah frame to Poinaré-transform all verties of the objets, from their referene frame to the referene frame of the observer, in order to find the time where the objet must emit a light ray to be visible for the observer. Figure 6: Shemati intersetion of the worldline of the point with the bakward light one of the observer. They interset at position x i at time t i, whih is the time we want to know Figure 5: Sheme of the different frames within the world While it is not neessary, in our approah the observer is not moving and thus the speed (and beta) is zero, however the objets an move freely. So we need to alulate a Lorentz-matrix Λ β and its inverse Λ β for the stati observer aswell as the orresponding matries for eah moving referene frame of the objets. γ γ v x Λ β = γ v y γ v z γ v x γ v y ˆ1 3 + γ2 1+γ v v 2 γ v z (5) The Poinaré-transformation transforms the position of a point from its referene frame to an other referene frame. We use use this transformation to transform the observer from its rest frame S into the global frame S (6) and then again to transform the new point from the global frame into the referene frame of the objet S (7). x obs = Λ β1 x obs + a 1 (6) x obs = Λ β2 (x obs a 2 ) (7) While in our approah we assume that a 1 and a 2 for the relative shift of the systems are null vetors (meaning the systems are overlaid), it is not neessary. Furthermore we assume that the oordinate systems are axis-aligned. Doing so, a Minkowski-diagram an be used to depit the position of the observer and the position of the point within the spaetime and relative to eah other. Minkowski-diagrams have the advantage that they often show a pituresque desription of movement of an objet in spaetime. They illustrate the mathematial properties of spae and time in the speial relativity with the time as y-axis and the spae as x-axis. Having transformed the observer from its referene frame to the referene frame of the point, it is then possible to alulate the time t i in whih the observer is able to see the point. This an be ahieved by alulating the intersetion of the worldline of the point with the bakward light one of the observer, as seen in Figure 6. Beause of the finite speed of light the observer an only see events happened in the past, thus it is only neessary to interset with the bakward one and to ignore the forward one. here returns the length of the differene of two vetors. x 0 ob j = t i = x 0 obs (x ob j,x obs) (8) Calulating the intersetion between the worldline of the point and the light one of the observer yields the time where a light ray must be emitted from this point in order to reah the observer at observation time (8). Having the time in whih the observer will see the point, the point itself is now transformed into the referene frame of the observer using the same transformations vie versa (9) and (10). x ob j = Λ β2 x ob j + a 2 (9) x ob j = Λ β1 (x ob j a 1 ) (10) With the last inverse transformation (10), the position of the point where it will be visible at observation time, was alulated. This transformation is needed, sine it is neessary to plae the objets to their right plae within the tessellation evaluation shader to reate the photo-objet. Nevertheless all these Poinaré-transformations have to be repeated if the observation time or the position of the observer or objets hanges. Until to this point, there are little hanges ompared to regular polygon rendering. However it is still unknown whih tessellation level is needed to see the orret relativisti distortions at high speed. The main and new part, the alulation of the tessellation level happens in the tessellation ontrol shader. Given a triangle as input we take a pair of verties and alulate the diretion vetor #» r = v #» 1 v #» 2 from one vertex to the other. From both verties a minimal distane δ is taken to reate another point into diretion to the other vertex, to get a pair of two tuple with points t 1 = ( v #» 1, v #» 1 + δ #» r ) and t 2 = ( v #» 2, v #» 2 δ #» r ). Afterwards all four points will be transformed into the rest frame of S using the double-staged Poinaré 4

5 transformation, reeiving the tuple t 1 and t 2 with the transformed points. With these two tuples it is now possible to alulate two tangents for whih we further alulate the angle between them (Figure 7). The angle between both tangents is an indiator how strong the Figure 7: Shema how to get the angle α from a tuple of verties of a triangle. With a distane δ two other points into the diretion of the other vertex are taken, getting two tuples. Poinaré-transforming all 4 points results in the tuples t 1 = (v 1,v 11 ) and t 2 = (v 2,v 22 ). With them, two tangents an be alulated, and interseting them yields the angle α. transformation is, for example how muh a line bends depending on veloity. A small angle means stronger bending, whereas a big angle means that the speial relativisti effets are not that strong. With help of the angle it is possible to ompute the level of tessellation. However it is important to onsider that the effets of relativity are not linear but inrease exponentially with higher speed, thus the angle dereases faster the faster the observer moves. These steps shall be repeated for all possible ombinations of pairs, getting the tessellation level for eah side of the triangle. Not in possession of a good fomula we used an approximation whih gave us aeptable results, by hoosing the appropriate tesselation levels when the angle is within an speifi range degree: tesselation level = degree: tesselation level = degree: tesselation level = degree: tesselation level = degree: tesselation level = otherwise Anyway this is just an approximation used in our approah, obviously a better solution is to develop a alulation speifiation to get the tessellation level diretly and by not using ranges Results Advaned polygon rendering indeed hugely inreases the quality of the image. The tessellation greatly works and tessellates the objet into smaller parts, so that the objet is rendered orretly even lose to the speed of light. This an be greatly seen in the omparison between regular polygon rendering of a mesh of a ube with only verties on the edges and our approah, the advaned polygon rendering, seen in Figure 8. It is learly visible in the wireframe of the ube, that the egdes of the ube beomes subdivided into smaller lines, whih are plaed to approximate a urvature. By using advaned polygon rendering we experiene a signifiant inrease of details and aurray to the effets of the speial relativity espeially urvatures beome muh better approximated, sine it is not needed to provide additional verties but only on the orners of the objets. However sometimes artefats are visible. So it happens that sometimes from a partiular angle the whole objet disappears, until the angle of view is hanged again. This espeially happens often with speed lose to the speed of light. So the reason of these ourrenes might be due to floating point underflows or invalid operations (for Figure 8: Comparison between regular polygon rendering and advaned polygon rendering whih depits the expeted urvature of the objet muh better. The ubes are rendered as wireframe, both with β = 0.9 example division by zero) in the shaders. Sometimes also the textures get broken so that for example parts of the borders of the textures appear in the middle of the objet. 3.3 Comparison In Figure 9 the results of advaned polygon rendering and the image based approah are diretly ompared. To do so, the amera was plaed in the sene desribed in Setion It s position was set to (0,0, 38) right beside the front ube. As an be expeted, the sene itself is orretly distorted with both methods. However, in the image based part, the bakground is distorted, too, whih leads to the big white spaes seen in the right part of the image. During the advaned polygon rendering, only the objets plaed inside the sene are altered. Another differene between the methods beomes only visible after a loser look: Looking at the outer edges of the heavily distorted ube, the image-based approah ontinues to show a smooth rounded edge, whereas in advaned polygon rendering, the edges are visibly divided into straight lines. This effet is due to the different approahes in rendering: In the image based approah, the images obtained through the amera are altered, whereas in the advaned polygon rendered method, the geometry of the objets themselves is hanged. 4 CONCLUSION AND FUTURE WORK Looking at the image based part, we implemented the state of the art as referene for omparison with the advaned polygon rendering. As seen in the previous setion, the referene was well met. We did further use this method to test dynami loading of senes and parameters with the help of Irrliht. Conerning Irrliht, we had to refrain from using the most reent version as it does not offer all the funtionality we needed. To use tesselation shaders, we had to revert to version as this was the only version that did natively support these shaders (see [4]). To be able to render the sene to a ubemap texture, further modifiations on the engine were needed, but we ould use a path provided by the ommunity [3]. Conerning advaned polygon rendering, several advanements with respet to lassial polygon rendering have been made: The usage of high level tesselation allowed a muh higher auray of the distortion in omparison to the formerly used simple polygon rendering. Anyway there is further researh needed to determine an appropriate δ and an approximation for the tessellation level. In our approah we used values for whih we obtained suitable results, whereby the results are nevertheless remarkable. Anyway a δ and the tessellation level with a greater mathematial bakground would 5

6 R EFERENCES β = 0.0 β = 0.0 () β = 0.9 (d) β = 0.9 (e) detailed view of the outer edge (f) detailed view of the outer edge [1] inner santum katdm3 ustom quake 3 level : Katsbits maps. inner-santum.php. [2] Irrliht engine - a free open soure 3d engine. soureforge.net/. [3] Irrliht with ubemap support. soureforge.net/forum/viewtopi.php?t= [4] Opengl tesselation. forum/viewtopi.php?f=2&t= [5] U. Kraus and M. Borhers. Visualisierung relativistisher Effekte: Fast Lihtshnell durh die Stadt. Physik in unserer Zeit, 2(2):64 69, [6] T. M uller and S. Boblest. Visual appearane of wireframe objets in speial relativity. European Journal of Physis, 35:065025, [7] T. M uller and D. Weiskopf. Speial-relativisti visualization. Computing in Siene & Engineering, 13(4):85 93, [8] T. M uller, S. Grottel, and D. Weiskopf. Speial relativisti visualization by loal ray traing. Visualization and Computer Graphis, IEEE Transations on, 16(6): , [9] T. M uller, A. King, and D. Adis. A trip to the end of the universe and the twin paradox. Amerian Journal of Physis, 76(4): , [10] C. M. Savage, A. Searle, and L. MCalman. Real time relativity: Exploratory learning of speial relativity. Amerian Journal of Physis, 75(9): , [11] E. Taylor. Spae-time software: Computer graphis utilities in speial relativity. Amerian Journal of Physis, 57: , [12] D. Weiskopf. A Survey of Visualization Methods for Speial Relativity. In H. Hagen, editor, Sientifi Visualization: Advaned Conepts, volume 1 of Dagstuhl Follow-Ups, pages Shloss Dagstuhl Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, Figure 9: Diret omparison between the two methods: Advaned Polygon Rendering on the left, image based on the right. be muh better for professional usage and probably inrease the quality even more. It is also neessary to find the reason why the artefats appear and the objets disappear sometimes and how this is avoidable. We also set the inner tesselation level to the maximum of the three alulated outer tessellation levels, here an other way might be more suitable. Furthermore it would be interesting to determine a maximum tessellation level whih do not let ollapse the performane of the image rendering unneessarily. Sine we used a maximum tessellation level of 32, even if tessellation levels up to 64 and even higher are possible, and worked with unompliated senes and meshes, a omparison of the osts is diffiult and has not been done yet. Therefore there is great potential to improve this method and obtain a better solution. By using greater and more ompliated senes a kind of benhmarking to find out whih method results in better performane with aeptable quality and under what irumstanes is needed and should be done, as the use of tessellation shader for geometry subdivision involves some more overhead ompared to the image-based approah or the simple version of polygon rendering. Another vital part in researhing speial relativisti simulation is the approah of loal ray traing. Implementing this third method and omparing these three methods with eah other is part of future work. ACKNOWLEDGMENTS The authors like to thank Dr. Thomas M uller, Dr. Sebastian Boblest and Dipl.-Inf Alexandros Panagiotidis for their support and help. We would like to further thank our former o-author Melanie Knupfer for her partiipation. 6