TITLE: Homotopic Fun in 5D Space NAME: Eric Fausett, Alexander Pasko, and Valery Adzhiev COUNTRY: Made in Japan by an American and Two Russians EMAIL: EFausett@mail.com WEBPAGE: http://www.hyperfun.org TOPIC: Robot COPYRIGHT: I SUBMIT TO THE STANDARD RAYTRACING COMPETITION COPYRIGHT. MPGFILE: homotopi.mpg ZIPFILE: homotopi.zip RENDERER USED: Povray 3.0 w/ Suzuki Isosurface Patch TOOLS USED: HyperFun, PhotoShop 4.0, SGI MediaConvert, and Visual C++ 5.0. CREATION TIME: 8 days (rendering) + 5 days (modeling) + 3 days (animation setup) HARDWARE USED: Dual Processor Pentium II 400 Mhz, Dual Processor Pentium II 350 Mhz, and a Single Processor Pentium 233 Mhz for rendering. SGI O2 for mpeg creation. ANIMATION DESCRIPTION: This is an animation of a very interesting shape shifting Robot. The Robot shape-shifts (metamorphosis) between 4 different 3D shapes. Really what you are seeing is a 3D slices of a 5 dimensional Robot as it moves passes along our 3D world. ********************************************************************************* * ANIMATION DESCRIPTION (TECHNICAL for those so inclined): ****IMPORTANT NOTE: This type of metamorphosis is very difficult if not impossible for current polygonal models, general B-rep based models, or CSG based models. Notice that each of the key 3D shapes are of a different topology. This is the benefit of using the function representation (F-rep). In the animation "Homotopic Fun in 5D space" we present a time-dependent version of bi-directional metamorphosis, which is a non-traditional operation in computer graphics and animation. It results in a smooth transformation (metamorphosis) between four key 3D shapes modeled using function representation (F-rep) 1 (a generalization of the implicit surface model). Each of the key 3D shapes is defined by a single real function f(x1, x2, x3) = 0, where f = 0 defines the objects surface (isosurface of the function). Algebraically, the model of the bi-directional metamorphosis is the bilinear interpolation between four real-valued functions by coordinates x4 and x5. Geometrically, it is a 5D object defined by the real function as f(x1, x2, x3, x4, x5) = 0. On the other hand, the metamorphosis can be considered a homotopy in the functional space, which is reflected in the animation's title. All 3D shapes were modeled using set-theoretic (CSG) operation with R-functions. The transformation applied to the key shapes can be described by the following expression: Meta5D = (Cat*(1.-x 4 )+Robot*x 4 )*(1.-x 5 ) + (NiHon*(1.-x 4 )+Rob_word*x 4 )*x 5 ; where Meta5D stands for the defining function of the bi-directional metamorphosis; Cat, Robot, NiHon, and Rob_Word each represent the individual defining functions of the key shapes, x 4 and x 5 are coordinates of 5D space and parameters of the interpolation. Therefore, the entire animation is described by a single function of five variables. Then, coordinates x1, x2, x3 are considered as "real life" Cartesian coordinates; x4 and x5 are dynamic ("time") coordinates. Each frame of the animation corresponds to a point in a 5D space and to a unique shape. To get the current values of x4 and x5, the trajectory in the plane (x4 , x5 ) is analytically defined. Then, by moving along the trajectory we project the 5D shape onto the 3D space to produce a single frame. ********************************************************************************* * VIEWING RECOMMENDATIONS: Windows Media Player for Windows 95/NT worked for me. The original animation is an uncompressed AVI at twice the size that comes with a nice soundtrack. For those interested contact me, or see information below. ********************************************************************************* * DESCRIPTION OF HOW THIS ANIMATION WAS CREATED: Modeling: - "Cat" The complete model was created using the HyperFun modeling language. HyperFun is a language for describing F-rep objects (and IsoSurfaces) which includes many built in operations and primitives including CSG operations using R-functions. The complete "Cat" model in HyperFun can be found at: http://www.u-aizu.ac.jp/labs/sw-sm/FrepWWW/HF_dor.html - "NiHon" Also created using the HyperFun modeling language. It is a 3D puzzle representing the word "Japan". First, the two projected 3D Chinese characters "Ni" and "Hon" are constructed independently as unions of blocks. Then, the solids are oriented along Z and X axes respectively and combined as NiHon = Ni & Hon, where & represents intersection operation. The idea of this puzzle construction is that the resulting 3D solid looks like a single initial 2D character "Ni" or "Hon" when projected along Z and X axes respectively onto a plane. - "Robot" and "Rob_Word" were also modeled using set-theoretic operations in HyperFun. Due to the current speed of the HyperFun interpreter, these models were translated by hand into C code. The C code was compiled into a dll for use with Suzuki's Isosurface Patch. The variables x 4 and x 5 which change the shape were made accessible through Suzuki's Isosurface interface. Next a path in the x 4 x 5 plane was created in the pov file and the movement of the shape on the checkerboard was set up to correspond to the the changing shape. The checkerboard and sky were created and textured so as to be aesthetically pleasing. The animation frames were then divided up between 3 machines (5 processors) which rendered the animation in parallel in about a week. The resulting images were compiled into an AVI and converted to mpeg on a Silicon Graphics O2 machine. (Didn't have such an application for Windows). I cheated on the fades and the lettering. These were all done in post-processing by hand using Adobe Photo-shop. All effects used are very straight forward and could have almost just as easily been done directly in POVRay. I didn't have the time, and took the fastest route however. The only post-processing done to the images was the fades and lettering however. The animation portion is all 100% POVRay. The original AVI file contains a nice "Homotopic" sound track, and gives the animation a very nice atmosphere. Due to size constrictions the sound track was left out for the competition. Also the image quality is quite bad due to the high compression ratio needed to keep the entire animation. For those interested I will provide a full copy of the AVI or higher quality MPEG with sound. The actual animation is also 2X as big and has been copied to full screen video tape. I hope to get these versions of the animation up on our web site soon. For more information on HyperFun and the HyperFun Project, see www.hyperfun.org. Last but not least, here are the credit lines that are unreadable in the mpeg: ********************************************************************************* * ********** CREDITS ********** ********************************************************************************* * Alexander Pasko Design 3D and 5D Modeling Music Selection Saying "That_s Right" Valery Adzhiev Design 3D Modeling Critical Head Shaking Eric Fausett Rendering Animation Production Audio/Video Editing Exhausting All Resources ********************************************************************************* * Three Dimensional Shapes contributed by: Kensuke Masuda Yukio Hashimoto Kou Setoguchi Masayuki Tetsuka Tetsurou Tobe Music: Aquarium Mitya's Waltz Composers: S. Shurakov and BG SoLyd Records, Russia Copyright _ 1993 (Used with permission) Equiptment and Facilities Provided by: University of Aizu Shape Modeling Lab ********************************************************************************* * Computers Used: Widows NT Workstations (Pentium II) Software: Povray 3.0 (Isosurface Patch by R. Suzuki) HyperFun for Windows (implemented by A. Osipov) Metamorphosis Stunts Performed by: Bi-Directional Metamorphosis (Cat*(1-x 4 )+Robot*x 4 )*(1-x 5 )+(NiHon*(1-x 4 )+Rob_word*x 4 )*x 5 Homotopic Fun in 5D space Copyright _ 1999 Alexander Pasko ********************************************************************************* * No robots or animals were hurt in the rendering of this animation.