====== Animation "Homotopic Fun in 5D Space" ======

[[http://www.hyperfun.org/App/H5D/H5D_center.jpg|{{http://www.hyperfun.org/App/H5D/H5D_center300.jpg}}]] \\ Click on an image to get its larger version

In the animation "Homotopic Fun in 5D space" (by  Pasko A., Adzhiev V.,  Fausett E. ) 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) and shown below:

[[http://www.hyperfun.org/App/H5D/Cat.jpg|{{http://www.hyperfun.org/App/H5D/Cat200.jpg}}]] [[http://www.hyperfun.org/App/H5D/Robot.jpg|{{http://www.hyperfun.org/App/H5D/Robot200.jpg}}]] \\ [[http://www.hyperfun.org/App/H5D/NiHon.jpg|{{http://www.hyperfun.org/App/H5D/NiHon200.jpg}}]] [[http://www.hyperfun.org/App/H5D/Rob_Let.jpg|{{http://www.hyperfun.org/App/H5D/Rob_Let200.jpg}}]]

The selected key-shapes are, to some extent, "cultural key signs" in Japan:

    * "Cat" (upper left) resembles the cult character of kids' animation.
    * "NiHon" (lower left) is a 3D puzzle representing the word "Japan". First, two 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 using the 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" (upper right)
    * "Rob_Let" (lower right) presents the word "robot". This object consists of five disjoint components of differing topology. 

The transformation applied to the key shapes can be described by the following expression:
<code>Meta5D = (Cat*(1.-x[4])+Robot*x[4])*(1.-x[5]) + (NiHon*(1.-x[4])+Rob_Let*x[4])*x[5];</code>

Therefore, the entire animation is described by a single function of five variables. Then, coordinates x[1], x[2], x[3] are considered as "real life" Cartesian coordinates; x[4] and x[5] are dynamic ("time") coordinates. Each frame of the animation presents a unique 3D shape corresponding to a point in the plane (x[4], x[5]). To get the current values of x[4] and x[5], the trajectory in this plane is analytically defined (see below).

[[http://www.hyperfun.org/App/H5D/Axpath800.jpg|{{http://www.hyperfun.org/App/H5D/Axpath400.jpg}}]]

Then, by moving along the trajectory we get the current 5D shape, which is projected to a 3D space to produce a single frame. The central shape, which is an equal mixture of the four initial shapes, is shown at the top of this page. Note that all topology changes are handled automatically. No polygonal surface was generated. The frames were rendered by direct ray tracing of the intermediate F-rep objects.

Click to view a streamed video version of the  animation [[http://www.hyperfun.org/App/H5D/H5D128.html|"Homotopic Fun in 5D Space"]] (WMV format, 128 Kbps)

This video has reduced quality because of compression. Do not look too close!

his animation received a prize of Dream Centenary CG Grand Prix (Japan, 1999). \\
Details can be found in the paper: Fausett E. , Pasko A., Adzhiev V., "Space-time and higher dimensional modeling for animation", Computer Animation 2000, IEEE Computer Society, ISBN 0-7695-0683-6, 2000, pp.140-145. \\ 
Electronic version: [[http://www.hyperfun.org/ST_Animation.zip|PostScript + zip]] (929K) and [[http://www.hyperfun.org/ST_Animation.pdf|PDF]] (675K)