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<H3> 7.6.5 Space Warping </H3>
<!--docid::SEC376::-->
<P>
<EM>Written by Jorrit Tyberghein,
<A HREF="mailto:jorrit.tyberghein@uz.kuleuven.ac.be">jorrit.tyberghein@uz.kuleuven.ac.be</A>. Mathematical typesetting for
TeX performed by Eric Sunshine, <A HREF="mailto:sunshine@sunshineco.com">sunshine@sunshineco.com</A>.</EM>
</P><P>
A little explanation about space warping in Crystal Space and how the space
warping matrix/vector work should be given.
</P><P>
Crystal Space always works with 3x3 matrices and one 3-element vector to
represent transformations. Let's say that the camera is given as <EM>Mc</EM> and
<EM>Vc</EM> (camera matrix and camera vector, respectively).
</P><P>
When going through a warping portal (mirror for example) there is also a
warping matrix and two vectors, <EM>Mw</EM>, <EM>Vw1</EM> and <EM>Vw2</EM>. <EM>Vw1</EM>
is the vector that is applied before <EM>Mw</EM> and <EM>Vw2</EM> is applied after
<EM>Mw</EM>. The warping transformation is a transformation in world space. For
example, if you have the following sector:
</P><P>
<TABLE><tr><td> </td><td class=example><pre> A
+---------+ z
| | ^
| | |
D | o | B o-->x
| |
| |
+---------+
C
</pre></td></tr></table></P><P>
With point <EM>o</EM> at (0,0,0) and the <EM>B</EM> side a mirror. Let's say that
<EM>B</EM> is 2 units to the right of <EM>o</EM>. The warping matrix/vector would
then be:
</P><P>
<TABLE><tr><td> </td><td class=example><pre> /-1 0 0 \ / 2 \ / 2 \
Mw = | 0 1 0 | Vw1 = | 0 | Vw2 = | 0 |
\ 0 0 1 / \ 0 / \ 0 /
</pre></td></tr></table></P><P>
The mirror swaps along the X-axis.
</P><P>
How is this transformation then used?
</P><P>
To know how this works we should understand that
<EM>Mc</EM> and <EM>Vc</EM>
(the camera transformation) is a transformation from world space to camera
space. Since the warping transformation is in world space we first have to
apply
<EM>Mw / Vw</EM> before <EM>Mc / Vc</EM>.
</P><P>
So we want to make a new camera transformation matrix/vector that we are then
going to use for the recursive rendering of the sector behind the mirror.
Let's call this
<EM>Mc'</EM> and <EM>Vc'</EM>.
</P><P>
The camera transformation is used like this in Crystal Space:
</P><P>
<BLOCKQUOTE>
<TABLE>
<TR><TD><EM>C = Mc * (W - Vc)</EM> </TD><TD> (Equation 1)</TD>
</TR></TABLE>
</BLOCKQUOTE>
<P>
Where <EM>C</EM> is the camera space coordinates and <EM>W</EM> is the world space
coordinates.
</P><P>
But first we want to transform world space using the warping transformation:
</P><P>
<BLOCKQUOTE>
<TABLE>
<TR><TD><EM>W' = Mw * (W - Vw1) + Vw2</EM> </TD><TD> (Equation 2)</TD>
</TR></TABLE>
</BLOCKQUOTE>
<P>
It is important to realize that the
<EM>Mw / Vwn</EM>
transformation is used a little differently here. The <EM>Vw1</EM> vector is
used to translate to the warping polygon first and <EM>Vw2</EM> is used to go
back when the matrix <EM>Mw</EM> has done its work. This is just how Crystal
Space does it. One could use other matrices/vectors to express the warping
transformations.
</P><P>
Combining equations (1) and (2), but replacing <EM>W</EM> by <EM>W'</EM> in (1),
gives:
</P><P>
<BLOCKQUOTE>
<EM>C = Mc * (Mw * (W - Vw1) + Vw2 - Vc)</EM><BR>
<EM>C = Mc * Mw * ((W - Vw1) - 1 / Mw * (Vc - Vw2))</EM><BR>
<EM>C = Mc * Mw * (W - (Vw1 + 1 / Mw * (Vc - Vw2)))</EM>
</BLOCKQUOTE>
<P>
And this is the new camera transformation:
</P><P>
<BLOCKQUOTE>
<EM>Mc' = Mc * Mw</EM><BR>
<EM>Vc' = Vw1 + 1 / Mw * (Vc - Vw2)</EM>
</BLOCKQUOTE>
<P>
In summary, the warping transformation works by first transforming world space
to a new warped world space. The new camera transformation is made by
combining the warping transformation with the old camera transformation.
</P><P>
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