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<h4 class="subsubsection"> 4.12.3.2 General Concepts </h4>
<p>This section describes general core concepts which are used by following
discussions.
</p>
<a name="1"></a>
<h4 class="subsubheading"> Polygon </h4>
<p>A polygon is represented as a set of 3D vertices oriented in clockwise order
(if the orientation is different, the polygon is not visible on screen).
There are three versions of any polygon:
</p>
<ol>
<li>
<em>Po</em> is the polygon in object space. Here the vertices of the polygon are
given in object space coordinates.
</li><li>
<em>Pw</em> is the polygon in world space. For a sector world space is equal to
object space but for a thing there is a transformation from object space to
world space given as:
<blockquote><p><em>Vw = Mow * Vo - Vow</em>
</p></blockquote>
<p>This means that the vertex in world space coordinates (<em>Vw</em>) is given as
the vertex in object space coordinates (<em>Vo</em>) transformed by the matrix
from object to world space (<em>Mow</em>) and the vector from object to world
space (<em>Vow</em>).
</p>
<p>This transformation is recomputed whenever a thing moves.
</p>
</li><li>
<em>Pc</em> is the polygon in camera space. Before viewing, every relevant
vertex is transformed from world space to camera space using the following
equation:
<blockquote><p><em>Vc = Mwc * (Vw - Vwc)</em>
</p></blockquote>
</li></ol>
<p>Note, all three versions of a polygon are represented by the same
polygon object. The change from object to world and world to
camera space happens inside the Vertex class that is referenced by
the polygon.
</p>
<p>Associated with every polygon there is also a plane normal or plane equation:
</p>
<blockquote><p><em>N: A * x + B * y + C * z + D = 0</em>
</p></blockquote>
<p>There are also three versions of this plane equation; one for object space,
one for world space, and one for camera space:
</p>
<blockquote><p><em>No: Ao * x + Bo * y + Co * z + Do = 0</em><br>
<em>Nw: Aw * x + Bw * y + Cw * z + Dw = 0</em><br>
<em>Nc: Ac * x + Bc * y + Cc * z + Dc = 0</em>
</p></blockquote>
<a name="2"></a>
<h4 class="subsubheading"> Camera </h4>
<p>A camera is represented as a matrix (<em>Mwc</em>) and a vector (<em>Vwc</em>). In
fact the camera describes the transformation between world and camera space.
The inverse transformation (<em>Mcw</em>) from camera space to world space is
also kept inside a <samp>‘csCamera’</samp> object because it can be used for movement
(for example, to move forward one would want to move forward along the Z-axis
in camera space and not in world space).
</p>
<a name="3"></a>
<h4 class="subsubheading"> Texture </h4>
<p>There is also a texture associated with every polygon.
The texture represents a 2-dimensional rectangular space
where (0,0) is one corner of the texture and (1,1) is the opposite corner. If
a texture is overlayed onto a rectangular polygon such that one corner is
mapped onto (0,0) in texture space and the opposite corner is mapped onto
(2,2) then the texture will be tiled four times across the surface of the
polygon.
</p>
<p>To transform object to texture space we have the following equation:
</p>
<blockquote><p><em>Vt = Mot * (Vo - Vot)</em>
</p></blockquote>
<p>Similarly for world space:
</p>
<blockquote><p><em>Vt = Mwt * (Vw - Vwt)</em>
</p></blockquote>
<p>And for camera space:
</p>
<blockquote><p><em>Vt = Mct * (Vc - Vct)</em>
</p></blockquote>
<p>So, just as there are three versions of the polygon (<em>Po</em>, <em>Pw</em>, and
<em>Pc</em>) and three plane equations, we also have three equations transforming
the various spaces to the texture. <em>Mot</em> and <em>Vot</em> (object to texture
space) are fixed (calculated at load time). <em>Mwt</em> and <em>Vwt</em> (world to
texture space) are the same for sectors but for things they are calculated
whenever a thing moves. <em>Mct</em> and <em>Vct</em> (camera to texture space) are
calculated every time the corresponding polygon is visible and needs to be
drawn on screen.
</p>
<p>Note that the texture transformation matrix is actually a transformation from
3D to 3D. We simply don't use the Z component in texture space (it is just
ignored).
</p>
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