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<TITLE>Appendix F - OpenGL Programming Guide (Addison-Wesley Publishing Company)</TITLE>
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<H1>
Appendix F<BR>
Calculating Normal Vectors</H1>
This appendix describes how to calculate normal vectors for surfaces. You
need to define normals to use OpenGL's lighting facility, which is described
in Chapter 6, "Lighting.""Normal Vectors" introduces normals and the OpenGL
command for specifying them. This appendix goes through the details of
calculating them. It has the following major sections:
<UL>"Finding Normals for Analytic Surfaces"
<BR>
<P>"Finding Normals from Polygonal Data"</UL>
Since normals are perpendicular to a surface, you can find the normal at
a particular point on a surface by first finding the flat plane that just
touches the surface at that point. The normal is the vector that's perpendicular
to that plane. On a perfect sphere, for example, the normal at a point
on the surface is in the same direction as the vector from the center of
the sphere to that point. For other types of surfaces, there are other,
better means for determining the normals, depending on how the surface
is specified.
<P>Recall that smooth curved surfaces are approximated by a large number
of small flat polygons. If the vectors perpendicular to these polygons
are used as the surface normals in such an approximation, the surface appears
faceted, since the normal direction is discontinuous across the polygonal
boundaries. In many cases, however, an exact mathematical description exists
for the surface, and true surface normals can be calculated at every point.
Using the true normals improves the rendering considerably, as shown in
Figure F-1 . Even if you don't have a mathematical description, you can
do better than the faceted look shown in the figure. The two major sections
in this appendix describe how to calculate normal vectors for these two
cases:
<UL>"Finding Normals for Analytic Surfaces" explains what to do when you
have a mathematical description of a surface.
<BR>
<P>"Finding Normals from Polygonal Data" covers the case when you have
only the polygonal data to describe a surface.</UL>
<IMG SRC="figures/polytrue.gif" ALT="[IMAGE]" NOSAVE >
<P><B>Figure F-1 : </B>Rendering with Polygonal Normals vs. True Normals
<BR>
<BR>
<P>
<HR>
<H2>
<A NAME="X"></A>Finding Normals for Analytic Surfaces</H2>
Analytic surfaces are smooth, differentiable surfaces that are described
by a mathematical equation (or set of equations). In many cases, the easiest
surfaces to find normals for are analytic surfaces for which you have an
explicit definition in the following form:
<P><B>V</B>(<B>s,t</B>) = [ <B>X</B>(<B>s,t</B>) <B>Y</B>(<B>s,t</B>) <B>Z</B>(<B>s,t</B>)
]
<P>where <I>s</I> and <B>t</B> are constrained to be in some domain, and
<B>X</B>, <B>Y</B>, and <B>Z</B> are differentiable functions of two variables.
To calculate the normal, find
<P><IMG SRC="figures/eqapf01.gif" ALT="[IMAGE]" NOSAVE >
<P>which are vectors tangent to the surface in the <I>s</I> and <I>t</I>
directions. The cross product
<P><IMG SRC="figures/eqapf02.gif" ALT="[IMAGE]" NOSAVE >
<P>is perpendicular to both, and hence to the surface. The following shows
how to calculate the cross product of two vectors. (Watch out for the degenerate
cases where the cross product has zero length!)
<P><IMG SRC="figures/eqapf03.gif" ALT="[IMAGE]" NOSAVE >
<P>You should probably normalize the resulting vector. To normalize a vector
[<B>x</B>,<B>y</B>,<B>z</B>], calculate its length
<P><IMG SRC="figures/eqapf04.gif" ALT="[IMAGE]" NOSAVE >
<P>and divide each component of the vector by the length.
<P>As an example of these calculations, consider the analytic surface
<P><B>V</B>(<B>s,t</B>) = [ <B>s</B>2<B> t</B>3 3-<B>st</B> ]
<P>From this we have
<P><IMG SRC="figures/eqapf05.gif" ALT="[IMAGE]" NOSAVE >
<P>So, for example, when<B> s</B>=1 and<B> t</B>=2, the corresponding point
on the surface is (1, 8, 1), and the vector (-24, 2, 24) is perpendicular
to the surface at that point. The length of this vector is 34, so the unit
normal vector is (-24/34, 2/34, 24/34) = (-0.70588, 0.058823, 0.70588).
<P>For analytic surfaces that are described implicitly, as <B>F</B>(<B>x,
y, z</B>) = 0, the problem is harder. In some cases, you can solve for
one of the variables, say <B>z</B> = <B>G</B>(<B>x, y</B>), and put it
in the explicit form given previously:
<P><IMG SRC="figures/eqapf06.gif" ALT="[IMAGE]" NOSAVE >
<P>Then continue as described earlier.
<P>If you can't get the surface equation in an explicit form, you might
be able to make use of the fact that the normal vector is given by the
gradient
<P><IMG SRC="figures/eqapf07.gif" ALT="[IMAGE]" NOSAVE >
<P>evaluated at a particular point (<I>x, y, z</I>). Calculating the gradient
might be easy, but finding a point that lies on the surface can be difficult.
As an example of an implicitly defined analytic function, consider the
equation of a sphere of radius 1 centered at the origin:
<P><I>x</I>2 + <I>y</I>2 + <I>z</I>2 - 1 = 0
<P>This means that
<P>F(<I>x</I>, <I>y</I>, <I>z</I>) = x2 + y2 + z2 - 1
<P>which can be solved for<I> z</I> to yield
<P><IMG SRC="figures/eqapf08.gif" ALT="[IMAGE]" NOSAVE >
<P>Thus, normals can be calculated from the explicit form
<P><IMG SRC="figures/eqapf09.gif" ALT="[IMAGE]" NOSAVE >
<P>as described previously.
<P>If you could not solve for <I>z</I>, you could have used the gradient
<P><IMG SRC="figures/eqapf10.gif" ALT="[IMAGE]" NOSAVE >
<P>as long as you could find a point on the surface. In this case, it's
not so hard to find a point - for example, (2/3, 1/3, 2/3) lies on the
surface. Using the gradient, the normal at this point is (4/3, 2/3, 4/3).
The unit-length normal is (2/3, 1/3, 2/3), which is the same as the point
on the surface, as expected.
<P>
<HR>
<H2>
Finding Normals from Polygonal Data</H2>
As mentioned previously, you often want to find normals for surfaces that
are described with polygonal data such that the surfaces appear smooth
rather than faceted. In most cases, the easiest way for you to do this
(though it might not be the most efficient way) is to calculate the normal
vectors for each of the polygonal facets and then to average the normals
for neighboring facets. Use the averaged normal for the vertex that the
neighboring facets have in common. Figure F-2 shows a surface and its polygonal
approximation. (Of course, if the polygons represent the exact surface
and aren't merely an approximation - if you're drawing a cube or a cut
diamond, for example - don't do the averaging. Calculate the normal for
each facet as described in the following paragraphs, and use that same
normal for each vertex of the facet.)
<P><IMG SRC="figures/normals.gif" ALT="[IMAGE]" NOSAVE >
<P><B>Figure F-2 : </B>Averaging Normal Vectors
<BR>
<BR>
<P>To find the normal for a flat polygon, take any three vertices <B>v</B>1,
<B>v</B>2, and <B>v</B>3 of the polygon that do not lie in a straight line.
The cross product
<P>[<I>v</I>1 - <I>v</I>2] ⌨️ 快捷键说明
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