Initial conversion to markdown

61-02.md

@@ -0,0 +1,127 @@
+  ------------------------ --------------------------------- --------------------
+  [Previous](61-01.html)   [Table of Contents](index.html)   [Next](61-03.html)
+  ------------------------ --------------------------------- --------------------
+
+I’ll be working in a left-handed coordinate system, whereby if you wrap
+the fingers of your left hand around the z axis with your thumb pointing
+in the positive z direction, your fingers will curl from the positive x
+axis to the positive y axis. The positive x axis runs left to right
+across the screen, the positive y axis runs bottom to top across the
+screen, and the positive z axis runs into the screen.
+
+For our purposes, *projection* is the process of mapping coordinates
+onto a line or surface. *Perspective projection* projects 3-D
+coordinates onto a viewplane, scaling coordinates according to their z
+distance from the viewpoint in order to provide proper perspective.
+*Objectspace* is the coordinate space in which an object is defined,
+independent of other objects and the world itself. *Worldspace* is the
+absolute frame of reference for a 3-D world; all objects’ locations and
+orientations are with respect to worldspace, and this is the frame of
+reference around which the viewpoint and view direction move.
+*Viewspace* is worldspace as seen from the viewpoint, looking in the
+view direction. *Screenspace* is viewspace after perspective projection
+and scaling to the screen.
+
+Finally, *transformation* is the process of converting points from one
+coordinate space into another; in our case, that’ll mean rotating and
+translating (moving) points from objectspace or worldspace to viewspace.
+
+For additional information, you might want to check out Foley & van
+Dam’s *Computer Graphics* (ISBN 0-201-12110-7), or the chapters in this
+book dealing with my X-Sharp 3-D graphics library.
+
+### The Dot Product {#Heading5}
+
+Now we’re ready to move on to the dot product. Given two vectors **U** =
+[u~1~ u~2~ u~3~] and **V** = [v~1~ v~2~ v~3~], their dot product,
+denoted by the symbol •, is calculated as:
+
+![](images/61-02d.jpg)
+
+(eq. 2)
+
+As you can see, the result is a scalar value (a single real-valued
+number), *not* another vector.
+
+Now that we know how to calculate a dot product, what does that get us?
+Not much. The dot product isn’t of much use for graphics until you start
+thinking of it this way
+
+![](images/61-03d.jpg)
+
+(eq. 3)
+
+where q is the angle between the two vectors, and the other two terms
+are the lengths of the vectors, as shown in Figure 61.1. Although it’s
+not immediately obvious, equation 3 has a wide variety of applications
+in 3-D graphics.
+
+#### Dot Products of Unit Vectors {#Heading6}
+
+The simplest case of the dot product is when both vectors are *unit
+vectors*; that is, when their lengths are both one, as calculated as in
+Equation 1. In this case, equation 3 simplifies to:
+
+![](images/61-04d.jpg)
+
+(eq. 4)
+
+In other words, the dot product of two unit vectors is the cosine of the
+angle between them.
+
+One obvious use of this is to find angles between unit vectors, in
+conjunction with an inverse cosine function or lookup table. A more
+useful application in 3-D graphics lies in lighting surfaces, where the
+cosine of the angle between incident light and the normal (perpendicular
+vector) of a surface determines the fraction of the light’s full
+intensity at which the surface is illuminated, as in
+
+![](images/61-05d.jpg)
+
+(eq. 5)
+
+![](images/61-01.jpg)\
+ **Figure 61.1**  *The dot product.*
+
+where **I**~s~ is the intensity of illumination of the surface, **I**~l~
+is the intensity of the light, and q is the angle between **-D**~l~
+(where **D**~l~ is the light direction vector) and the surface normal.
+If the inverse light vector and the surface normal are both unit
+vectors, then this calculation can be performed with four multiplies and
+three additions—and no explicit cosine calculations—as
+
+![](images/61-06d.jpg)
+
+(eq. 6)
+
+where **N**~s~ is the surface unit normal and **D**~l~ is the light unit
+direction vector, as shown in Figure 61.2.
+
+### Cross Products and the Generation of Polygon Normals {#Heading7}
+
+One question equation 6 begs is where the surface unit normal comes
+from. One approach is to store the end of a surface normal as an extra
+data point with each polygon (with the start being some point that’s
+already in the polygon), and transform it along with the rest of the
+points. This has the advantage that if the normal starts out as a unit
+normal, it will end up that way too, if only rotations and translations
+(but not scaling and shears) are performed.
+
+The problem with having an explicit normal is that it will remain a
+normal—that is, perpendicular to the surface—only through viewspace.
+Rotation, translation, and scaling preserve right angles, which is why
+normals are still normals in viewspace, but perspective projection does
+not preserve angles, so vectors that were surface normals in viewspace
+are no longer normals in screenspace.
+
+![](images/61-02.jpg)\
+ **Figure 61.2**  *The dot product as used in calculating lighting
+intensity.*
+
+  ------------------------ --------------------------------- --------------------
+  [Previous](61-01.html)   [Table of Contents](index.html)   [Next](61-03.html)
+  ------------------------ --------------------------------- --------------------
+
+* * * * *
+
+Graphics Programming Black Book © 2001 Michael Abrash