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