Normal quotes

61-02.md

@@ -2,7 +2,7 @@
   [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
+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
@@ -15,7 +15,7 @@ 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
+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
@@ -23,16 +23,16 @@ 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
+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
+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** =
+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:
 
@@ -44,7 +44,7 @@ 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
+Not much. The dot product isn't of much use for graphics until you start
 thinking of it this way
 
 ![](images/61-03d.jpg)
@@ -52,7 +52,7 @@ thinking of it this way
 (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
+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.
 
@@ -73,7 +73,7 @@ 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
+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)
@@ -101,7 +101,7 @@ direction vector, as shown in Figure 61.2.
 
 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
+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