Normal quotes

61-03.md

@@ -3,21 +3,21 @@
   ------------------------ --------------------------------- --------------------
 
 Why does this matter? It matters because, on average, half the polygons
-in any scene are facing away from the viewer, and hence shouldn’t be
-drawn. One way to identify such polygons is to see whether they’re
+in any scene are facing away from the viewer, and hence shouldn't be
+drawn. One way to identify such polygons is to see whether they're
 facing toward or away from the viewer; that is, whether their normals
-have negative z values (so they’re visible) or positive z values (so
-they should be culled). However, we’re talking about screenspace normals
+have negative z values (so they're visible) or positive z values (so
+they should be culled). However, we're talking about screenspace normals
 here, because the perspective projection can shift a polygon relative to
 the viewpoint so that although its viewspace normal has a negative z,
 its screenspace normal has a positive z, and vice-versa, as shown in
-Figure 61.3. So we need screenspace normals, but those can’t readily be
+Figure 61.3. So we need screenspace normals, but those can't readily be
 generated by transformation from worldspace.
 
 ![](images/61-03.jpg)\
  **Figure 61.3**  *A problem with determining front/back visibility.*
 
-The solution is to use the cross product of two of the polygon’s edges
+The solution is to use the cross product of two of the polygon's edges
 to generate a normal. The formula for the cross product is:
 
 ![](images/61-07d.jpg)
@@ -34,29 +34,29 @@ screenspace normals we need by taking the cross product of two adjacent
 polygon edges, as shown in Figure 61.4.
 
   ------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-  ![](images/i.jpg)   *In fact, we can cull with only one-third the work needed to generate a full cross product; because we’re interested only in the sign of the z component of the normal, we can skip entirely calculating the x and y components. The only caveat is to be careful that neither edge you choose is zero-length and that the edges aren’t collinear, because the dot product can’t produce a normal in those cases.*
+  ![](images/i.jpg)   *In fact, we can cull with only one-third the work needed to generate a full cross product; because we're interested only in the sign of the z component of the normal, we can skip entirely calculating the x and y components. The only caveat is to be careful that neither edge you choose is zero-length and that the edges aren't collinear, because the dot product can't produce a normal in those cases.*
   ------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 
 ![](images/61-04.jpg)\
  **Figure 61.4**  *How the cross product of polygon edge vectors
 generates a polygon normal.*
 
-Perhaps the most often asked question about cross products is “Which way
-do normals generated by cross products go?” In a left-handed coordinate
+Perhaps the most often asked question about cross products is "Which way
+do normals generated by cross products go?" In a left-handed coordinate
 system, curl the fingers of your left hand so the fingers curl through
 an angle of less than 180 degrees from the first vector in the cross
 product to the second vector. Your thumb now points in the direction of
 the normal.
 
 If you take the cross product of two orthogonal (right-angle) unit
-vectors, the result will be a unit vector that’s orthogonal to both of
-them. This means that if you’re generating a new coordinate space—such
+vectors, the result will be a unit vector that's orthogonal to both of
+them. This means that if you're generating a new coordinate space—such
 as a new viewing frame of reference—you only need to come up with unit
 vectors for two of the axes for the new coordinate space, and can then
 use their cross product to generate the unit vector for the third axis.
-If you need unit normals, and the two vectors being crossed aren’t
-orthogonal unit vectors, you’ll have to normalize the resulting vector;
-that is, divide each of the vector’s components by the length of the
+If you need unit normals, and the two vectors being crossed aren't
+orthogonal unit vectors, you'll have to normalize the resulting vector;
+that is, divide each of the vector's components by the length of the
 vector, to make it a unit long.
 
 ### Using the Sign of the Dot Product {#Heading8}
@@ -65,10 +65,10 @@ The dot product is the cosine of the angle between two vectors, scaled
 by the magnitudes of the vectors. Magnitudes are always positive, so the
 sign of the cosine determines the sign of the result. The dot product is
 positive if the angle between the vectors is less than 90 degrees,
-negative if it’s greater than 90 degrees, and zero if the angle is
+negative if it's greater than 90 degrees, and zero if the angle is
 exactly 90 degrees. This means that just the sign of the dot product
 suffices for tests involving comparisons of angles to 90 degrees, and
-there are more of those than you’d think.
+there are more of those than you'd think.
 
 Consider, for example, the process of backface culling, which we
 discussed above in the context of using screenspace normals to determine
@@ -81,12 +81,12 @@ in objectspace, if we transform the viewpoint into that frame of
 reference), given only a vertex and a normal for each polygon and a
 location for the viewer.
 
-Here’s the trick: Calculate the vector from the viewpoint to any vertex
-in the polygon and take its dot product with the polygon’s normal, as
+Here's the trick: Calculate the vector from the viewpoint to any vertex
+in the polygon and take its dot product with the polygon's normal, as
 shown in Figure 61.5. If the polygon is facing the viewpoint, the result
 is negative, because the angle between the two vectors is greater than
 90 degrees. If the polygon is facing away, the result is positive, and
-if the polygon is edge-on, the result is 0. That’s all there is to
+if the polygon is edge-on, the result is 0. That's all there is to
 it—and this sort of backface culling happens before any transformation
 or projection at all is performed, saving a great deal of work for the
 half of all polygons, on average, that are culled.
@@ -95,7 +95,7 @@ Backface culling with the dot product is just a special case of
 determining which side of a plane any point (in this case, the
 viewpoint) is on. The same trick can be applied whenever you want to
 determine whether a point is in front of or behind a plane, where a
-plane is described by any point that’s on the plane (which I’ll call the
+plane is described by any point that's on the plane (which I'll call the
 plane origin), plus a plane normal. One such application is in clipping
 a line (such as a polygon edge) to a plane. Just do a dot product
 between the plane normal and the vector from one line endpoint to the