Normal quotes

51-02.md

@@ -2,7 +2,7 @@
   [Previous](51-01.html)   [Table of Contents](index.html)   [Next](51-03.html)
   ------------------------ --------------------------------- --------------------
 
-For a single convex polyhedron, removal of polygons that aren’t facing
+For a single convex polyhedron, removal of polygons that aren't facing
 the viewer would solve all hidden surface problems. In a convex
 polyhedron, any polygon facing the viewer can never be obscured by any
 other polygon in that polyhedron; this falls out of the definition of a
@@ -13,7 +13,7 @@ the viewer, everything will work out properly, with no additional
 checking for overlap and hidden surfaces needed.
 
 Unfortunately, backface removal completely solves the hidden surface
-problem for convex polyhedrons *only*, and only if there’s a single
+problem for convex polyhedrons *only*, and only if there's a single
 convex polyhedron involved; when convex polyhedrons overlap, other
 methods must be used. Nonetheless, backface removal does instantly halve
 the number of polygons to be handled in rendering any particular scene.
@@ -22,20 +22,20 @@ built out of convex polyhedrons. In this chapter, though, we have only
 one convex polyhedron to deal with, so backface removal alone will do
 the trick.
 
-Given that I’ve convinced you that backface removal would be a handy
+Given that I've convinced you that backface removal would be a handy
 thing to have, how do we actually do it? A logical approach, often
 implemented in the PC literature, would be to calculate the plane
 equation for the plane in which the polygon lies, and see which way the
 normal (perpendicular) vector to the plane points. That works, but
-there’s a more efficient way to calculate the normal to the polygon: as
-the cross-product of two of the polygon’s edges.
+there's a more efficient way to calculate the normal to the polygon: as
+the cross-product of two of the polygon's edges.
 
 The cross-product of two vectors is defined as the vector shown in
 Figure 51.1. One interesting property of the cross-product vector is
 that it is perpendicular to the plane in which the two original vectors
 lie. If we take the cross-product of the vectors that form two edges of
 a polygon, the result will be a vector perpendicular to the polygon;
-then, we’ll know that the polygon is visible if and only if the
+then, we'll know that the polygon is visible if and only if the
 cross-product vector points toward the viewer. We need one more thing to
 make the cross-product approach work, though. The cross-product can
 actually point either way, depending on which edges of the polygon we
@@ -46,22 +46,22 @@ cross-product.
 ![](images/51-01.jpg)\
  **Figure 51.1**  *The cross-product of two vectors.*
 
-We’ll define only convex polygons, with the vertices defined in
-clockwise order, as viewed from the outside; that is, if you’re looking
+We'll define only convex polygons, with the vertices defined in
+clockwise order, as viewed from the outside; that is, if you're looking
 at the visible side of the polygon, the vertices will appear in the
 polygon definition in clockwise order. With those assumptions, the
 cross-product becomes a quick and easy indicator of polygon orientation
-with respect to the viewer; we’ll calculate it as the cross-product of
+with respect to the viewer; we'll calculate it as the cross-product of
 the first and last vectors in a polygon, as shown in Figure 51.2, and if
-it’s pointing toward the viewer, we’ll know that the polygon is visible.
-Actually, we don’t even have to calculate the entire cross-product
+it's pointing toward the viewer, we'll know that the polygon is visible.
+Actually, we don't even have to calculate the entire cross-product
 vector, because the Z component alone suffices to tell us which way the
 polygon is facing: positive Z means visible, negative Z means not. The Z
 component can be calculated very efficiently, with only two multiplies
 and a subtraction.
 
 The question remains of the proper space in which to perform backface
-removal. There’s a temptation to perform it in view space, which is,
+removal. There's a temptation to perform it in view space, which is,
 after all, the space defined with respect to the viewer, but view space
 is not a good choice. Screen space—the space in which perspective
 projection has been performed—is the best choice. The purpose of
@@ -77,15 +77,15 @@ normal.*
 Backface removal may also be performed using the polygon vertices in
 screen coordinates, which are integers. This is less accurate than using
 the screen space coordinates, which are floating point, but is, by the
-same token, faster. In Listing 51.3, which we’ll discuss shortly,
+same token, faster. In Listing 51.3, which we'll discuss shortly,
 backface removal is performed in screen coordinates in the interests of
 speed.
 
 Backface removal, as implemented in Listing 51.3, will not work reliably
-if the polygon is not convex, if the vertices don’t appear in clockwise
+if the polygon is not convex, if the vertices don't appear in clockwise
 order, if either the first or last edge in a polygon has zero length, or
 if the first and last edges are collinear. These latter two points are
-the reason it’s preferable to work in screen space rather than screen
+the reason it's preferable to work in screen space rather than screen
 coordinates (which suffer from rounding problems), speed considerations
 aside.