Normal quotes

65-02.md

@@ -10,13 +10,13 @@ clipped by (intersects) the plane at all and, if it is clipped,
 calculating the point of intersection.
 
 Before we can intersect a line segment with a plane, we must first
-define how we’ll represent the line segment and the plane. The segment
+define how we'll represent the line segment and the plane. The segment
 will be represented in the obvious way by the (x,y,z) coordinates of its
 two endpoints; this extends well to polygons, where each vertex is an
 (x,y,z) point. Planes can be described in many ways, among them are
 three points on the plane, a point on the plane and a unit normal, or a
-unit normal and a distance from the origin along the normal; we’ll use
-the latter definition. Further, we’ll define the normal to point to the
+unit normal and a distance from the origin along the normal; we'll use
+the latter definition. Further, we'll define the normal to point to the
 inside (unclipped side) of the plane. The structures for points,
 polygons, and planes are shown in Listing 65.1.
 
@@ -58,8 +58,8 @@ polygons, and planes are shown in Listing 65.1.
 Given a line segment, and a plane to which to clip the segment, the
 first question is whether the segment is entirely on the inside or the
 outside of the plane, or intersects the plane. If the segment is on the
-inside, then the segment is not clipped by the plane, and we’re done. If
-it’s on the outside, then it’s entirely clipped, and we’re likewise
+inside, then the segment is not clipped by the plane, and we're done. If
+it's on the outside, then it's entirely clipped, and we're likewise
 done. If it intersects the plane, then we have to remove the clipped
 portion of the line by replacing the endpoint on the outside of the
 plane with the point of intersection between the line and the plane.
@@ -68,7 +68,7 @@ The way to answer this question is to find out which side of the plane
 each endpoint is on, and the dot product is the right tool for the job.
 As you may recall from Chapter 61, dotting any vector with a unit normal
 returns the length of the projection of that vector onto the normal.
-Therefore, if we take any point and dot it with the plane normal we’ll
+Therefore, if we take any point and dot it with the plane normal we'll
 find out how far from the origin the point is, as measured along the
 plane normal. Another way to think of this is to say that the dot
 product of a point and the plane normal returns how far from the origin
@@ -84,24 +84,24 @@ of the point and the normal; a simple comparison of the two values
 suffices to tell us which side of the plane the point is on. If the dot
 product of the point and the plane normal is greater than the plane
 distance, then the point is in front of the plane (inside the volume
-being clipped to); if it’s less, then the point is outside the volume
+being clipped to); if it's less, then the point is outside the volume
 and should be clipped.
 
 After we do this twice, once for each line endpoint, we know everything
 necessary to categorize our line segment. If both endpoints are on the
-same side of the plane, there’s nothing more to do, because the line is
-either completely inside or completely outside; otherwise, it’s on to
+same side of the plane, there's nothing more to do, because the line is
+either completely inside or completely outside; otherwise, it's on to
 the next step, clipping the line to the plane by replacing the outside
 vertex with the point of intersection of the line and the plane.
 Happily, it turns out that we already have all of the information we
 need to do this.
 
 From our earlier tests, we already know the length from the plane,
-measured along the normal, to the inside endpoint; that’s just the
+measured along the normal, to the inside endpoint; that's just the
 distance, along the normal, of the inside endpoint from the origin (the
 dot product of the endpoint with the normal), minus the plane distance,
 as shown in Figure 65.1. We also know the length of the line segment,
-again measured as projected onto the normal; that’s the difference
+again measured as projected onto the normal; that's the difference
 between the distances along the normal of the inside and outside
 endpoints from the origin. The ratio of these two lengths is the
 fraction of the segment that remains after clipping. If we scale the x,
@@ -114,18 +114,18 @@ point of intersection.
 Line clipping is fine for wireframe rendering, but what we really want
 to do is polygon rendering of solid models, which requires polygon
 clipping. As with line segments, the clipping process with polygons is
-to determine if they’re inside, outside, or partially inside the clip
+to determine if they're inside, outside, or partially inside the clip
 volume, lopping off any vertices that are outside the clip volume and
 substituting vertices at the intersection between the polygon and the
 clip plane, as shown in Figure 65.2.
 
 An easy way to clip a polygon is to decompose it into a set of edges,
-and clip each edge separately as a line segment. Let’s define a polygon
+and clip each edge separately as a line segment. Let's define a polygon
 as a set of vertices that wind clockwise around the outside of the
 polygonal area, as viewed from the front side of the polygon; the edges
 are implicitly defined by the order of the vertices. Thus, an edge is
 the line segment described by the two adjacent vertices that form its
-endpoints. We’ll clip a polygon by clipping each edge individually,
+endpoints. We'll clip a polygon by clipping each edge individually,
 emitting vertices for the resulting polygon as appropriate, depending on
 the clipping state of the edge. If the start point of the edge is
 inside, that point is added to the output polygon. Then, if the start