1 .TH ARITH3 3
2 .SH NAME
3 add3, sub3, neg3, div3, mul3, eqpt3, closept3, dot3, cross3, len3, dist3, unit3, midpt3, lerp3, reflect3, nearseg3, pldist3, vdiv3, vrem3, pn2f3, ppp2f3, fff2p3, pdiv4, add4, sub4 \- operations on 3-d points and planes
4 .SH SYNOPSIS
5 .PP
6 .B
7 #include <draw.h>
8 .PP
9 .B
10 #include <geometry.h>
11 .PP
12 .B
13 Point3 add3(Point3 a, Point3 b)
14 .PP
15 .B
16 Point3 sub3(Point3 a, Point3 b)
17 .PP
18 .B
19 Point3 neg3(Point3 a)
20 .PP
21 .B
22 Point3 div3(Point3 a, double b)
23 .PP
24 .B
25 Point3 mul3(Point3 a, double b)
26 .PP
27 .B
28 int eqpt3(Point3 p, Point3 q)
29 .PP
30 .B
31 int closept3(Point3 p, Point3 q, double eps)
32 .PP
33 .B
34 double dot3(Point3 p, Point3 q)
35 .PP
36 .B
37 Point3 cross3(Point3 p, Point3 q)
38 .PP
39 .B
40 double len3(Point3 p)
41 .PP
42 .B
43 double dist3(Point3 p, Point3 q)
44 .PP
45 .B
46 Point3 unit3(Point3 p)
47 .PP
48 .B
49 Point3 midpt3(Point3 p, Point3 q)
50 .PP
51 .B
52 Point3 lerp3(Point3 p, Point3 q, double alpha)
53 .PP
54 .B
55 Point3 reflect3(Point3 p, Point3 p0, Point3 p1)
56 .PP
57 .B
58 Point3 nearseg3(Point3 p0, Point3 p1, Point3 testp)
59 .PP
60 .B
61 double pldist3(Point3 p, Point3 p0, Point3 p1)
62 .PP
63 .B
64 double vdiv3(Point3 a, Point3 b)
65 .PP
66 .B
67 Point3 vrem3(Point3 a, Point3 b)
68 .PP
69 .B
70 Point3 pn2f3(Point3 p, Point3 n)
71 .PP
72 .B
73 Point3 ppp2f3(Point3 p0, Point3 p1, Point3 p2)
74 .PP
75 .B
76 Point3 fff2p3(Point3 f0, Point3 f1, Point3 f2)
77 .PP
78 .B
79 Point3 pdiv4(Point3 a)
80 .PP
81 .B
82 Point3 add4(Point3 a, Point3 b)
83 .PP
84 .B
85 Point3 sub4(Point3 a, Point3 b)
86 .SH DESCRIPTION
87 These routines do arithmetic on points and planes in affine or projective 3-space.
88 Type
89 .B Point3
90 is
91 .IP
92 .EX
93 .ta 6n
94 typedef struct Point3 Point3;
95 struct Point3{
96 	double x, y, z, w;
97 };
98 .EE
99 .PP
100 Routines whose names end in
101 .B 3
102 operate on vectors or ordinary points in affine 3-space, represented by their Euclidean
103 .B (x,y,z)
104 coordinates.
105 (They assume
106 .B w=1
107 in their arguments, and set
108 .B w=1
109 in their results.)
110 .TF reflect3
111 .TP
112 Name
113 Description
114 .TP
115 .B add3
116 Add the coordinates of two points.
117 .TP
118 .B sub3
119 Subtract coordinates of two points.
120 .TP
121 .B neg3
122 Negate the coordinates of a point.
123 .TP
124 .B mul3
125 Multiply coordinates by a scalar.
126 .TP
127 .B div3
128 Divide coordinates by a scalar.
129 .TP
130 .B eqpt3
131 Test two points for exact equality.
132 .TP
133 .B closept3
134 Is the distance between two points smaller than 
135 .IR eps ?
136 .TP
137 .B dot3
138 Dot product.
139 .TP
140 .B cross3
141 Cross product.
142 .TP
143 .B len3
144 Distance to the origin.
145 .TP
146 .B dist3
147 Distance between two points.
148 .TP
149 .B unit3
150 A unit vector parallel to
151 .IR p .
152 .TP
153 .B midpt3
154 The midpoint of line segment 
155 .IR pq .
156 .TP
157 .B lerp3
158 Linear interpolation between 
159 .I p
160 and
161 .IR q .
162 .TP
163 .B reflect3
164 The reflection of point
165 .I p
166 in the segment joining 
167 .I p0
168 and
169 .IR p1 .
170 .TP
171 .B nearseg3
172 The closest point to 
173 .I testp
174 on segment
175 .IR "p0 p1" .
176 .TP
177 .B pldist3
178 The distance from 
179 .I p
180 to segment
181 .IR "p0 p1" .
182 .TP
183 .B vdiv3
184 Vector divide \(em the length of the component of 
185 .I a
186 parallel to
187 .IR b ,
188 in units of the length of
189 .IR b .
190 .TP
191 .B vrem3
192 Vector remainder \(em the component of 
193 .I a
194 perpendicular to
195 .IR b .
196 Ignoring roundoff, we have 
197 .BR "eqpt3(add3(mul3(b, vdiv3(a, b)), vrem3(a, b)), a)" .
198 .PD
199 .PP
200 The following routines convert amongst various representations of points
201 and planes.  Planes are represented identically to points, by duality;
202 a point
203 .B p
204 is on a plane
205 .B q
206 whenever
207 .BR p.x*q.x+p.y*q.y+p.z*q.z+p.w*q.w=0 .
208 Although when dealing with affine points we assume
209 .BR p.w=1 ,
210 we can't make the same assumption for planes.
211 The names of these routines are extra-cryptic.  They contain an
212 .B f
213 (for `face') to indicate a plane,
214 .B p
215 for a point and
216 .B n
217 for a normal vector.
218 The number
219 .B 2
220 abbreviates the word `to.'
221 The number
222 .B 3
223 reminds us, as before, that we're dealing with affine points.
224 Thus
225 .B pn2f3
226 takes a point and a normal vector and returns the corresponding plane.
227 .TF reflect3
228 .TP
229 Name
230 Description
231 .TP
232 .B pn2f3
233 Compute the plane passing through
234 .I p
235 with normal
236 .IR n .
237 .TP
238 .B ppp2f3
239 Compute the plane passing through three points.
240 .TP
241 .B fff2p3
242 Compute the intersection point of three planes.
243 .PD
244 .PP
245 The names of the following routines end in
246 .B 4
247 because they operate on points in projective 4-space,
248 represented by their homogeneous coordinates.
249 .TP
250 pdiv4
251 Perspective division.  Divide
252 .B p.w
253 into
254 .IR p 's
255 coordinates, converting to affine coordinates.
256 If
257 .B p.w
258 is zero, the result is the same as the argument.
259 .TP
260 add4
261 Add the coordinates of two points.
262 .PD
263 .TP
264 sub4
265 Subtract the coordinates of two points.
266 .SH SOURCE
267 .B \*9/src/libgeometry
268 .SH "SEE ALSO
269 .MR matrix (3)