1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
|
// Source: https://github.com/golang/geo
/*
Copyright 2014 Google Inc. All rights reserved.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
package r3
import (
"fmt"
"math"
"github.com/golang/geo/s1"
)
// Vector represents a point in ℝ³.
type Vector struct {
X, Y, Z float64
}
// ApproxEqual reports whether v and ov are equal within a small epsilon.
func (v Vector) ApproxEqual(ov Vector) bool {
const epsilon = 1e-16
return math.Abs(v.X-ov.X) < epsilon && math.Abs(v.Y-ov.Y) < epsilon && math.Abs(v.Z-ov.Z) < epsilon
}
func (v Vector) String() string { return fmt.Sprintf("(%0.24f, %0.24f, %0.24f)", v.X, v.Y, v.Z) }
// Norm returns the vector's norm.
func (v Vector) Norm() float64 { return math.Sqrt(v.Dot(v)) }
// Norm2 returns the square of the norm.
func (v Vector) Norm2() float64 { return v.Dot(v) }
// Normalize returns a unit vector in the same direction as v.
func (v Vector) Normalize() Vector {
if v == (Vector{0, 0, 0}) {
return v
}
return v.Mul(1 / v.Norm())
}
// IsUnit returns whether this vector is of approximately unit length.
func (v Vector) IsUnit() bool {
const epsilon = 5e-14
return math.Abs(v.Norm2()-1) <= epsilon
}
// Abs returns the vector with nonnegative components.
func (v Vector) Abs() Vector { return Vector{math.Abs(v.X), math.Abs(v.Y), math.Abs(v.Z)} }
// Add returns the standard vector sum of v and ov.
func (v Vector) Add(ov Vector) Vector { return Vector{v.X + ov.X, v.Y + ov.Y, v.Z + ov.Z} }
// Sub returns the standard vector difference of v and ov.
func (v Vector) Sub(ov Vector) Vector { return Vector{v.X - ov.X, v.Y - ov.Y, v.Z - ov.Z} }
// Mul returns the standard scalar product of v and m.
func (v Vector) Mul(m float64) Vector { return Vector{m * v.X, m * v.Y, m * v.Z} }
// Dot returns the standard dot product of v and ov.
func (v Vector) Dot(ov Vector) float64 { return v.X*ov.X + v.Y*ov.Y + v.Z*ov.Z }
// Cross returns the standard cross product of v and ov.
func (v Vector) Cross(ov Vector) Vector {
return Vector{
v.Y*ov.Z - v.Z*ov.Y,
v.Z*ov.X - v.X*ov.Z,
v.X*ov.Y - v.Y*ov.X,
}
}
// Distance returns the Euclidean distance between v and ov.
func (v Vector) Distance(ov Vector) float64 { return v.Sub(ov).Norm() }
// Angle returns the angle between v and ov.
func (v Vector) Angle(ov Vector) s1.Angle {
return s1.Angle(math.Atan2(v.Cross(ov).Norm(), v.Dot(ov))) * s1.Radian
}
// Axis enumerates the 3 axes of ℝ³.
type Axis int
// The three axes of ℝ³.
const (
XAxis Axis = iota
YAxis
ZAxis
)
// Ortho returns a unit vector that is orthogonal to v.
// Ortho(-v) = -Ortho(v) for all v.
func (v Vector) Ortho() Vector {
ov := Vector{0.012, 0.0053, 0.00457}
switch v.LargestComponent() {
case XAxis:
ov.Z = 1
case YAxis:
ov.X = 1
default:
ov.Y = 1
}
return v.Cross(ov).Normalize()
}
// LargestComponent returns the axis that represents the largest component in this vector.
func (v Vector) LargestComponent() Axis {
t := v.Abs()
if t.X > t.Y {
if t.X > t.Z {
return XAxis
}
return ZAxis
}
if t.Y > t.Z {
return YAxis
}
return ZAxis
}
// SmallestComponent returns the axis that represents the smallest component in this vector.
func (v Vector) SmallestComponent() Axis {
t := v.Abs()
if t.X < t.Y {
if t.X < t.Z {
return XAxis
}
return ZAxis
}
if t.Y < t.Z {
return YAxis
}
return ZAxis
}
// Cmp compares v and ov lexicographically and returns:
//
// -1 if v < ov
// 0 if v == ov
// +1 if v > ov
//
// This method is based on C++'s std::lexicographical_compare. Two entities
// are compared element by element with the given operator. The first mismatch
// defines which is less (or greater) than the other. If both have equivalent
// values they are lexicographically equal.
func (v Vector) Cmp(ov Vector) int {
if v.X < ov.X {
return -1
}
if v.X > ov.X {
return 1
}
// First elements were the same, try the next.
if v.Y < ov.Y {
return -1
}
if v.Y > ov.Y {
return 1
}
// Second elements were the same return the final compare.
if v.Z < ov.Z {
return -1
}
if v.Z > ov.Z {
return 1
}
// Both are equal
return 0
}
|
