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
|
// Geometric Tools, LLC
// Copyright (c) 1998-2014
// Distributed under the Boost Software License, Version 1.0.
// http://www.boost.org/LICENSE_1_0.txt
// http://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
//
// File Version: 5.0.1 (2010/10/01)
#include "Wm5MathematicsPCH.h"
#include "Wm5IntrCircle2Circle2.h"
namespace Wm5
{
//----------------------------------------------------------------------------
template <typename Real>
IntrCircle2Circle2<Real>::IntrCircle2Circle2 (const Circle2<Real>& circle0,
const Circle2<Real>& circle1)
:
mCircle0(&circle0),
mCircle1(&circle1)
{
}
//----------------------------------------------------------------------------
template <typename Real>
const Circle2<Real>& IntrCircle2Circle2<Real>::GetCircle0 () const
{
return *mCircle0;
}
//----------------------------------------------------------------------------
template <typename Real>
const Circle2<Real>& IntrCircle2Circle2<Real>::GetCircle1 () const
{
return *mCircle1;
}
//----------------------------------------------------------------------------
template <typename Real>
bool IntrCircle2Circle2<Real>::Find ()
{
// The two circles are |X-C0| = R0 and |X-C1| = R1. Define U = C1 - C0
// and V = Perp(U) where Perp(x,y) = (y,-x). Note that Dot(U,V) = 0 and
// |V|^2 = |U|^2. The intersection points X can be written in the form
// X = C0+s*U+t*V and X = C1+(s-1)*U+t*V. Squaring the circle equations
// and substituting these formulas into them yields
// R0^2 = (s^2 + t^2)*|U|^2
// R1^2 = ((s-1)^2 + t^2)*|U|^2.
// Subtracting and solving for s yields
// s = ((R0^2-R1^2)/|U|^2 + 1)/2
// Then replace in the first equation and solve for t^2
// t^2 = (R0^2/|U|^2) - s^2.
// In order for there to be solutions, the right-hand side must be
// nonnegative. Some algebra leads to the condition for existence of
// solutions,
// (|U|^2 - (R0+R1)^2)*(|U|^2 - (R0-R1)^2) <= 0.
// This reduces to
// |R0-R1| <= |U| <= |R0+R1|.
// If |U| = |R0-R1|, then the circles are side-by-side and just tangent.
// If |U| = |R0+R1|, then the circles are nested and just tangent.
// If |R0-R1| < |U| < |R0+R1|, then the two circles to intersect in two
// points.
Vector2<Real> U = mCircle1->Center - mCircle0->Center;
Real USqrLen = U.SquaredLength();
Real R0 = mCircle0->Radius, R1 = mCircle1->Radius;
Real R0mR1 = R0 - R1;
if (USqrLen < Math<Real>::ZERO_TOLERANCE
&& Math<Real>::FAbs(R0mR1) < Math<Real>::ZERO_TOLERANCE)
{
// Circles are essentially the same.
mIntersectionType = IT_OTHER;
mQuantity = 0;
return true;
}
Real R0mR1Sqr = R0mR1*R0mR1;
if (USqrLen < R0mR1Sqr)
{
mIntersectionType = IT_EMPTY;
mQuantity = 0;
return false;
}
Real R0pR1 = R0 + R1;
Real R0pR1Sqr = R0pR1*R0pR1;
if (USqrLen > R0pR1Sqr)
{
mIntersectionType = IT_EMPTY;
mQuantity = 0;
return false;
}
if (USqrLen < R0pR1Sqr)
{
if (R0mR1Sqr < USqrLen)
{
Real invUSqrLen = ((Real)1)/USqrLen;
Real s = ((Real)0.5)*((R0*R0-R1*R1)*invUSqrLen+(Real)1);
Vector2<Real> tmp = mCircle0->Center + s*U;
// In theory, discr is nonnegative. However, numerical round-off
// errors can make it slightly negative. Clamp it to zero.
Real discr = R0*R0*invUSqrLen - s*s;
if (discr < (Real)0)
{
discr = (Real)0;
}
Real t = Math<Real>::Sqrt(discr);
Vector2<Real> V(U.Y(), -U.X());
mQuantity = 2;
mPoint[0] = tmp - t*V;
mPoint[1] = tmp + t*V;
}
else
{
// |U| = |R0-R1|, circles are tangent.
mQuantity = 1;
mPoint[0] = mCircle0->Center + (R0/R0mR1)*U;
}
}
else
{
// |U| = |R0+R1|, circles are tangent.
mQuantity = 1;
mPoint[0] = mCircle0->Center + (R0/R0pR1)*U;
}
mIntersectionType = IT_POINT;
return true;
}
//----------------------------------------------------------------------------
template <typename Real>
int IntrCircle2Circle2<Real>::GetQuantity () const
{
return mQuantity;
}
//----------------------------------------------------------------------------
template <typename Real>
const Vector2<Real>& IntrCircle2Circle2<Real>::GetPoint (int i) const
{
return mPoint[i];
}
//----------------------------------------------------------------------------
template <typename Real>
const Circle2<Real>& IntrCircle2Circle2<Real>::GetIntersectionCircle () const
{
return *mCircle0;
}
//----------------------------------------------------------------------------
//----------------------------------------------------------------------------
// Explicit instantiation.
//----------------------------------------------------------------------------
template WM5_MATHEMATICS_ITEM
class IntrCircle2Circle2<float>;
template WM5_MATHEMATICS_ITEM
class IntrCircle2Circle2<double>;
//----------------------------------------------------------------------------
}
|
