ios - 三角测量 - 如何找到 3 个球体的交点(iBeacons)
<p><p>在给定这 3 个球体方程的情况下，计算 3 个球体交集的 iOSobjective-c方法是什么:</p>
<pre><code>1) (x -x1)^2 + (y - y1)^2 + (z - z1)^2 = (r1)^2
2) (x -x2)^2 + (y - y2)^2 + (z - z2)^2 = (r2)^2
3) (x -x3)^2 + (y - y3)^2 + (z - z3)^2 = (r3)^2
</code></pre>
<p>//我想要的方法</p>
<pre><code>-(NSArray*) findIntersectionOfThreeSpheresWithFirstSphereRadius: (float) r1 xCoordinate: (float) x1 yCoordinate:(float) y1 zCoordinate:(float) z1 andWithSecondSphereRadius: (float) r2 xCoordinate: (float) x2 yCoordinate:(float) y2 zCoordinate:(float) z2 andWithThirdSphereRadius: (float) r3 xCoordinate: (float) x3 yCoordinate:(float) y3 zCoordinate:(float) z3
</code></pre></p>
<br><hr><h1><strong>Best Answer-推荐答案</ strong></h1><br>
<p><p>我在任何地方都找不到这个问题的答案 - 所以我自己解决了。如果您觉得此回答有帮助，请 +1 回答!</p>
<pre><code>-(NSArray*) findIntersectionOfThreeSpheresWithFirstSphereRadius: (float) r1 xCoordinate: (float) x1 yCoordinate:(float) y1 zCoordinate:(float) z1 andWithSecondSphereRadius: (float) r2 xCoordinate: (float) x2 yCoordinate:(float) y2 zCoordinate:(float) z2 andWithThirdSphereRadius: (float) r3 xCoordinate: (float) x3 yCoordinate:(float) y3 zCoordinate:(float) z3
{
double x1Answer;
double y1Answer;
double z1Answer;
double x2Answer;
double y2Answer;
double z2Answer;
double k = 0.0;
double k2 = 0.0;
double l = 0.0;
double l2 = 0.0;
double m = 0.0;
double m2 = 0.0;
double n = 0.0;
double n2 = 0.0;
double p = 0.0;
double p2 = 0.0;
double q = 0.0;
double q2 = 0.0;
double u = 0.0;
double u2 = 0.0;
double w = 0.0;
double w2 = 0.0;
double j = 0.0;
double j2 = 0.0;
double h = 0.0;
double h2 = 0.0;
double v = 0.0;
double v2 = 0.0;
double c = 0.0;
double c2 = 0.0;
double d = 0.0;
double d2 = 0.0;
double f = 0.0;
double f2 = 0.0;
double t = 0.0;
double t2 = 0.0;
double s = 0.0;
double s2 = 0.0;
double A = 0.0;
double B = 0.0;
double C = 0.0;
double i1 = 0.0;
double b1 = 0.0;
double i2 = 0.0;
double b2 = 0.0;
double i3 = 0.0;
double b3 = 0.0;
double i4 = 0.0;
double b4 = 0.0;
BOOL a1SpecialCase = NO;
BOOL a2SpecialCase = NO;
BOOL a3SpecialCase = NO;
BOOL a4SpecialCase = NO;
BOOL b1SpecialCase = NO;
BOOL b2SpecialCase = NO;
BOOL oneAnswer = NO;
BOOL twoAnswers = NO;
if (z1-z2 != 0)
{
k2 = 2* (x2-x1)/(2*z1-2*z2);
l2 = 2* (y2-y1)/(2*z1-2*z2);
m2 = (x1*x1 - x2*x2 + y1*y1 - y2*y2 + z1*z1 - z2*z2 - r1*r1 + r2*r2)/(2*z1-2*z2);
}
else
{
a1SpecialCase = YES;
k = 2* (x2-x1);
l = 2* (y2-y1);
m = x1*x1 - x2*x2 + y1*y1 - y2*y2 + z1*z1 - z2*z2 - r1*r1 + r2*r2;
}
if (y1-y2 != 0)
{
n2 = 2* (x2-x1)/(2*y1-2*y2);
p2 = 2* (z2-z1)/(2*y1-2*y2);
q2 = (x1*x1 - x2*x2 + y1*y1 - y2*y2 + z1*z1 - z2*z2 - r1*r1 + r2*r2)/(2*y1-2*y2);
}
else
{
a2SpecialCase = YES;
n = 2* (x2-x1);
p = 2* (z2-z1);
q = x1*x1 - x2*x2 + y1*y1 - y2*y2 + z1*z1 - z2*z2 - r1*r1 + r2*r2;
}
if (z1-z3 != 0)
{
u2 = 2* (x3-x1)/(2*z1-2*z3);
w2 = 2* (y3-y1)/(2*z1-2*z3);
j2 = (x1*x1 - x3*x3 + y1*y1 - y3*y3 + z1*z1 - z3*z3 - r1*r1 + r3*r3)/(2*z1-2*z3);
}
else
{
a3SpecialCase = YES;
u = 2* (x3-x1);
w = 2* (y3-y1);
j = x1*x1 - x3*x3 + y1*y1 - y3*y3 + z1*z1 - z3*z3 - r1*r1 + r3*r3;
}
if (y1-y3 != 0)
{
h2 = 2* (x3-x1)/(2*y1-2*y3);
v2 = 2* (z3-z1)/(2*y1-2*y3);
c2 = (x1*x1 - x3*x3 + y1*y1 - y3*y3 + z1*z1 - z3*z3 - r1*r1 + r3*r3)/(2*y1-2*y3);
}
else
{
a4SpecialCase = YES;
h = 2* (x3-x1);
v = 2* (z3-z1);
c = (x1*x1 - x3*x3 + y1*y1 - y3*y3 + z1*z1 - z3*z3 - r1*r1 + r3*r3);
}
if ( (!a1SpecialCase&& !a2SpecialCase && !a3SpecialCase && !a4SpecialCase) || ((a1SpecialCase&&a3SpecialCase) && (a2SpecialCase&&a4SpecialCase)) ||((a1SpecialCase&&a3SpecialCase) && (!a2SpecialCase&&!a4SpecialCase)) || ((!a1SpecialCase&&!a3SpecialCase) && (a2SpecialCase&&a4SpecialCase)) )
{
if (l2-w2 !=0)
{
d2 = (u2-k2)/(l2-w2);
f2 = (j2-m2)/(l2-w2);
}
else
{
b1SpecialCase = YES;
d =u2-k2;
f =j2-m2;
}
if (p2-v2 !=0)
{
t2 = (h2-n2)/(p2-v2);
s2 = (c2-q2)/(p2-v2);
}
else
{
b2SpecialCase = YES;
t = h2-n2;
s = c2-q2;
}
if (!b1SpecialCase&&!b2SpecialCase)
{
A = 1+ d2*d2 + t2*t2;
B = d2*f2- 2*x1 -d2*y1 +f2*d2 - y1*d2 +s2*t2 -z1*t2 +s2*t2 - z1*t2;
C = x1*x1+f2*f2 - y1*f2 -y1*f2 + y1*y1 + s2*s2 - s2*z1 - z1*s2+ z1*z1 - r1*r1;
twoAnswers = YES;
x1Answer = (-B + sqrt(B * B - 4 * A * C)) / (2 * A);
x2Answer = (-B - sqrt(B * B - 4 * A * C)) / (2 * A);
y1Answer = d2*x1Answer + f2;
y2Answer = d2*x2Answer + f2;
z1Answer = t2*x1Answer +s2;
z2Answer = t2*x2Answer +s2;
}
else if (b1SpecialCase&&b2SpecialCase)
{
//Figure Out Later if this ever gets called
}
else if (b1SpecialCase)
{
oneAnswer = YES;
x1Answer = -f/d;
z1Answer = -f*t*t/d + s2;
if (a1SpecialCase)
{
y1Answer = -m+k*f/d;
}
else
{
y1Answer = (z1Answer-m2-k2*x1Answer)/l2;
}
}
else if (b2SpecialCase)
{
oneAnswer = YES;
x1Answer = -s/t;
y1Answer = d2*x1Answer + f2;
if (a2SpecialCase)
{
z1Answer = (-1-n*x1Answer)/p;
}
else
{
z1Answer = (y1Answer-q2-n2*x1Answer)/p2;
}
}
}
else
{
if ( (!a1SpecialCase&&!a3SpecialCase))
{
if (a2SpecialCase)
{
if (l2-w2 !=0)
{
d2 = (u2-k2)/(l2-w2);
f2 = (j2-m2)/(l2-w2);
i1 = -q/p;
b1 = -n/p;
A = 1+ b1*b1 + d2*d2;
B = -2*x1 + f2*d2- d2*y1 +f2*d2 - y1*d2 + b1*i1 - b1*z1 +b1*i1 -z1*b1;
C = x1*x1 + f2*f2 - f2*y1 - y1*f2 + y1*y1 +i1*i1 - i1*z1- z1*i1 +z1*z1- r1*r1;
twoAnswers = YES;
x1Answer = (-B + sqrt(B * B - 4 * A * C)) / (2 * A);
x2Answer = (-B - sqrt(B * B - 4 * A * C)) / (2 * A);
y1Answer = d2*x1Answer + f2;
y2Answer = d2*x2Answer + f2;
z1Answer = m2+l2*y1Answer + x1Answer*k2;
z2Answer = m2+l2*y2Answer + x2Answer*k2;
}
else
{
b1SpecialCase = YES;
oneAnswer = YES;
x1Answer = -f/d;
z1Answer = (-n*x1Answer - q)/p;
y1Answer = (z1Answer-m2-k2*x1Answer)/l2;
}
}
else if (a4SpecialCase)
{
if (l2-w2 !=0)
{
d2 = (u2-k2)/(l2-w2);
f2 = (j2-m2)/(l2-w2);
i2 = -h/v;
b2 = -c/p;
A = 1+ b2*b2 + d2*d2;
B = -2*x1 + f2*d2- d2*y1 +f2*d2 - y1*d2 + b2*i2 - b2*z1 +b2*i2 -z1*b2;
C = x1*x1 + f2*f2 - f2*y1 - y1*f2 + y1*y1 +i2*i2 - i2*z1- z1*i2 +z1*z1- r1*r1;
twoAnswers = YES;
x1Answer = (-B + sqrt(B * B - 4 * A * C)) / (2 * A);
x2Answer = (-B - sqrt(B * B - 4 * A * C)) / (2 * A);
y1Answer = d2*x1Answer + f2;
y2Answer = d2*x2Answer + f2;
z1Answer = m2+l2*y1Answer + x1Answer*k2;
z2Answer = m2+l2*y2Answer + x2Answer*k2;
}
else
{
b1SpecialCase = YES;
oneAnswer = YES;
x1Answer = -f/d;
z1Answer = (-n*x1Answer - q)/p;
y1Answer = (z1Answer-m2-k2*x1Answer)/l2;
}
}
}
else if (!a2SpecialCase&&!a4SpecialCase)
{
if (a1SpecialCase)
{
if (p2-v2 !=0)
{
t2 = (h2-n2)/(p2-v2);
s2 = (c2-q2)/(p2-v2);
b3 = -k/l;
i3 = m/l;
A = 1+ b3*b3 + t2*t2;
B = -2*x1 -b3*i3-b3*y1-b3*i3 -y1*b3 + s2*t2 - z1*t2+ s2*t2-z1*t2;
C = x1*x1 + i3*i3 - i3*y1 - y1*i3 + y1*y1 +s2*s2 - z1*s2- z1*s2 +z1*z1- r1*r1;
twoAnswers = YES;
x1Answer = (-B + sqrt(B * B - 4 * A * C)) / (2 * A);
x2Answer = (-B - sqrt(B * B - 4 * A * C)) / (2 * A);
y1Answer = (-m-k*x1Answer)/l;
y2Answer = (-m-k*x2Answer)/l;
z1Answer = u2*x1Answer+w2*y1Answer +j2;
z2Answer = u2*x2Answer+w2*y2Answer +j2;
}
else
{
b2SpecialCase = YES;
oneAnswer = YES;
x1Answer = -s/t;
y1Answer = (-m-k*x1Answer)/l;
z1Answer = u2*x1Answer+w2*y1Answer+j2;
}
}
else if (a3SpecialCase)
{
if (p2-v2 !=0)
{
t2 = (h2-n2)/(p2-v2);
s2 = (c2-q2)/(p2-v2);
b4 = -u/w;
i4 = j/w;
A = 1+ b4*b4 + t2*t2;
B = -2*x1 -b4*i4-b4*y1-b4*i4 -y1*b3 + s2*t2 - z1*t2+ s2*t2-z1*t2;
C = x1*x1 + i4*i4 - i4*y1 - y1*i4 + y1*y1 +s2*s2 - z1*s2- z1*s2 +z1*z1- r1*r1;
twoAnswers = YES;
x1Answer = (-B + sqrt(B * B - 4 * A * C)) / (2 * A);
x2Answer = (-B - sqrt(B * B - 4 * A * C)) / (2 * A);
y1Answer = (-j-u*x1Answer)/w;
y2Answer = (-j-u*x2Answer)/w;
z1Answer = k2*x1Answer+l2*y1Answer;
z2Answer = k2*x2Answer+l2*y2Answer;
}
else
{
b2SpecialCase = YES;
oneAnswer = YES;
x1Answer = -s/t;
y1Answer = (-j-u*x1Answer)/w;
z1Answer = k2*x1Answer+l2*y1Answer+m2;
}
}
}
else if(a1SpecialCase&&a2SpecialCase&&a3SpecialCase)
{
if (u!=0)
{
oneAnswer = YES;
y1Answer = (k*j/u -m)/(l-k*w/u);
x1Answer = (-m-l*y1Answer)/k;
z1Answer = y1Answer-c2-h2*x1Answer;
}
else
{
oneAnswer = YES;
y1Answer =-j/w;
x1Answer = (-m-l*y1Answer)/k;
z1Answer = y1Answer-c2-h2*x1Answer;
}
}
else if(a2SpecialCase&&a3SpecialCase&&a4SpecialCase)
{
if (n!=0)
{
z1Answer = (h*q/n -c)/(v-h*p/n);
x1Answer = (-c-v*z1Answer)/h;
y1Answer = (z1Answer-m2-k2*x1Answer)/l2;
}
else
{
z1Answer = -1/p;
x1Answer = (-c-v*z1Answer)/h;
y1Answer = (z1Answer-m2-k2*x1Answer)/l2;
}
}
else if(a3SpecialCase&&a4SpecialCase&&a1SpecialCase)
{
if (u!=0)
{
oneAnswer = YES;
y1Answer = (k*j/u -m)/(l-k*w/u);
x1Answer = (-m-l*y1Answer)/k;
z1Answer = (y1Answer-q2-n2*x1Answer)/p2;
}
else
{
oneAnswer = YES;
y1Answer =-j/w;
x1Answer = (-m-l*y1Answer)/k;
z1Answer = (y1Answer-q2-n2*x1Answer)/p2;
}
}
else if(a4SpecialCase&&a1SpecialCase&&a2SpecialCase)
{
if (n!=0)
{
oneAnswer = YES;
z1Answer = (h*q/n -c)/(v-h*p/n);
x1Answer = (-c-v*z1Answer)/h;
y1Answer = (z1Answer- u2*x1Answer-j2)/w;
}
else
{
oneAnswer = YES;
z1Answer = -1/p;
x1Answer = (-c-v*z1Answer)/h;
y1Answer = (z1Answer- u2*x1Answer-j2)/w;
}
}
}
if (!isnan(x1Answer)&&!isnan(y1Answer)&&!isnan(z1Answer))
{
NSLog(@"Answer: One Answer :%i Coordinate1 (%f, %f, %f) and Coordinate 2 (%f, %f, %f) and r1: %f, r2: %f, r3: %f", oneAnswer, x1Answer, y1Answer, z1Answer, x2Answer, y2Answer, z2Answer, r1, r2, r3);
NSArray *answer;
if (twoAnswers)
{
answer = ,, , nil];
}
else if (oneAnswer)
{
answer = ,, , nil];
}
return answer;
}
else
{
return nil;
}
}
</code></pre></p>
<p style="font-size: 20px;">关于ios - 三角测量 - 如何找到 3 个球体的交点(iBeacons)，我们在Stack Overflow上找到一个类似的问题：
<a href="https://stackoverflow.com/questions/24696902/" rel="noreferrer noopener nofollow" style="color: red;">
https://stackoverflow.com/questions/24696902/
</a>
</p>
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