确定的点的坐标基于其已知的差异从三个其他点 [英] Determining The Coordinates Of A Point Based On Its Known Difference From Three Other Points

问题描述

我有三个点的平面上的坐标。让我们称他们为X1，Y1，X2，Y2，Y3 X3

我需要计算X4，Y4但我知道的是：

X1，Y1为350台，距离X4，Y4 X2，Y2是200个单位在从X4，Y4距离 X3，Y3是50个单位的距离X4，Y4

我知道确切的值对于X1，Y1，X2，Y2，和X3，Y3

如何确定X4的确切位置，Y4？

解决方案

 （X  -  X1）^ 2 +（Y  -  Y1）^ 2 = R1 ^ 2 ----- -  P
（X  -  X 2）^ 2 +（Y  -  Y2）^ 2 = R2 ^ 2 ------ q
（X  -  X 3）^ 2 +（Y  -  Y3）^ 2 = R3 ^ 2 ------ř
 

解决了这3个圆的交点。

 对 -  q -----升
 对 -   -  [R -----ñ
 

使用解方程（L）和（n） 克莱姆法则 。

的 GET_POINT（X1，Y1，R1，X2，Y2，R2，X3，Y3，R3）：
    A = X1  -  X2
    B = Y1  -  Y2
    D = X1  -  X3
    E = Y1  -  Y3

    T =（R1 * R1  -  X1 * X1  -  Y1 * Y1）
    C =（R2 * R2  -  X2 * X2  -  Y2 * Y2） -  T的
    F =（R3 * R3  -  X3 * X3  -  Y3 * Y3） -  T的

    一λ+ B Y = C / 2 //这是方程式L
    深x + E Y = F / 2 //这是等式'N'

    //克莱姆法则

    MX =（权证 -  B F）/ 2
    我=（A F  -  D C）/ 2
    M = AE  -  DB

    X =的Mx / M
    Y =我/ M

    回报（X，Y）
 

I have the coordinates of three points on a plane. Let's call them X1,Y1, X2,Y2, X3 Y3.

I need to calculate X4,Y4 but all I know is:

X1,Y1 is 350 units in distance from X4,Y4 X2,Y2 is 200 units in distance from X4,Y4 X3,Y3 is 50 units in distance from X4,Y4

I Know The Exact Values For X1,Y1, X2,Y2, and X3,Y3

How can I determine the exact location of X4,Y4?

解决方案

(x - x1)^2 + (y - y1)^2 = r1^2  ------ p
(x - x2)^2 + (y - y2)^2 = r2^2  ------ q
(x - x3)^2 + (y - y3)^2 = r3^2  ------ r

Solve for intersection point of these 3 circles.

 p - q     ----- l 
 p - r     ----- n

Solve equation (l) and (n) using Cramer's rule.

GET_POINT(x1,y1,r1,x2,y2,r2,x3,y3,r3):
    A = x1 - x2
    B = y1 - y2
    D = x1 - x3
    E = y1 - y3

    T = (r1*r1 - x1*x1 - y1*y1)
    C = (r2*r2 - x2*x2 - y2*y2) - T
    F = (r3*r3 - x3*x3 - y3*y3) - T

    A x + B y = C/2  // this is equation 'l'
    D x + E y = F/2  // this is equation 'n'

    // Cramer's Rule

    Mx = (C E  - B F) /2
    My = (A F  - D C) /2
    M  = AE - DB

    x = Mx/M
    y = My/M

    return (x,y)

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