找到圆上的切点？ [英] Find a tangent point on circle?

问题描述

给定一条带有第一终点P（x1，y1）的线，另一个终点是未知的，与仅在一个点（切线）T（x2，y2）处具有半径R的原点相交的圆相交。任何人都知道如何获得T点？

解决方案

给定一条带第一个终点P（x1，y1）的行结束点是未知的，与仅在一点（切线）T（x2，y2）处位于具有半径R的原点的圆相交。任何人都知道如何获得T点？

其他一些解决方案看起来有点像矫枉过正。我认为最简单的方法是注意到这是一个直角三角形，顶点P，T和O（原点）。角度PTO是直角，因为切线总是与半径成直角。

您知道 TO ，因为它的长度 r 并且在原点处有一个顶点;你知道 OP ，因为你知道 O 和 P 是。给定直角三角形的两边，很容易找到第三边的长度和方向。这是作业，所以我将其余的部分作为练习留给读者。

  __...---- --__ T（x2，y2）
 _.-'' - （+）
， - '| ---- 
，'| ---- 
，'| '---- 
 / | `---- 
 / | `。 ---- 
 / | \ ---- 
 | | | ---- 
 | | | ---- 
 | | | ---- 
 | （+）---------------------------------------------（+ ）P（x1，y1）
 | ''
 | O | 
 | 。'
 \ / 
 \，'
`/ 
'。 ，'
' - 。 _，'
'-._ _，（+）T'（x3，y3）
'`--......--'
  TO 有两个可能的方向，因为T'点也是有效的切点，所以你将有两个全等的三角形。
 
Given a line with first end point P(x1,y1) another end point is unknown, intersect with a circle that located at origin with radius R at only one point(tangent) T(x2,y2). Anyone know how to get the point T? Thanks in advance!
解决方案

  
Given a line with first end point P(x1,y1) another end point is unknown, intersect with a circle that located at origin with radius R at only one point(tangent) T(x2,y2). Anyone know how to get the point T?
Some of the other solutions seem a little like overkill. I think the simplest way is just to notice that this is a right triangle, with vertices P, T, and O (the origin). The angle PTO is the right angle, because a tangent line is always at a right angle to a radius.

You know the length of TO because it's of length r and has a vertex at the origin; you know OP because you know where O and P is. Given two sides of a right triangle, it's easy to find the length and direction of the third side. This is homework, so I'll leave the rest as an exercise to the reader.
                    __...------__    T(x2, y2)                      
               _.-''             -(+)
            ,-'                   |----             
          ,'                     |     ----
        ,'                      |       '  ----
       /                       |         `     ----       
      /                       |           `.       ----   
     /                       |             \           ----
    |                       |               |              ----
    |                      |                 |                  ----
    |                     |                  |                      ----
    |                   (+)---------------------------------------------(+) P (x1,y1)
    |                                        .'        
    |                    O                   |         
     |                                      .'         
      \                                     /          
       \                                  ,'           
        `                                /             
         '.                            ,'              
           '-.                      _,'             
              '-._              _,(+)  T'(x3, y3)                   
                  '`--......---'                       
There are two possible directions for TO, since the point T' is also a valid tangent point, so you will have two congruent triangles.

                        
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