你如何在数学上从一个点找到圆中的外点？

问题描述

试图找到一组点，使一个点看似连接到一个圆，如下图所示，但是没有重叠，它与圆的边缘对齐。抱歉，如果这很难理解，很难解释。

解决方法

如果你想要这样的东西：

那么下面的python代码就是这样做的。

import numpy as np

# function that calcuates the points at which the lines are tangent to the circle
def tangent_points(Point,Circle):
    Rot_90 = np.array([[0,-1],[1,0]]) 
    O = Circle[0]
    r = Circle[1]
    unit_OP = Point - O
    OP = np.sqrt( unit_OP.dot(unit_OP) )
    unit_OP = unit_OP / OP
    a = r**2 / OP
    unit_perp = Rot_90.dot(unit_OP)
    b = np.sqrt(r**2 - a**2)
    return O + a*unit_OP + b*unit_perp,O + a*unit_OP - b*unit_perp

# Test example 
O = np.array([0,0])
r = 2
P = np.array([7,5])   
Circ = (O,r)
  
T1,T2 = tangent_points(P,Circ)

# plotting preparations:

# prepare circle
s = np.linspace( 0,2 * np.pi,150 )
xc = O[0] + r*np.cos( s )
yc = O[1] + r*np.sin( s )

# prepare tangents
s = np.linspace( 0,1,150)  
L1 = P[:,np.newaxis]*(1-s[np.newaxis,:]) + T1[:,np.newaxis]*s[np.newaxis,:]
L2 = P[:,:]) + T2[:,:]

# actual plotting
# setup a figure environment
figure,axes = plt.subplots()
axes.set_aspect( 1 )
# plotting circle and center
axes.plot( xc,yc )
axes.plot(O[0],O[1],'bo')
# plotting point ouside circle
axes.plot(P[0],P[1],'bo')

# plotting tagnetn lines
axes.plot( L1[0],L1[1] )
axes.plot( L2[0],L2[1] )

# plotting tangent points
axes.plot(T1[0],T1[1],'ro')
axes.plot(T2[0],T2[1],'ro')

plt.show()

algebra geometry math