给定椭圆的焦点(x,y),方向线(ax + by + c)和偏心率e ,任务是使用椭圆的焦点,方向线和偏心率找到椭圆方程。
例子:
Input: x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = 0.5
Output: 1.75 x^2 + 1.75 y^2 + -5.50 x + -2.50 y + 0.50 xy + 1.75 = 0
Input: x1 = -1, y1 = 1, a = 1, b = -1, c = 3, e = 0.5
Output: 1.75 x^2 + 1.75 y^2 + 2.50 x + -2.50 y + 0.50 xy + 1.75 = 0
令P(x,y)是椭圆上焦点为S(x1,y1)的任何点,Directrix是直线ax + by + c = 0且偏心率是e。
从直线上的P垂直绘制PM。然后通过定义椭圆距离SP = e * PM => SP ^ 2 =(e * PM)^ 2
(x – x1)^2 + (y – y1)^2 = e * ( ( a*x + b*y + c ) / (sqrt( a*a + b*b )) ) ^ 2
let ( a*a + b*b ) = t
x^2 + x1^2 – 2*x1*x + y^2 + y1^2 – 2*y1*y = e * ( ( a*x + b*y + c ) ^ 2 )/ t
在交叉乘以上方,我们得到
t*x^2 + t*x1^2 – 2*t*x1*x + t*y^2 + t*y1^2 – 2*t*y1*y = e * ( ( a*x + b*y + c ) ^ 2 )
t*x^2 + t*x1^2 – 2*t*x1*x + t*y^2 + t*y1^2 – 2*t*y1*y = e*a^2*x^2 + e*b^2*y^2 + 2*e*a*x*b*y + e*c^2 + 2*e*c*(a*x + b*y)
t*x^2 + t*x1^2 – 2*t*x1*x + t*y^2 + t*y1^2 – 2*t*y1*y = e*a^2*x^2 + e*b^2*y^2 + 2*e*a*x*b*y + e*c^2 + 2*e*c*a*x + 2*e*c*b*y
t*x^2 – e*a^2*x^2 + t*y^2 – e*b^2*y^2 – 2*t*x1*x – 2*e*c*a*x – 2*t*y1*y – 2*e*c*b*y – 2*e*a*x*b*y – e*c^2 + t*x1^2 + t*y1^2 =0
可以将其与以下一般形式进行比较:
a*x^2 + 2*h*x*y + b*y^2 + 2*g*x + 2*f*y + c = 0
下面是上述方法的实现:
C++
// C++ program to find equation of an ellipse
// using focus and directrix.
#include
#include
#include
#include
using namespace std;
// Function to find equation of ellipse.
void equation_ellipse(float x1, float y1,
float a, float b,
float c, float e)
{
float t = a * a + b * b;
float a1 = t - e * (a * a);
float b1 = t - e * (b * b);
float c1 = (-2 * t * x1) - (2 * e * c * a);
float d1 = (-2 * t * y1) - (2 * e * c * b);
float e1 = -2 * e * a * b;
float f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1);
cout << fixed;
cout << setprecision(2);
cout << "Equation of ellipse is \n"
<< a1
<< " x^2 + " << b1 << " y^2 + "
<< c1 << " x + " << d1 << " y + "
<< e1 << " xy + " << f1 << " = 0";
}
// Driver Code
int main()
{
float x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = 0.5 * 0.5;
equation_ellipse(x1, y1, a, b, c, e);
return 0;
}
Java
// Java program to find equation of an ellipse
// using focus and directrix.
import java.util.*;
class solution
{
// Function to find equation of ellipse.
static void equation_ellipse(float x1, float y1,
float a, float b,
float c, float e)
{
float t = a * a + b * b;
float a1 = t - e * (a * a);
float b1 = t - e * (b * b);
float c1 = (-2 * t * x1) - (2 * e * c * a);
float d1 = (-2 * t * y1) - (2 * e * c * b);
float e1 = -2 * e * a * b;
float f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1);
System.out.println("Equation of ellipse is ");
System.out.print(a1+" x^2 + "+ b1 + " y^2 + "+ c1 + " x + "
+ d1 + " y + " + e1 + " xy + " + f1 + " = 0");
}
// Driver Code
public static void main(String arr[])
{
float x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = (float)0.5 * (float)0.5;
equation_ellipse(x1, y1, a, b, c, e);
}
}
//This code is contributed by Surendra_Gaangwar
Python3
# Python3 program to find equation of an ellipse
# using focus and directrix.
# Function to find equation of ellipse.
def equation_ellipse(x1, y1, a, b, c, e) :
t = a * a + b * b
a1 = t - e * (a * a)
b1 = t - e * (b * b)
c1 = (-2 * t * x1) - (2 * e * c * a)
d1 = (-2 * t * y1) - (2 * e * c * b)
e1 = -2 * e * a * b
f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1)
print("Equation of ellipse is",a1,"x^2 +", b1 ,"y^2 +",
c1, "x +" ,d1 ,"y +", e1 ,"xy +" , f1 ,"= 0")
# Driver Code
if __name__ == "__main__" :
x1, y1, a, b, c, e = 1, 1, 1, -1, 3, 0.5 * 0.5
equation_ellipse(x1, y1, a, b, c, e)
# This code is contributed by Ryuga
C#
// C# program to find equation of an ellipse
// using focus and directrix.
class solution
{
// Function to find equation of ellipse.
static void equation_ellipse(float x1, float y1,
float a, float b,
float c, float e)
{
float t = a * a + b * b;
float a1 = t - e * (a * a);
float b1 = t - e * (b * b);
float c1 = (-2 * t * x1) - (2 * e * c * a);
float d1 = (-2 * t * y1) - (2 * e * c * b);
float e1 = -2 * e * a * b;
float f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1);
System.Console.WriteLine("Equation of ellipse is ");
System.Console.WriteLine(a1+" x^2 + "+ b1 + " y^2 + "+ c1 + " x + "
+ d1 + " y + " + e1 + " xy + " + f1 + " = 0");
}
// Driver Code
public static void Main()
{
float x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = (float)0.5 * (float)0.5;
equation_ellipse(x1, y1, a, b, c, e);
}
}
//This code is contributed by mits
PHP
Javascript
Equation of ellipse is
1.75 x^2 + 1.75 y^2 + -5.50 x + -2.50 y + 0.50 xy + 1.75 = 0