Dado el foco (x, y), la directriz (ax + by + c) y la excentricidad e de una elipse, la tarea es encontrar la ecuación de la elipse usando su foco, directriz y excentricidad.
Ejemplos:
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
Sea P(x, y) cualquier punto de la elipse cuyo foco S(x1, y1), la directriz es la recta ax + by + c = 0 y la excentricidad es e.
Dibuje PM perpendicular desde P en la directriz. Entonces, por definición de distancia de elipse SP = e * PM => SP^2 = (e * PM)^2
(x – x1)^2 + (y – y1)^2 = e * ( ( a*x + b*y + c ) / (raíz cuadrada( a*a + b*b )) ) ^ 2
sea ( 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
al multiplicar en cruz arriba obtenemos
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
Esto se puede comparar con una forma general que es:
a*x^2 + 2*h*x*y + b*y^2 + 2*g*x + 2*f*y + c = 0
A continuación se muestra la implementación del enfoque anterior:
C++
// C++ program to find equation of an ellipse // using focus and directrix. #include <bits/stdc++.h> #include <iomanip> #include <iostream> #include <math.h> 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
<?php // PHP program to find equation of // an ellipse using focus and directrix. // Function to find equation of ellipse. function 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); $fixed; // echo setprecision(2); echo "Equation of ellipse is \n" , $a1, " x^2 + ", $b1 , " y^2 + ", $c1 , " x + " , $d1 , " y + ", $e1 , " xy + " , $f1 , " = 0"; } // Driver Code $x1 = 1; $y1 = 1; $a = 1; $b = -1; $c = 3; $e = 0.5 * 0.5; equation_ellipse($x1, $y1, $a, $b, $c, $e); // This code is contributed by jit_t ?>
Javascript
<script> // Javascript program to find equation // of an ellipse using focus and directrix. // Function to find equation of ellipse. function equation_ellipse(x1, y1, a, b, c, e) { var t = a * a + b * b; var a1 = t - e * (a * a); var b1 = t - e * (b * b); var c1 = (-2 * t * x1) - (2 * e * c * a); var d1 = (-2 * t * y1) - (2 * e * c * b); var e1 = -2 * e * a * b; var f1 = (-e * c * c) + (t * x1 * x1) + (t * y1 * y1); document.write("Equation of ellipse is " + "<br>"); document.write(a1+" x^2 + "+ b1 + " y^2 + "+ c1 + " x + " + d1 + " y + " + e1 + " xy + " + f1 + " = 0"); } // Driver Code var x1 = 1, y1 = 1, a = 1, b = -1, c = 3, e = 0.5 * 0.5; equation_ellipse(x1, y1, a, b, c, e); // This code is contributed by Khushboogoyal499 </script>
Equation of ellipse is 1.75 x^2 + 1.75 y^2 + -5.50 x + -2.50 y + 0.50 xy + 1.75 = 0
Complejidad de tiempo: O(1)
Espacio Auxiliar: O(1)
Publicación traducida automáticamente
Artículo escrito por Amber_Saxena y traducido por Barcelona Geeks. The original can be accessed here. Licence: CCBY-SA