/*  Bilinear Interpolation  (15 points)


MATH BACKGROUND (Linear Interpolation):
=======================================
	
      ^
f(x2) |..........*
      |         /.
      |        / .
p=f(x)|.......?
      |      /.  .
      |     / .  .
      |    /  .  .
f(x1) |...*   .  .
      |   .   .  .
      -------------------->
          x1  x  x2

Linear interpolation is a common method for estimating the value of a 
function at a point x for which no data is available, but data exists 
for points located on either side of x on the horizontal axis.
Given two points x1 and x2 and the value of the 
function evaluated at these points f(x1) and f(x2), using
linear interpolation one can estimate the value of the function
p=f(x) at another point x that lies between x1 and x2. 

The linear interpolation formula is as follows:
	
                (x - x1)             (x2 - x)
	p = f(x) = ---------- f(x2)  +  ---------- f(x1)
	            (x2 - x1)            (x2 - x1)
	

In other words, the estimation is performed by fitting a straight line
between the points (x1, f(x1)) and (x2, f(x2)) and using this line to 
estimate the values of the function in the interval (x1, x2).
Note that we don't even need to know the analytical form of the 
function; we only need to know its values at two points.


MATH BACKGROUND (Bilinear Interpolation):
=========================================


      ^
      |    Q12     R2   Q22
   y2 |...*.......o....*
      |   .       .    .
      |   .       .P   .
    y |...........?.....
      |   .       .    .
      |   .       .    .
      |   .Q11    .R1  .Q21
   y1 |...*.......o....*
      |   .       .    .
      |   .       .    .
      ---------------------->
          x1      x    x2


Bilinear interpolation is a natural extension of linear interpolation that 
can be used with functions of two variables, e.g., f(x,y).
Once again, we don't need to know the analytical form of the function.
We only need to know its value at four points that form a rectangle,
which contains the point of interest (x,y).

To find an estimate for the value of the function at the point (x,y), where
x1<x<x2, and y1<y<y2, we can perform linear interpolation twice 
in the x direction followed by a third linear interpolation in the y direction.

For example, in the image above, we can interpolate the function between 
the points Q11=f(x1,y1) and Q21=f(x2, y1) to find an approximation for 
its value at the point R1=(x,y1). We can also perform another linear 
interpolation between the points Q12=f(x1,y2) and Q22=f(x2,y2) to find 
an interpolated value of the function at the point R2=(x,y2). Finally, we 
can perform a third linear interpolation between R1 and R2
to find an estimate for the value of the function at the point P=(x,y).


PROBLEM DESCRIPTION:
====================

Write a complete C program that approximates and prints the value of an 
analytically unknown function at the point (x,y) using bilinear interpolation. 
The program must prompt the user to enter the following double numbers:

	* x1, x2, y1, and y2; 

	* Q11, Q12, Q21, Q22, which correspond to the value of 
	  the function at (x1,y1), (x1,y2), (x2,y1), and (x2,y2).

    * x,  y -- the coordinates of the point for at which we 
               would like to approximate the value of the function

	
SAMPLE RUN #1:
==============
	
Please enter x1:    12
Please enter x2:    13
Please enter y1:    18
Please enter y2:    19
Please enter x:     12.6
Please enter y:     18.7
Please enter Q11=f(x1,y1): 135
Please enter Q12=f(x1,y2): 85
Please enter Q21=f(x2,y1): 147
Please enter Q22=f(x2,y2): 215

The interpolated value of the function at R1=(x,y1) is 142.2
The interpolated value of the function at R2=(x,y2) is 163

The interpolated value of the function at (x=12.6, y=18.7) is equal to 156.76

SAMPLE RUN #2:
==============

Please enter x1:    10
Please enter x2:    20
Please enter y1:    40
Please enter y2:    50
Please enter x:     14
Please enter y:     48
Please enter Q11=f(x1,y1): 1.5
Please enter Q12=f(x1,y2): 3.3
Please enter Q21=f(x2,y1): 2.1
Please enter Q22=f(x2,y2): 4.5

The interpolated value of the function at R1=(x,y1) is 1.74
The interpolated value of the function at R2=(x,y2) is 3.78

The interpolated value of the function at (x=14, y=48) is equal to 3.372

*/ 

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

double linearInterpolation(double x1, double f_x1, double x2, double f_x2, double x)
{
	double result = (x - x1)/(x2-x1)*f_x2  + (x2-x)/(x2-x1)*f_x1;
	return result;
}

int main(void)
{
	double x1, x2;
    double y1, y2; 
    double x,  y;
	double Q11, Q12, Q21, Q22;
	
	printf("Please enter x1:    "); 	scanf("%lf", &x1);
	printf("Please enter x2:    "); 	scanf("%lf", &x2);
	printf("Please enter y1:    "); 	scanf("%lf", &y1);
	printf("Please enter y2:    "); 	scanf("%lf", &y2);
	  
	printf("Please enter x:     "); 	scanf("%lf", &x);
	printf("Please enter y:     "); 	scanf("%lf", &y);
	  
	printf("Please enter Q11=f(x1,y1): ");	scanf("%lf", &Q11);
	printf("Please enter Q12=f(x1,y2): ");	scanf("%lf", &Q12);
	printf("Please enter Q21=f(x2,y1): ");	scanf("%lf", &Q21);
	printf("Please enter Q22=f(x2,y2): ");	scanf("%lf", &Q22);

	double R1 =  linearInterpolation(x1, Q11, x2, Q21, x);
	double R2 =  linearInterpolation(x1, Q12, x2, Q22, x);

	printf("The nterpolated value of the function at R1=(x,y1) is %g\n", R1);
	printf("The nterpolated value of the function at R2=(x,y2) is %g\n", R2);

	double  P =  linearInterpolation(y1,  R1, y2,  R2, y);
	
    printf("\nThe interpolated value of the function at (x=%g, y=%g) is equal to %g\n\n", x, y, P);

    system("pause");
}