That's a paid service, thank you. In fact, I am looking for a matrix that
transforms one parallelogram into another, not necessarily the unit square. But
the price might get to high...

I am looking for a way to transform the unit square to a given
parallelogram using java.awt.geom.AffineTransform. Two methods
where suggested to me by John Bollinger in this group. One of
them is very succinct, but the other, more complex one, I do not
get right. Here is a class Parallelogram in which I implement
both the succesfull and the failing method. The comments in the code
elaborate on the attemps. Can someone review the code and
hopefully fix it? Thanks in advance.

-Martijn

//class Parallelogram
//Find the matrix that transforms the unit square (0,0), (0,1),
//(1,1), (1,0) to a predefined parallelogram. Two methods are
//given, getAffineTransform_1() and getAffineTransform_2(). As of
//this attempt, only the first method yields the desired result.
public class Parallelogram extends java.awt.Component
{

//data members point1..point4, vertices of parallelogram
java.awt.Point point1;
java.awt.Point point2;
java.awt.Point point3;
java.awt.Point point4;

//data members u1..u4, vertices of the unit square
java.awt.Point u1=new java.awt.Point(0,0);
java.awt.Point u2=new java.awt.Point(0,1);
java.awt.Point u3=new java.awt.Point(1,1);
java.awt.Point u4=new java.awt.Point(1,0);

//constructor
Parallelogram(int a,int b,int c,int d,int e,int f)
{
 point1=new java.awt.Point(a,b);
 point2=new java.awt.Point(c,d);
 point3=new java.awt.Point(e,f);
 point4=new java.awt.Point(e-c+a,f-d+b);
}

//getAffineTransform_1
//Very cool and very sure way to find the matrix that
//transforms the unit square to a given parallelogram.
//Works like a breeze. See function getAffineTransform_2
//for a hairier beast.
java.awt.geom.AffineTransform getAffineTransform_1()
{
 return new java.awt.geom.AffineTransform
 (
  point3.x-point2.x,
  point3.y-point2.y,
  point2.x-point1.x,
  point2.y-point1.y,
  point1.x,
  point1.y
 );
}

//getAffineTransform_2
//Cookbook approach to find the matrix that transforms
//the unit square to a given parellelogram, based on the
//work of John Bollinger. See his post earlier in this
//thread for an explanation.
//
//PROBLEM: IT DOES NOT WORK! WHERE DO I GO WRONG?
//
//My mathematical skills don't go anywhere near this, so
//I relied on 'cut-and-paste' (sort of) to write this
//method. There must be a typo somewhere, but I cannot
//find it. Please help.
java.awt.geom.AffineTransform getAffineTransform_2()
{
 int vxx = point4.x - point1.x;
 int vxy = point4.y - point1.y;
 int vyx = point2.x - point1.x;
 int vyy = point2.y - point1.y;
 double lx = Math.sqrt(vxx * vxx + vxy * vxy);
 double ly = Math.sqrt(vyx * vyx + vyy * vyy);
 double l = (vxx * vyx + vxy * vyy) / lx;
 double h = Math.sqrt(ly * ly - l * l);
 double theta = Math.atan2(vxy / lx, vxx / lx);
 java.awt.geom.AffineTransform at=new java.awt.geom.AffineTransform();
 at.translate(point1.x, point1.y);
 at.rotate(theta);
 at.shear(l / h, 0);
 at.scale(l, h);
 return at;
}

//paint with AffineTransform
public void paint
(
 java.awt.Graphics a,
 java.awt.geom.AffineTransform b
)
{

 //data members p1..p4, will hold the transformed unit square
 java.awt.Point p1=new java.awt.Point();
 java.awt.Point p2=new java.awt.Point();
 java.awt.Point p3=new java.awt.Point();
 java.awt.Point p4=new java.awt.Point();

 //transform the unit square using AffineTransform
 b.transform(u1,p1);
 b.transform(u2,p2);
 b.transform(u3,p3);
 b.transform(u4,p4);

 //paint the transformed unit square
 a.setColor(java.awt.Color.red);
 a.drawLine(p1.x,p1.y,p2.x,p2.y);
 a.setColor(java.awt.Color.black);
 a.drawLine(p2.x,p2.y,p3.x,p3.y);
 a.setColor(java.awt.Color.green);
 a.drawLine(p3.x,p3.y,p4.x,p4.y);
 a.setColor(java.awt.Color.blue);
 a.drawLine(p4.x,p4.y,p1.x,p1.y);
}

//paint
public void paint(java.awt.Graphics a)
{

 //paint the original parallelogram
 a.setColor(java.awt.Color.red);
 a.drawLine(point1.x,point1.y,point2.x,point2.y);
 a.setColor(java.awt.Color.black);
 a.drawLine(point2.x,point2.y,point3.x,point3.y);
 a.setColor(java.awt.Color.green);
 a.drawLine(point3.x,point3.y,point4.x,point4.y);
 a.setColor(java.awt.Color.blue);
 a.drawLine(point4.x,point4.y,point1.x,point1.y);

 //paint with getAffineTransform_1(). This one works!
 paint(a,getAffineTransform_1());

 //paint with getAffineTransform_2(). Close....
 paint(a,getAffineTransform_2());
}

//main
static void main(String[]a)
{
 java.awt.Frame frame=new java.awt.Frame("Parallelogram");
 frame.add(new Parallelogram(13,21,55,34,144,89));
 frame.setSize(300,300);
 frame.show();
}
}