square to trapezoid

Question

I know that transforming a square into a trapezoid is a linear transformation, and can be done using the projective matrix, but I\'m having a little trouble figuring out how

Answer 1

Java implementation with minimal dependencies

For those with limited knowledge and time looking for a quick and dirty solution there is a working and quite reliable Java implementation in the Wii-interact project.

The transormation is in the The Homography source file. It boils down to constructing and solving the matrix:

/**
* Please note that Dr. John Zelle assisted us in developing the code to
* handle the matrices involved in solving for the homography mapping.
* 
**/
Matrix A = new Matrix(new double[][]{
                {x1, y1, 1, 0,  0,  0, -xp1*x1, -xp1*y1},
                {0,  0,  0, x1, y1, 1, -yp1*x1, -yp1*y1},
                {x2, y2, 1, 0,  0,  0, -xp2*x2, -xp2*y2},
                {0,  0,  0, x2, y2, 1, -yp2*x2, -yp2*y2},
                {x3, y3, 1, 0,  0,  0, -xp3*x3, -xp3*y3},
                {0,  0,  0, x3, y3, 1, -yp3*x3, -yp3*y3},
                {x4, y4, 1, 0,  0,  0, -xp4*x4, -xp4*y4},
                {0,  0,  0, x4, y4, 1, -yp4*x4, -yp4*y4}
        });

        Matrix XP = new Matrix(new double[][]
                          {{xp1}, {yp1}, {xp2}, {yp2}, {xp3}, {yp3}, {xp4}, {yp4}});
        Matrix P = A.solve(XP);
        transformation = new Matrix(new double[][]{
                {P.get(0, 0), P.get(1, 0), P.get(2,0)},
                {P.get(3, 0), P.get(4, 0), P.get(5,0)},
                {P.get(6, 0), P.get(7, 0), 1}
        });

Usage: the following method does the final transformation:

public Point2D.Double transform(Point2D.Double point) {
    Matrix p = new Matrix(new double[][]{{point.getX()}, {point.getY()}, {1}});
    Matrix result = transformation.times(p);
    double z = result.get(2, 0);
    return new Point2D.Double(result.get(0, 0) / z, result.get(1, 0) / z);
}

The Matrix class dependency comes from JAMA: Java Matrix Package

License

1. Wii-interact GNU GPL v3
2. JAMA public domain