test whether a 2D transformation matrix is orthogonal

Question

We have found that HP's printer drivers fail to handle PlgBlt()s properly for many of their printers.

Our intention is to handle any orthogonal rotations ourselves, and only have the printer handle scale and translation (which it seems to handle correctly).

I have a 2D matrix available to me at the point in code where I am about to "draw" the bitmap to the printer's DC.

I am weak as hell in math, and I understand just enough about matrix math to use them to transform 2D or 3D coordinates. But the underlying math is opaque to me.

So, what I need is to detect if a given 2D matrix is orthogonal in its transformation (the rotation aspect anyway). I suppose another way to ask this question would be: How do I get the rotation vector back out of the 2D matrix? If I knew the angle of rotation in radians or degrees, I could say whether its orthogonal or not (0,90,180,270).

Presumably, the code is generic on this subject, but below is the basics of the code we use, in case that helps:

typedef double ThreeByThreeMatrix[3][3];    // 3x3 for an X, Y coordinate transformation matrix

Then there's a wrapper for that which handles most obvious operations:

class Simple2DTransform
{
public:

    ////////////////////////////////////////////////////
    // Construction
    ////////////////////////////////////////////////////

    // we always begin life as an identity matrix (you can then apply scale, skew, etc.)
    Simple2DTransform() 
    {
        Reset(); 
    }

    ////////////////////////////////////////////////////
    // Operators
    ////////////////////////////////////////////////////

    bool operator == (const Simple2DTransform & rhs) const
    {
        return memcmp(m_matrix, rhs.m_matrix, sizeof(m_matrix)) == 0;
    }

    Simple2DTransform & operator *= (const Simple2DTransform & rhs)
    {
        return *this = GetCrossProduct(rhs);
    }

    ////////////////////////////////////////////////////
    // Setup
    ////////////////////////////////////////////////////

    // reset to the identity matrix
    Simple2DTransform & Reset()
    {
        memcpy(m_matrix, GetIdentityMatrix(), sizeof(m_matrix)); 
        return *this;
    }

    // combine with the specified translation
    Simple2DTransform & Translate(double x_shift, double y_shift)
    {
        Simple2DTransform transform;
        translate(x_shift, y_shift, transform.m_matrix);
        return *this *= transform;
    }

    // combine with the specified operations (these are cumulative operations, so rotating twice by 1 degree gives a total of 2 degrees rotation)
    Simple2DTransform & Rotate(double radians)
    {
        Simple2DTransform transform;
        rotate(radians, transform.m_matrix);
        return *this *= transform;
    }

    // apply a heterogeneous scale factor
    Simple2DTransform & Scale(double x_scale, double y_scale)
    {
        Simple2DTransform transform;
        scale(x_scale, y_scale, transform.m_matrix);
        return *this *= transform;
    }

    // apply a homogeneous scale factor
    Simple2DTransform & Scale(double scale) 
    {
        return Scale(scale, scale); 
    }

    // apply a skew
    Simple2DTransform & Skew(double skew)
    {
        Simple2DTransform transform;
        skew_y(skew, transform.m_matrix);
        return *this *= transform;
    }

    ////////////////////////////////////////////////////
    // Queries
    ////////////////////////////////////////////////////

    // return the cross product of this and the given matrix
    Simple2DTransform GetCrossProduct(const Simple2DTransform & rhs) const
    {
        Simple2DTransform result;
        GEMM(m_matrix, rhs.m_matrix, result.m_matrix);
        return result;
    }

    // returns the inverse of ourselves
    Simple2DTransform GetInverse() const
    {
        // note: invert mucks with both matrices, so we use a copy of ourselves for it
        Simple2DTransform original(*this), inverse;
        invert(original.m_matrix, inverse.m_matrix);
        return inverse;
    }

    // derivative values
    double GetCoefficient(int i, int j) const
    {
        return m_matrix[i][j];
    }

    // return the cross product
    double GetDeterminate() const
    {
        return m_matrix[0][0]*m_matrix[1][1] - m_matrix[0][1]*m_matrix[1][0];
    }

    // returns the square root of the determinate (this ignores heterogeneous scaling factors)
    double GetScale() const
    {
        return sqrt(abs(GetDeterminate()));
    }

    // returns true if there is a scale factor
    bool IsStretched() const
    {
        return (abs(abs(m_matrix[0][0]) - abs(m_matrix[1][1])) > 1.0e-7
             || abs(abs(m_matrix[0][1]) - abs(m_matrix[1][0])) > 1.0e-7);
    }

    // true if we're the identity matrix
    bool IsIdentity() const
    {
        return memcmp(m_matrix, GetIdentityMatrix(), sizeof(m_matrix)) == 0;
    }

    // returns true if this represents the same transformation as the given subtable
    bool IsSubtableEqual(const SUBTABLE * pSubTable) const
    {
        ASSERT(pSubTable);
        if (abs(pSubTable->tran1 - m_matrix[0][0]) > 1.0e-7)
            return false;
        if (abs(pSubTable->tran2 - m_matrix[1][0]) > 1.0e-7)
            return false;
        if (abs(pSubTable->tran3 - m_matrix[0][1]) > 1.0e-7)
            return false;
        if (abs(pSubTable->tran4 - m_matrix[1][1]) > 1.0e-7)
            return false;
        return true;
    }

    ////////////////////////////////////////////////////
    // Application / Execution
    ////////////////////////////////////////////////////

    void Transform(const SimplePoint & point, SimplePoint & newpoint) const
    {
        newpoint.x = point.x * m_matrix[0][0] + point.y * m_matrix[1][0] + m_matrix[2][0];
        newpoint.y = point.x * m_matrix[0][1] + point.y * m_matrix[1][1] + m_matrix[2][1];
    }

    void Transform(SimplePoint & point) const
    {
        SimplePoint newpoint;
        Transform(point, newpoint);
        point = newpoint;
    }

    void Transform(const SimpleRect & rect, SimpleRect & newrect) const
    {
        newrect.minX = rect.minX * m_matrix[0][0] + rect.minY * m_matrix[1][0] + m_matrix[2][0];
        newrect.minY = rect.minX * m_matrix[0][1] + rect.minY * m_matrix[1][1] + m_matrix[2][1];
        newrect.maxX = rect.maxX * m_matrix[0][0] + rect.maxY * m_matrix[1][0] + m_matrix[2][0];
        newrect.maxY = rect.maxX * m_matrix[0][1] + rect.maxY * m_matrix[1][1] + m_matrix[2][1];
    }

    void Transform(SimpleRect & rect) const
    {
        SimpleRect newrect;
        Transform(rect, newrect);
        rect = newrect;
    }

    void Transform(CPoint & point) const
    {
        SimplePoint newpoint(point);
        Transform(newpoint);
        point.x = (int)(newpoint.x > 0.0 ? newpoint.x + 0.5 : newpoint.x - 0.5);
        point.y = (int)(newpoint.y > 0.0 ? newpoint.y + 0.5 : newpoint.y - 0.5);
    }

    void Transform(SimplePoint point, CPoint & transformed) const
    {
        Transform(point);
        transformed.x = (int)(point.x > 0.0 ? point.x + 0.5 : point.x - 0.5);
        transformed.y = (int)(point.y > 0.0 ? point.y + 0.5 : point.y - 0.5);
    }

    void Transform(CPoint point, SimplePoint & transformed) const
    {
        transformed = point;
        Transform(transformed);
    }

    void Transform(double dx, double dy, double & x, double & y) const
    {
        SimplePoint point(dx, dy);
        Transform(point);
        x = point.x;
        y = point.y;
    }

    void Transform(double & x, double & y) const
    {
        SimplePoint point(x, y);
        Transform(point);
        x = point.x;
        y = point.y;
    }

    SimplePoint GetTransformed(SimplePoint point) const
    {
        Transform(point);
        return point;
    }

    CPoint GetTransformed(CPoint point) const
    {
        Transform(point);
        return point;
    }

    SimpleRect GetTransformed(const CRect & rect) const
    {
        return SimpleRect(GetTransformed(SimplePoint(rect.left, rect.bottom)), GetTransformed(SimplePoint(rect.right, rect.top)));
    }

    SimpleRect GetTransformed(double x1, double y1, double x2, double y2) const
    {
        return SimpleRect(GetTransformed(SimplePoint(x1, y1)), GetTransformed(SimplePoint(x2, y2)));
    }

    double GetTransformedX(double x, double y) const
    {
        return GetTransformed(SimplePoint(x, y)).x;
    }

    double GetTransformedY(double x, double y) const
    {
        return GetTransformed(SimplePoint(x, y)).y;
    }

    double GetTransformedX(int x, int y) const
    {
        return GetTransformed(SimplePoint(x, y)).x;
    }

    double GetTransformedY(int x, int y) const
    {
        return GetTransformed(SimplePoint(x, y)).y;
    }

    double GetTransformedX(const SimplePoint & point) const
    {
        return GetTransformed(point).x;
    }

    double GetTransformedY(const SimplePoint & point) const
    {
        return GetTransformed(point).y;
    }

    int GetTransformedIntX(double x, double y) const
    {
        CPoint point;
        Transform(SimplePoint(x, y), point);
        return point.x;
    }

    int GetTransformedIntY(double x, double y) const
    {
        CPoint point;
        Transform(SimplePoint(x, y), point);
        return point.y;
    }

    int GetTransformedIntX(const SimplePoint & point) const
    {
        CPoint pt;
        Transform(point, pt);
        return pt.x;
    }

    int GetTransformedIntY(const SimplePoint & point) const
    {
        CPoint pt;
        Transform(point, pt);
        return pt.y;
    }

protected:
    ////////////////////////////////////////////////////
    // Static Class Operations
    ////////////////////////////////////////////////////

    static const ThreeByThreeMatrix & GetIdentityMatrix();

    ////////////////////////////////////////////////////
    // Instance Variables
    ////////////////////////////////////////////////////

    ThreeByThreeMatrix  m_matrix;
};

Answer 1

Let A = (0, 0), B = (1, 0). Transform both through the matrix to get A' and B'. Measure the angle of the vector B' - A'. That should give you the angle.

To measure the angle of the vector, you can use atan2 (B'.y - A'.y, B'.x - A'.x)