affine_inverse.h

// Description: Inverse affine transformation
// Documentation: affine_algorithms.txt

#ifndef PASTELMATH_AFFINE_INVERSE_H
#define PASTELMATH_AFFINE_INVERSE_H

#include "pastel/math/affine/affine_transformation.h"

namespace Pastel
{

    //! Returns the inverse of an affine transformation.
    template <typename Real, int M, int N>
    AffineTransformation<Real, M, N> inverse(
        const AffineTransformation<Real, M, N>& that)
    {
        // (forall x)
        //
        // C(Ax + b) + d = x
        // <=>
        // CAx + (Cb + d) - x = 0
        // <=>
        // (CA - I)x + (Cb + d) = 0
        // <=>
        // CA - I = 0 and Cb + d = 0
        // <=>
        // C = A^{-1} and d = -A^{-1}b

        // Compute C = A^{-1}.
        Matrix<Real, M, N> inverseMatrix(
            inverse(that.matrix()));

        // Compute d = -A^{-1}b.
        Vector<Real, N> inverseTranslation =
            (-inverseMatrix).eval() * that.translation();

        AffineTransformation<Real, M, N> result(
            std::move(inverseMatrix),
            std::move(inverseTranslation));

        return result;
    }

    //! Returns the inverse of an orthogonal affine transformation.
    template <typename Real, int M, int N>
    AffineTransformation<Real, M, N> orthogonalInverse(
        const AffineTransformation<Real, M, N>& that)
    {
        // C = A^{-1} = A^T and
        // d = -A^{-1}b = -A^T b

        AffineTransformation<Real, M, N> result(
            transpose(that.matrix()),
            -transpose(that.matrix()) * that.translation());

        return result;
    }

}

#endif