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|---|---|---|---|
| 1 | /** | ||
| 2 | * \file | ||
| 3 | * \brief 3x3 affine transformation matrix. | ||
| 4 | *//* | ||
| 5 | * Authors: | ||
| 6 | * Lauris Kaplinski <lauris@kaplinski.com> (Original NRAffine definition and related macros) | ||
| 7 | * Nathan Hurst <njh@mail.csse.monash.edu.au> (Geom::Affine class version of the above) | ||
| 8 | * Michael G. Sloan <mgsloan@gmail.com> (reorganization and additions) | ||
| 9 | * Krzysztof Kosiński <tweenk.pl@gmail.com> (removal of boilerplate, docs) | ||
| 10 | * | ||
| 11 | * This code is in public domain. | ||
| 12 | */ | ||
| 13 | |||
| 14 | #ifndef LIB2GEOM_SEEN_AFFINE_H | ||
| 15 | #define LIB2GEOM_SEEN_AFFINE_H | ||
| 16 | |||
| 17 | #include <boost/operators.hpp> | ||
| 18 | #include <2geom/forward.h> | ||
| 19 | #include <2geom/point.h> | ||
| 20 | #include <2geom/utils.h> | ||
| 21 | |||
| 22 | namespace Geom { | ||
| 23 | |||
| 24 | /** | ||
| 25 | * @brief 3x3 matrix representing an affine transformation. | ||
| 26 | * | ||
| 27 | * Affine transformations on elements of a vector space are transformations which can be | ||
| 28 | * expressed in terms of matrix multiplication followed by addition | ||
| 29 | * (\f$x \mapsto A x + b\f$). They can be thought of as generalizations of linear functions | ||
| 30 | * (\f$y = a x + b\f$) to vector spaces. Affine transformations of points on a 2D plane preserve | ||
| 31 | * the following properties: | ||
| 32 | * | ||
| 33 | * - Colinearity: if three points lie on the same line, they will still be colinear after | ||
| 34 | * an affine transformation. | ||
| 35 | * - Ratios of distances between points on the same line are preserved | ||
| 36 | * - Parallel lines remain parallel. | ||
| 37 | * | ||
| 38 | * All affine transformations on 2D points can be written as combinations of scaling, rotation, | ||
| 39 | * shearing and translation. They can be represented as a combination of a vector and a 2x2 matrix, | ||
| 40 | * but this form is inconvenient to work with. A better solution is to represent all affine | ||
| 41 | * transformations on the 2D plane as 3x3 matrices, where the last column has fixed values. | ||
| 42 | * \f[ A = \left[ \begin{array}{ccc} | ||
| 43 | c_0 & c_1 & 0 \\ | ||
| 44 | c_2 & c_3 & 0 \\ | ||
| 45 | c_4 & c_5 & 1 \end{array} \right]\f] | ||
| 46 | * | ||
| 47 | * We then interpret points as row vectors of the form \f$[p_X, p_Y, 1]\f$. Applying a | ||
| 48 | * transformation to a point can be written as right-multiplication by a 3x3 matrix | ||
| 49 | * (\f$p' = pA\f$). This subset of matrices is closed under multiplication - combination | ||
| 50 | * of any two transforms can be expressed as the multiplication of their matrices. | ||
| 51 | * In this representation, the \f$c_4\f$ and \f$c_5\f$ coefficients represent | ||
| 52 | * the translation component of the transformation. | ||
| 53 | * | ||
| 54 | * Matrices can be multiplied by other more specific transformations. When multiplying, | ||
| 55 | * the transformations are applied from left to right, so for example <code>m = a * b</code> | ||
| 56 | * means: @a m first transforms by a, then by b. | ||
| 57 | * | ||
| 58 | * @ingroup Transforms | ||
| 59 | */ | ||
| 60 | class Affine | ||
| 61 | : boost::equality_comparable< Affine // generates operator!= from operator== | ||
| 62 | , boost::multipliable1< Affine | ||
| 63 | , MultipliableNoncommutative< Affine, Translate | ||
| 64 | , MultipliableNoncommutative< Affine, Scale | ||
| 65 | , MultipliableNoncommutative< Affine, Rotate | ||
| 66 | , MultipliableNoncommutative< Affine, HShear | ||
| 67 | , MultipliableNoncommutative< Affine, VShear | ||
| 68 | , MultipliableNoncommutative< Affine, Zoom | ||
| 69 | > > > > > > > > | ||
| 70 | { | ||
| 71 | Coord _c[6]; | ||
| 72 | public: | ||
| 73 | 1238 | Affine() { | |
| 74 | 1238 | _c[0] = _c[3] = 1; | |
| 75 | 1238 | _c[1] = _c[2] = _c[4] = _c[5] = 0; | |
| 76 | 1238 | } | |
| 77 | |||
| 78 | /** @brief Create a matrix from its coefficient values. | ||
| 79 | * It's rather inconvenient to directly create matrices in this way. Use transform classes | ||
| 80 | * if your transformation has a geometric interpretation. | ||
| 81 | * @see Translate | ||
| 82 | * @see Scale | ||
| 83 | * @see Rotate | ||
| 84 | * @see HShear | ||
| 85 | * @see VShear | ||
| 86 | * @see Zoom */ | ||
| 87 | 9000 | Affine(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4, Coord c5) { | |
| 88 | 9000 | _c[0] = c0; _c[1] = c1; | |
| 89 | 9000 | _c[2] = c2; _c[3] = c3; | |
| 90 | 9000 | _c[4] = c4; _c[5] = c5; | |
| 91 | 9000 | } | |
| 92 | |||
| 93 | /** @brief Access a coefficient by its index. */ | ||
| 94 | 16484712 | inline Coord operator[](unsigned i) const { return _c[i]; } | |
| 95 | 6003 | inline Coord &operator[](unsigned i) { return _c[i]; } | |
| 96 | |||
| 97 | /// @name Combine with other transformations | ||
| 98 | /// @{ | ||
| 99 | Affine &operator*=(Affine const &m); | ||
| 100 | // implemented in transforms.cpp | ||
| 101 | Affine &operator*=(Translate const &t); | ||
| 102 | Affine &operator*=(Scale const &s); | ||
| 103 | Affine &operator*=(Rotate const &r); | ||
| 104 | Affine &operator*=(HShear const &h); | ||
| 105 | Affine &operator*=(VShear const &v); | ||
| 106 | Affine &operator*=(Zoom const &); | ||
| 107 | /// @} | ||
| 108 | |||
| 109 | bool operator==(Affine const &o) const { | ||
| 110 | for(unsigned i = 0; i < 6; ++i) { | ||
| 111 | if ( _c[i] != o._c[i] ) return false; | ||
| 112 | } | ||
| 113 | return true; | ||
| 114 | } | ||
| 115 | |||
| 116 | /// @name Get the parameters of the matrix's transform | ||
| 117 | /// @{ | ||
| 118 | Point xAxis() const; | ||
| 119 | Point yAxis() const; | ||
| 120 | Point translation() const; | ||
| 121 | Coord expansionX() const; | ||
| 122 | Coord expansionY() const; | ||
| 123 | Point expansion() const { return Point(expansionX(), expansionY()); } | ||
| 124 | /// @} | ||
| 125 | |||
| 126 | /// @name Modify the matrix | ||
| 127 | /// @{ | ||
| 128 | void setXAxis(Point const &vec); | ||
| 129 | void setYAxis(Point const &vec); | ||
| 130 | |||
| 131 | void setTranslation(Point const &loc); | ||
| 132 | |||
| 133 | void setExpansionX(Coord val); | ||
| 134 | void setExpansionY(Coord val); | ||
| 135 | void setIdentity(); | ||
| 136 | /// @} | ||
| 137 | |||
| 138 | /// @name Inspect the matrix's transform | ||
| 139 | /// @{ | ||
| 140 | bool isIdentity(Coord eps = EPSILON) const; | ||
| 141 | |||
| 142 | bool isTranslation(Coord eps = EPSILON) const; | ||
| 143 | bool isScale(Coord eps = EPSILON) const; | ||
| 144 | bool isUniformScale(Coord eps = EPSILON) const; | ||
| 145 | bool isRotation(Coord eps = EPSILON) const; | ||
| 146 | bool isHShear(Coord eps = EPSILON) const; | ||
| 147 | bool isVShear(Coord eps = EPSILON) const; | ||
| 148 | |||
| 149 | bool isNonzeroTranslation(Coord eps = EPSILON) const; | ||
| 150 | bool isNonzeroScale(Coord eps = EPSILON) const; | ||
| 151 | bool isNonzeroUniformScale(Coord eps = EPSILON) const; | ||
| 152 | bool isNonzeroRotation(Coord eps = EPSILON) const; | ||
| 153 | bool isNonzeroNonpureRotation(Coord eps = EPSILON) const; | ||
| 154 | Point rotationCenter() const; | ||
| 155 | bool isNonzeroHShear(Coord eps = EPSILON) const; | ||
| 156 | bool isNonzeroVShear(Coord eps = EPSILON) const; | ||
| 157 | |||
| 158 | bool isZoom(Coord eps = EPSILON) const; | ||
| 159 | bool preservesArea(Coord eps = EPSILON) const; | ||
| 160 | bool preservesAngles(Coord eps = EPSILON) const; | ||
| 161 | bool preservesDistances(Coord eps = EPSILON) const; | ||
| 162 | bool flips() const; | ||
| 163 | |||
| 164 | bool isSingular(Coord eps = EPSILON) const; | ||
| 165 | /// @} | ||
| 166 | |||
| 167 | /// @name Compute other matrices | ||
| 168 | /// @{ | ||
| 169 | 1211 | Affine withoutTranslation() const { | |
| 170 | 1211 | Affine ret(*this); | |
| 171 |
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1211 | ret.setTranslation(Point(0,0)); |
| 172 | 1211 | return ret; | |
| 173 | } | ||
| 174 | Affine inverse() const; | ||
| 175 | /// @} | ||
| 176 | |||
| 177 | /// @name Compute scalar values | ||
| 178 | /// @{ | ||
| 179 | Coord det() const; | ||
| 180 | Coord descrim2() const; | ||
| 181 | Coord descrim() const; | ||
| 182 | /// @} | ||
| 183 | inline static Affine identity(); | ||
| 184 | }; | ||
| 185 | |||
| 186 | /** @brief Print out the Affine (for debugging). | ||
| 187 | * @relates Affine */ | ||
| 188 | ✗ | inline std::ostream &operator<< (std::ostream &out_file, const Geom::Affine &m) { | |
| 189 | ✗ | out_file << "A: " << m[0] << " C: " << m[2] << " E: " << m[4] << "\n"; | |
| 190 | ✗ | out_file << "B: " << m[1] << " D: " << m[3] << " F: " << m[5] << "\n"; | |
| 191 | ✗ | return out_file; | |
| 192 | } | ||
| 193 | |||
| 194 | // Affine factories | ||
| 195 | Affine from_basis(const Point &x_basis, const Point &y_basis, const Point &offset=Point(0,0)); | ||
| 196 | Affine elliptic_quadratic_form(Affine const &m); | ||
| 197 | |||
| 198 | /** Given a matrix (ignoring the translation) this returns the eigen | ||
| 199 | * values and vectors. */ | ||
| 200 | class Eigen{ | ||
| 201 | public: | ||
| 202 | Point vectors[2]; | ||
| 203 | double values[2]; | ||
| 204 | Eigen(Affine const &m); | ||
| 205 | Eigen(double M[2][2]); | ||
| 206 | }; | ||
| 207 | |||
| 208 | /** @brief Create an identity matrix. | ||
| 209 | * This is a convenience function identical to Affine::identity(). */ | ||
| 210 | inline Affine identity() { | ||
| 211 | Affine ret(Affine::identity()); | ||
| 212 | return ret; // allow NRVO | ||
| 213 | } | ||
| 214 | |||
| 215 | /** @brief Create an identity matrix. | ||
| 216 | * @return The matrix | ||
| 217 | * \f$\left[\begin{array}{ccc} | ||
| 218 | 1 & 0 & 0 \\ | ||
| 219 | 0 & 1 & 0 \\ | ||
| 220 | 0 & 0 & 1 \end{array}\right]\f$. | ||
| 221 | * @relates Affine */ | ||
| 222 | inline Affine Affine::identity() { | ||
| 223 | Affine ret(1.0, 0.0, | ||
| 224 | 0.0, 1.0, | ||
| 225 | 0.0, 0.0); | ||
| 226 | return ret; // allow NRVO | ||
| 227 | } | ||
| 228 | |||
| 229 | bool are_near(Affine const &a1, Affine const &a2, Coord eps=EPSILON); | ||
| 230 | |||
| 231 | } // end namespace Geom | ||
| 232 | |||
| 233 | #endif // LIB2GEOM_SEEN_AFFINE_H | ||
| 234 | |||
| 235 | /* | ||
| 236 | Local Variables: | ||
| 237 | mode:c++ | ||
| 238 | c-file-style:"stroustrup" | ||
| 239 | c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +)) | ||
| 240 | indent-tabs-mode:nil | ||
| 241 | fill-column:99 | ||
| 242 | End: | ||
| 243 | */ | ||
| 244 | // vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 : | ||
| 245 |