text: /*************************************************************************/
  /*  transform_2d.h                                                       */
  /*************************************************************************/
  /*                       This file is part of:                           */
  /*                           GODOT ENGINE                                */
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  /* Copyright (c) 2007-2019 Juan Linietsky, Ariel Manzur.                 */
  /* Copyright (c) 2014-2019 Godot Engine contributors (cf. AUTHORS.md).   */
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  #pragma once
  
  #include "core/math/rect2.h" // also includes vector2, math_funcs, and ustring
  
  struct GODOT_EXPORT Transform2D {
      // Warning #1: basis of Transform2D is stored differently from Basis. In terms of elements array, the basis matrix looks like "on paper":
      // M = (elements[0][0] elements[1][0])
      //     (elements[0][1] elements[1][1])
      // This is such that the columns, which can be interpreted as basis vectors of the coordinate system "painted" on the object, can be accessed as elements[i].
      // Note that this is the opposite of the indices in mathematical texts, meaning: $M_{12}$ in a math book corresponds to elements[1][0] here.
      // This requires additional care when working with explicit indices.
      // See https://en.wikipedia.org/wiki/Row-_and_column-major_order for further reading.
  
      // Warning #2: 2D be aware that unlike 3D code, 2D code uses a left-handed coordinate system: Y-axis points down,
      // and angle is measure from +X to +Y in a clockwise-fashion.
  
      Vector2 elements[3];
  
      _FORCE_INLINE_ real_t tdotx(const Vector2 &v) const { return elements[0][0] * v.x + elements[1][0] * v.y; }
      _FORCE_INLINE_ real_t tdoty(const Vector2 &v) const { return elements[0][1] * v.x + elements[1][1] * v.y; }
  
      const Vector2 &operator[](int p_idx) const { return elements[p_idx]; }
      Vector2 &operator[](int p_idx) { return elements[p_idx]; }
  
      _FORCE_INLINE_ Vector2 get_axis(int p_axis) const {
          ERR_FAIL_INDEX_V(p_axis, 3, Vector2());
          return elements[p_axis];
      }
      _FORCE_INLINE_ void set_axis(int p_axis, const Vector2 &p_vec) {
          ERR_FAIL_INDEX(p_axis, 3);
          elements[p_axis] = p_vec;
      }
  
      void invert();
      Transform2D inverse() const;
  
      void affine_invert();
      Transform2D affine_inverse() const;
  
      void set_rotation(real_t p_rot);
      real_t get_rotation() const;
      real_t get_skew() const;
      void set_skew(float p_angle);
      _FORCE_INLINE_ void set_rotation_and_scale(real_t p_rot, const Size2 &p_scale);
      _FORCE_INLINE_ void set_rotation_scale_and_skew(real_t p_rot, const Size2 &p_scale, float p_skew);
      void rotate(real_t p_phi);
  
      void scale(const Size2 &p_scale);
      void scale_basis(const Size2 &p_scale);
      void translate(real_t p_tx, real_t p_ty);
      void translate(const Vector2 &p_translation);
  
      real_t basis_determinant() const;
  
      Size2 get_scale() const;
      void set_scale(const Size2 &p_scale);
  
      _FORCE_INLINE_ const Vector2 &get_origin() const { return elements[2]; }
      _FORCE_INLINE_ void set_origin(const Vector2 &p_origin) { elements[2] = p_origin; }
  
      Transform2D scaled(const Size2 &p_scale) const;
      Transform2D basis_scaled(const Size2 &p_scale) const;
      Transform2D translated(const Vector2 &p_offset) const;
      Transform2D rotated(real_t p_phi) const;
  
      Transform2D untranslated() const;
  
      void orthonormalize();
      Transform2D orthonormalized() const;
      bool is_equal_approx(const Transform2D &p_transform) const;
  
      bool operator==(const Transform2D &p_transform) const;
      bool operator!=(const Transform2D &p_transform) const;
  
      void operator*=(const Transform2D &p_transform);
      Transform2D operator*(const Transform2D &p_transform) const;
  
      Transform2D interpolate_with(const Transform2D &p_transform, real_t p_c) const;
  
      _FORCE_INLINE_ Vector2 basis_xform(const Vector2 &p_vec) const;
      _FORCE_INLINE_ Vector2 basis_xform_inv(const Vector2 &p_vec) const;
      _FORCE_INLINE_ Vector2 xform(const Vector2 &p_vec) const;
      _FORCE_INLINE_ Vector2 xform_inv(const Vector2 &p_vec) const;
      _FORCE_INLINE_ Rect2 xform(const Rect2 &p_rect) const;
      _FORCE_INLINE_ Rect2 xform_inv(const Rect2 &p_rect) const;
      _FORCE_INLINE_ void xform(Vector2 *p_array,int size) const;
      _FORCE_INLINE_ void xform_inv(Vector2 *p_array,int size) const;
  
      operator String() const;
  
      Transform2D(real_t xx, real_t xy, real_t yx, real_t yy, real_t ox, real_t oy) {
  
          elements[0][0] = xx;
          elements[0][1] = xy;
          elements[1][0] = yx;
          elements[1][1] = yy;
          elements[2][0] = ox;
          elements[2][1] = oy;
      }
  
      Transform2D(real_t p_rot, const Vector2 &p_pos);
      Transform2D() {
          elements[0][0] = 1.0;
          elements[1][1] = 1.0;
      }
  };
  
  Vector2 Transform2D::basis_xform(const Vector2 &p_vec) const {
  
      return Vector2(
              tdotx(p_vec),
              tdoty(p_vec));
  }
  
  Vector2 Transform2D::basis_xform_inv(const Vector2 &p_vec) const {
  
      return Vector2(
              elements[0].dot(p_vec),
              elements[1].dot(p_vec));
  }
  
  Vector2 Transform2D::xform(const Vector2 &p_vec) const {
  
      return Vector2(
                     tdotx(p_vec),
                     tdoty(p_vec)) +
             elements[2];
  }
  Vector2 Transform2D::xform_inv(const Vector2 &p_vec) const {
  
      Vector2 v = p_vec - elements[2];
  
      return Vector2(
              elements[0].dot(v),
              elements[1].dot(v));
  }
  Rect2 Transform2D::xform(const Rect2 &p_rect) const {
  
      Vector2 x = elements[0] * p_rect.size.x;
      Vector2 y = elements[1] * p_rect.size.y;
      Vector2 pos = xform(p_rect.position);
  
      Rect2 new_rect;
      new_rect.position = pos;
      new_rect.expand_to(pos + x);
      new_rect.expand_to(pos + y);
      new_rect.expand_to(pos + x + y);
      return new_rect;
  }
  
  void Transform2D::set_rotation_and_scale(real_t p_rot, const Size2 &p_scale) {
  
      elements[0][0] = Math::cos(p_rot) * p_scale.x;
      elements[1][1] = Math::cos(p_rot) * p_scale.y;
      elements[1][0] = -Math::sin(p_rot) * p_scale.y;
      elements[0][1] = Math::sin(p_rot) * p_scale.x;
  }
  
  void Transform2D::set_rotation_scale_and_skew(real_t p_rot, const Size2 &p_scale, float p_skew) {
  
      elements[0][0] = Math::cos(p_rot) * p_scale.x;
      elements[1][1] = Math::cos(p_rot + p_skew) * p_scale.y;
      elements[1][0] = -Math::sin(p_rot + p_skew) * p_scale.y;
      elements[0][1] = Math::sin(p_rot) * p_scale.x;
  }
  
  Rect2 Transform2D::xform_inv(const Rect2 &p_rect) const {
  
      Vector2 ends[4] = {
          xform_inv(p_rect.position),
          xform_inv(Vector2(p_rect.position.x, p_rect.position.y + p_rect.size.y)),
          xform_inv(Vector2(p_rect.position.x + p_rect.size.x, p_rect.position.y + p_rect.size.y)),
          xform_inv(Vector2(p_rect.position.x + p_rect.size.x, p_rect.position.y))
      };
  
      Rect2 new_rect;
      new_rect.position = ends[0];
      new_rect.expand_to(ends[1]);
      new_rect.expand_to(ends[2]);
      new_rect.expand_to(ends[3]);
  
      return new_rect;
  }
  
  void Transform2D::xform(Vector2 *p_array,int size) const {
      for (int i = 0; i < size; ++i) {
          p_array[i] = xform(p_array[i]);
      }
  }
  
  void Transform2D::xform_inv(Vector2 *p_array,int size) const {
      for (int i = 0; i < size; ++i) {
          p_array[i] = xform_inv(p_array[i]);
      }
  }
  
  