13. vcsl: Co-ordinate systems

Chapter summary: Metric entities, co-ordinate systems and transformations.

This non-core Level 2 library vcsl is intended to provide an environment for representing co-ordinate systems, transformations, dimensions, and metric units.

This includes classes for

• Radians, degrees, and meters.
• 2D and 3D Cartesian co-ordinate systems
• Various polar co-ordinate systems
• Various geographic co-ordinate systems
• Various transformations between those

13.1 Contents

13.1.1 Metric and Dimensional classes

• vcsl_dimension - Abstract dimension
  • vcsl_length
  • vcsl_angle
  • vcsl_area
  • vcsl_volume
• vcsl_unit - Abstract unit associated to a dimension
  • vcsl_length_unit
    • vcsl_meter - This is the standard length unit
  • vcsl_angle_unit
    • vcsl_radian - This is the standard angular unit
    • vcsl_degree
  • vcsl_area_unit
    • vcsl_squaremeter - This is the standard area unit
  • vcsl_volume_unit
    • vcsl_cubicmeter - This is the standard volume unit
    • vcsl_liter

13.1.2 Co-ordinate Systems and Transformations

• vcsl_coordinate_system
  • vcsl_planar
    • vcsl_cartesian_2d
    • vcsl_polar
  • vcsl_spatial
    • vcsl_cartesian_3d
    • vcsl_cylindrical
    • vcsl_spherical
    • vcsl_geographic
      • vcsl_geocentric - Intended to represent a location relative to the earth
      • vcsl_geodetic - Intended to represent a location relative to the earth
      • vcsl_lambertian - Lambert Conformal Conic Projection
      • vcsl_utm - Universal Transverse Mercator projection
• vcsl_transformation
  • vcsl_planar_transformation
    • vcsl_rotation_2d - Rotation about an axis through the origin
      • vcsl_displacement_2d - Rotation about a general axis
    • vcsl_scale_2d
    • vcsl_translation_2d
    • vcsl_composition_2d - Composition of transformations
  • vcsl_spatial_transformation
    • vcsl_cylindrical_to_cartesian_3d
    • vcsl_rotation_3d - Rotation about an axis through the origin
      • vcsl_displacement_3d - Rotation about a general axis
    • vcsl_scale_3d
    • vcsl_translation_3d
    • vcsl_composition_3d - Composition of transformations
  • vcsl_3d_to_2d_transformation
    • vcsl_perspective
• vcsl_graph - Represent multiple co-ordinate systems and transformation between them.

13.2 Examples

The following example shows how to use and build a set of co-ordinate systems.

#include <vcsl/vcsl_cartesian_3d.h>
#include <vcsl/vcsl_rotation_3d.h>
#include <vcsl/vcsl_graph.h>
...
  //: Convenient indices for vectors representing points, etc.
  enum AXES { X, Y, Z, T };

  //: Graph of defined CS (Co-ordinate system)
  // All CS must be added to this graph as they are created.
  vcsl_graph_sptr graphCS = new vcsl_graph;

  //: World Coordinate System
  // Equivalent to the Top view, by definition.
  //
  // A CS for which no parent is defined is absolute.  Logically,
  // there must be one absolute CS which all other directly or
  // indirectly relate to.
  vcsl_spatial_sptr WCS = new vcsl_cartesian_3d;

  // Add WCS to the global map of coordinate systems.
  WCS->set_graph(graphCS);

  // New CS can be defined relative to any point (translation) or
  // axis (rotation) in the parent CS.  But it is so common to do
  // so relative to the X, Y, or Z axes, that it is convenient to
  // define the corresponding vectors:

  //: x-axis vector
  vnl_vector_fixed <double, 3> xA;
  xA[X]=1; xA[Y] = 0; xA[Z] = 0;

  //: y-axis vector
  vnl_vector_fixed <double, 3> yA;
  yA[X]=0; yA[Y] = 1; yA[Z] = 0;

  //: z-axis vector
  vnl_vector_fixed <double, 3> zA;
  zA[X]=0; zA[Y] = 0; zA[Z] = 1;

  //: 90 degree rotation about WCS y-axis
  // Transforms from WCS to right CS
  vcsl_rotation_3d rightXF;
  rightXF.set_static(vnl_math::pi_over_2, yA);

  //: WCS rotated 90 degrees about the y-axis to produce right hand view/CS
  vcsl_spatial_sptr right = new vcsl_cartesian_3d;
  right->set_graph(graphCS);
  right->set_unique(WCS, &rightXF);

  //: Corner of a box with opposite corner at origin of WCS.
  vnl_vector_fixed <double, 3> corner;
  corner[X] = 1; corner[Y] = 2; corner[Z] = 3;

  // By inspection, corner should be (-3,2,1) in 'right' CS
  vnl_vector<double> cornerXF = WCS->from_local_to_cs(corner, right, 0);

  vcl_cout << cornerXF.x() << ", " << cornerXF.y() << ", " << cornerXF.z() << '\n';

13.3 Further Work

1. Write conversion between classic coordinate systems in the same style as the class vcsl_cylindrical_to_cartesian_3d (singleton pattern)
   • vcsl_spherical_to_cartesian_3d
   • vcsl_polar_to_cartesian_2d
   • vcsl_geocentric_to_cartesian_3d
   • vcsl_geodetic_to_cartesian_3d
   • vcsl_lambertian_to_cartesian_3d
   • vcsl_utm_to_cartesian_3d
2. Add other units and dimensions
3. In vcsl_transformation, add

   //: May `this' have a matrix representation ? virtual bool is_linear(void) const=0; //: Homogeneous matrix of `this' at time `time' // REQUIRE: is_valid() // REQUIRE: is_linear() virtual vnl_matrix<double> matrix_value(const double time, bool type) const;

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