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Isotropy

What is isotropy?

In computational geometry things are often symmetric and invariant to direction and orientation.  This invariance to direction is called isotropy.  In such situations it is convenient to parameterize direction or orientation and write code that is invariant to the direction or orientation in which it is applied.  To do this effectively we provide an internally consistent set of isotropic data types to represent program data that describes orientations and directions.  These data types are:

• direction_1d - has one of the following 2 states: LOW, HIGH
• orientation_2d - has one of the following 2 states: HORIZONTAL, VERTICAL
• direction_2d - has one of the following 4 states: WEST, EAST, SOUTH, NORTH
• orientation_3d - has one of the following 3 states: HORIZONTAL, VERTICAL, PROXIMAL
• direction_3d - has one of the following 6 states: WEST, EAST, SOUTH, NORTH, DOWN, UP

The isotropic types create a system and interact with each other in various ways, such as casting.  Together they create a language for describing isotropic situations programmatically.  For instance, to get the positive direction_2d from an orientation_2d you would call a member function of orientation_2d and pass a direction_1d:

orientation_2d orient = HORIZONTAL;
direction_2d dir = orient.get_direction(direction_1d(HIGH));
assert(dir == EAST);

The motivation for providing isotropic data types is to encourage programming at a higher level of abstraction where program behavior is controlled by program data passed into function calls rather than flow control syntax.  Isotropic programming style is particularly applicable to working with points, intervals and rectangles.  Often times the implementation of such logic is identical when implemented for the x or y coordinates, except that the names of functions and data members are changed in a mechanical way leading to code duplication and bloat that results in copy-paste programming errors and maintenance problems where changes made to a given code block relating to x coordiantes are not duplicated to the code block that refers to y.  Isotropy therefore represents an opportunity to refactor and improve the quality of low level geometry code especially in regard to inter-relating coordinates, points, intervals and rectangles.

direction_1d

The direction_1d data type has two possible states.  These are the positive and negative directions on a continuum such as the number line.   These states can be described by one of several direction_1d_enum values:  We make clockwise and counterclockwise winding orientation of polygons a direction 1d value instead of providing a separate winding_orientation data type.  This is because winding orientation can be thought of as positive and negative directions in a 1d (although cyclic) space.  We assign counterclockwise to be the positive direction of travel in the 1d cyclic space to conform with the mathematical convention frequently described as the "right hand rule" which assigns positive normal value to counterclockwise and negative normal value to clockwise as well as the common convention that counterclockwise polygon winding corresponds to positive polygonal regions where as clockwise polygon winding corresponds to hole (negative) polygonal regions.

enum direction_1d_enum {LOW = 0, HIGH = 1,
                        LEFT = 0, RIGHT = 1,
                        CLOCKWISE = 0, COUNTERCLOCKWISE = 1,
                        REVERSE = 0, FORWARD = 1,
                        NEGATIVE = 0, POSITIVE = 1 };

Member Functions

orientation_2d

The orientation_2d data type has two possible states.  These are the horizontal and vertical axis of a 2d Cartesian coordinate system.   These states can be described by one of the two orientation_2d_enum values:

enum orientation_2d_enum { HORIZONTAL = 0, VERTICAL = 1 };

Member Functions

direction_2d

The direction_2d data type has four possible states.  These are the cardinal directions of the 2D Cartesian coordinate system.   These states can be described by one of several direction_2d_enum values:

enum direction_2d_enum { WEST = 0, EAST = 1, SOUTH = 2, NORTH = 3 };

Member Functions

orientation_3d

The orientation_3d data type has three possible states.  These are the horizontal, vertical and proximal (x, y, z) axis of a 3d Cartesian coordinate system.   These states can be described by one of the orientation_2d_enum values or by the orientation_3d_enum value:

enum orientation_3d_enum { PROXIMAL = 2 };

Member Functions

direction_3d

The direction_3d data type has six possible states.  These are the cardinal directions of the 3D Cartesian coordinate system.   These states can be described by one of the direction_2d_enum values or the direction_3d_enum values:

enum direction_3d_enum { DOWN = 4, UP = 5 };

Member Functions