# Parma_Polyhedra_Library::Generator Class Reference

A line, ray, point or closure point. More...

List of all members.

## Public Types

enum | Type { LINE, RAY, POINT, CLOSURE_POINT }
| The generator type. More...

## Public Member Functions

| Generator (const Generator &g)
| Ordinary copy-constructor.
| ~Generator ()
| Destructor.
Generator & | operator= (const Generator &g)
| Assignment operator.
dimension_type | space_dimension () const
| Returns the dimension of the vector space enclosing `*this`.
Type | type () const
| Returns the generator type of `*this`.
bool | is_line () const
| Returns `true` if and only if `*this` is a line.
bool | is_ray () const
| Returns `true` if and only if `*this` is a ray.
bool | is_point () const
| Returns `true` if and only if `*this` is a point.
bool | is_closure_point () const
| Returns `true` if and only if `*this` is a closure point.
const Integer & | coefficient (Variable v) const
| Returns the coefficient of `v` in `*this`.
const Integer & | divisor () const
| If `*this` is either a point or a closure point, returns its divisor.
bool | OK () const
| Checks if all the invariants are satisfied.

## Static Public Member Functions

Generator | line (const LinExpression &e)
| Shorthand for Generator Generator::line(const LinExpression& e).
Generator | ray (const LinExpression &e)
| Shorthand for Generator Generator::ray(const LinExpression& e).
Generator | point (const LinExpression &e=LinExpression::zero(), const Integer &d=Integer_one())
| Shorthand for Generator Generator::point(const LinExpression& e, const Integer& d).
Generator | closure_point (const LinExpression &e=LinExpression::zero(), const Integer &d=Integer_one())
| Shorthand for Generator Generator::closure_point(const LinExpression& e, const Integer& d).
const Generator & | zero_dim_point ()
| Returns the origin of the zero-dimensional space .
const Generator & | zero_dim_closure_point ()
| Returns, as a closure point, the origin of the zero-dimensional space .

## Related Functions

(Note that these are not member functions.)
std::ostream & | operator<< (std::ostream &s, const Generator &g)
| Output operator.
void | swap (Parma_Polyhedra_Library::Generator &x, Parma_Polyhedra_Library::Generator &y)
| Specializes `std::swap`.

## Detailed Description

A line, ray, point or closure point.

An object of the class Generator is one of the following:

- a line ;

- a ray ;

- a point ;

- a closure point ;

where is the dimension of the space and, for points and closure points, is the divisor.

A note on terminology.
  As observed in Section Representations of Convex Polyhedra, there are cases when, in order to represent a polyhedron using the generator system , we need to include in the finite set even points of that are not vertices of . This situation is even more frequent when working with NNC polyhedra and it is the reason why we prefer to use the word `point' where other libraries use the word `vertex'.

How to build a generator.
  Each type of generator is built by applying the corresponding function (`line`, `ray`, `point` or `closure_point`) to a linear expression, representing a direction in the space; the space-dimension of the generator is defined as the space-dimension of the corresponding linear expression. Linear expressions used to define a generator should be homogeneous (any constant term will be simply ignored). When defining points and closure points, an optional Integer argument can be used as a common divisor for all the coefficients occurring in the provided linear expression; the default value for this argument is 1.

  In all the following examples it is assumed that variables `x`, `y` and `z` are defined as follows:

        Variable x(0);
        Variable y(1);
        Variable z(2);

Example 1
  The following code builds a line with direction and having space-dimension :

        Generator l = line(x - y - z);

  As mentioned above, the constant term of the linear expression is not relevant. Thus, the following code has the same effect:

        Generator l = line(x - y - z + 15);

  By definition, the origin of the space is not a line, so that the following code throws an exception:

        Generator l = line(0*x);

Example 2
  The following code builds a ray with the same direction as the line in Example 1:

        Generator r = ray(x - y - z);

  As is the case for lines, when specifying a ray the constant term of the linear expression is not relevant; also, an exception is thrown when trying to build a ray from the origin of the space.

Example 3
  The following code builds the point :

        Generator p = point(1*x + 0*y + 2*z);

  The same effect can be obtained by using the following code:

        Generator p = point(x + 2*z);

  Similarly, the origin can be defined using either one of the following lines of code:

        Generator origin3 = point(0*x + 0*y + 0*z);
        Generator origin3_alt = point(0*z);

  Note however that the following code would have defined a different point, namely :

        Generator origin2 = point(0*y);

  The following two lines of code both define the only point having space-dimension zero, namely . In the second case we exploit the fact that the first argument of the function `point` is optional.

        Generator origin0 = Generator::zero_dim_point();
        Generator origin0_alt = point();

Example 4
  The point specified in Example 3 above can also be obtained with the following code, where we provide a non-default value for the second argument of the function `point` (the divisor):

        Generator p = point(2*x + 0*y + 4*z, 2);

  Obviously, the divisor can be usefully exploited to specify points having some non-integer (but rational) coordinates. For instance, the point can be specified by the following code:

        Generator q = point(-15*x + 32*y + 21*z, 10);

  If a zero divisor is provided, an exception is thrown.

Example 5
  Closures points are specified in the same way we defined points, but invoking their specific constructor function. For instance, the closure point is defined by

        Generator c = closure_point(1*x + 0*y + 2*z);

  For the particular case of the (only) closure point having space-dimension zero, we can use any of the following:

        Generator closure_origin0 = Generator::zero_dim_closure_point();
        Generator closure_origin0_alt = closure_point();

How to inspect a generator
  Several methods are provided to examine a generator and extract all the encoded information: its space-dimension, its type and the value of its integer coefficients.

Example 6
  The following code shows how it is possible to access each single coefficient of a generator. If `g1` is a point having coordinates , we construct the closure point `g2` having coordinates .

        if (g1.is_point()) {
          cout << "Point g1: " << g1 << endl;
          LinExpression e;
          for (int i = g1.space_dimension() - 1; i >= 0; i--)
            e += (i + 1) * g1.coefficient(Variable(i)) * Variable(i);
          Generator g2 = closure_point(e, g1.divisor());
          cout << "Closure point g2: " << g2 << endl;
        }
        else
          cout << "Generator g1 is not a point." << endl;

  Therefore, for the point

        Generator g1 = point(2*x - y + 3*z, 2);

  we would obtain the following output:

        Point g1: p((2*A - B + 3*C)/2)
        Closure point g2: cp((2*A - 2*B + 9*C)/2)

  When working with (closure) points, be careful not to confuse the notion of coefficient with the notion of coordinate: these are equivalent only when the divisor of the (closure) point is 1.

## Member Enumeration Documentation

enum Parma_Polyhedra_Library::Generator::Type

The generator type.

Enumeration values:

  LINE | The generator is a line.
  RAY | The generator is a ray.
  POINT | The generator is a point.
  CLOSURE_POINT | The generator is a closure point.

## Member Function Documentation

Generator line | ( | const LinExpression & | e | ) | `[static]`

Shorthand for Generator Generator::line(const LinExpression& e).

Exceptions:

  | std::invalid_argument | Thrown if the homogeneous part of `e` represents the origin of the vector space.

Generator ray | ( | const LinExpression & | e | ) | `[static]`

Shorthand for Generator Generator::ray(const LinExpression& e).

Exceptions:

  | std::invalid_argument | Thrown if the homogeneous part of `e` represents the origin of the vector space.

Generator point | ( | const LinExpression & | e = `LinExpression::zero()`,
|  | const Integer & | d = `Integer_one()`
| ) | `[static]`

Shorthand for Generator Generator::point(const LinExpression& e, const Integer& d).

Both `e` and `d` are optional arguments, with default values LinExpression::zero() and Integer_one(), respectively.

Exceptions:

  | std::invalid_argument | Thrown if `d` is zero.

Generator closure_point | ( | const LinExpression & | e = `LinExpression::zero()`,
|  | const Integer & | d = `Integer_one()`
| ) | `[static]`

Shorthand for Generator Generator::closure_point(const LinExpression& e, const Integer& d).

Both `e` and `d` are optional arguments, with default values LinExpression::zero() and Integer_one(), respectively.

Exceptions:

  | std::invalid_argument | Thrown if `d` is zero.

const Integer& Parma_Polyhedra_Library::Generator::coefficient | ( | Variable | v | ) | const

Returns the coefficient of `v` in `*this`.

Exceptions:

  | std::invalid_argument | Thrown if the index of `v` is greater than or equal to the space-dimension of `*this`.

const Integer& Parma_Polyhedra_Library::Generator::divisor | ( |  | ) | const

If `*this` is either a point or a closure point, returns its divisor.

Exceptions:

  | std::invalid_argument | Thrown if `*this` is neither a point nor a closure point.