Kernel Representations

Our object of study is the $$*d*-dimensional affine Euclidean space,
where $$*d* is a parameter of our geometry. Objects in that space are
sets of points. A common way to represent the points is the use of
Cartesian
coordinates, which assumes a reference
frame (an origin and $$*d* orthogonal axes). In that framework, a point
is represented by a $$*d*-tuple
$(c$_{0},c_{1},...,c_{d-1}),
and so are vectors in the underlying linear space. Each point is
represented uniquely by such
Cartesian
coordinates.

Another way to represent points is by homogeneous coordinates. In that
framework, a point is represented by a $$*(d+1)*-tuple
$(h$_{0},h_{1},...,h_{d}).
Via the formulae
$c$_{i}=h_{i}/h_{d},
the corresponding point with
Cartesian
coordinates
$(c$_{0},c_{1},...,c_{d-1})
can be computed. Note that homogeneous coordinates are not unique.
For $lambda\; !=\; 0$,
the tuples
$(h$_{0},h_{1}
,...,h_{d})
and
$(lambda\; h$_{0},lambda h_{1},...,lambda
h_{d}) represent the same point.
For a point with
Cartesian
coordinates
$(c$_{0},c_{1},...,c_{d-1}) a
possible homogeneous representation is
$(c$_{0},c_{1},...,c_{d-1},1).
Homogeneous
coordinates in fact allow to represent
objects in a more general space, the projective space
$P$_{d}.
In CGAL, we do not compute in projective geometry. Rather, we use
homogeneous coordinates to avoid division operations,
since the additional coordinate can serve as a common denominator.

CGAL offers two families of concrete models for the concept
representation class, one based on the
Cartesian
representation of points and one based on the homogeneous
representation of points. The interface of the kernel objects is
designed such that it works well with both
Cartesian
and homogeneous representation, for
example, points have a constructor with a range of coordinates plus a
common denominator (the $$*d+1* homogeneous coordinates of the point).
The common interfaces parameterized with a representation class allow
one to develop code independent of the chosen representation. We said
``families'' of models, because both families are parameterized too.
A user can choose the number type used to represent the coordinates
and the linear algebra module used to calculate the result of
predicates and constructions.

For reasons that will become evident later, a representation class
provides two typenames for number types,
namely *R::FT* and *R::RT* and one typename for the linear
algebra module *R::LA*. The type *R::FT* must fulfill the
requirements on what is called a *field type* in CGAL. This
roughly means that *R::FT* is a type for which operations $$*+*,
$$*-*, $$*** and $$*/* are defined with semantics (approximately)
corresponding to those of a field in a mathematical sense. Note that,
strictly speaking, the built-in type *int* does not fullfil the
requirements on a field type, since *int*s correspond to elements
of a ring rather than a field, especially operation $$*/* is not the
inverse of $$***. The requirements on the type *R::RT* are
weaker. This type must fulfill the requirements on what is called an
*Euclidean ring type* in CGAL. This roughly means that
*R::RT* is a type for which operations $$*+*, $$*-*, $$*** are
defined with semantics (approximately) corresponding to those of a
ring in a mathematical sense. A very limited division operation $$*/*
must be available as well. It must work for exact (i.e., no
remainder) integer divisions only. Furthermore, both number types
should fulfill CGAL's requirements on a number type.

When you choose
Cartesian
representation you have
to declare at least the type of the coordinates. A number type used
with the *Cartesian_d* representation class should be a *field
type* as described above. Both *Cartesian<FieldNumberType>::FT*
and *Cartesian<FieldNumberType>::RT* are mapped to number type
*FieldNumberType*.
*Cartesian_d<FieldNumberType,LinearAlgebra>::LA* is mapped to the
type *LinearAlgebra*. *Cartesian<FieldNumberType>* uses
reference counting internally to save copying costs.

The type *LinearAlgebra* must me a linear algebra module working
on numbers of type *RingNumberType*. Again the second parameter
defaults to module delivered with the kernel so for short one can just
write *Homogeneous_d<RingNumberType>* when replacing the default
is no issue.

However, some operations provided by this kernel involve division
operations, for example computing squared distances or returning a
Cartesian
coordinate. To keep the requirements on
the number type parameter of *Homogeneous* low, the number type
*Quotient<RingNumberType>* is used instead. This number type
turns a ring type into a field type. It maintains numbers as
quotients, i.e. a numerator and a denominator. Thereby, divisions are
circumvented. With *Homogeneous_d<RingNumberType>*,
*Homogeneous_d<RingNumberType>::FT* is equal to
*Quotient<RingNumberType>* while
*Homogeneous_d<RingNumberType>::RT* is equal to
*RingNumberType*.
*Homogeneous_d<RingNumberType,LinearAlgebra>::LA* is mapped to the
type *LinearAlgebra*.

The use of representation classes not only avoids problems, it also
makes all CGAL classes very uniform. They **always** consist of:

- The
*capitalized base name*of the geometric object, such as*Point*,*Segment*,*Triangle*. - Followed by
*_d*. - A
*representation class*as parameter, which itself is parameterized with a number type, such as*Cartesian_d<double>*or*Homogeneous_d<leda_integer>*.

Algorithms and data structures in the basic library of CGAL are parameterized by a traits class that subsumes the objects on which the algorithm or data structure operates as well as the operations to do so. For most of the algorithms and data structures in the basic library you can use a kernel as a traits class. For some algorithms you even do not have to specify the kernel; it is detected automatically using the types of the geometric objects passed to the algorithm. In some other cases, the algorithms or data structures needs more than is provided by a kernel. In these cases, a kernel can not be used as a traits class.

If you start with integral Cartesian coordinates, many geometric computations will involve integral numerical values only. Especially, this is true for geometric computations that evaluate only predicates, which are tantamount to determinant computations. Examples are triangulation of point sets and convex hull computation.

The dimension $$*d* of our affine space determines the dimension of the
matrix computions in the mathematical evaluation of predicates. As
rounding errors accumulate fast the homogeneous represenation used
with multi-precision integers is the kernel of choice for well-behaved
algorithms. Note, that unless you use an arbitrary precision integer
type, incorrect results might arise due to overflow.

If new points are to be constructed, for example the
intersection
point of two lines, computation of
Cartesian
coordinates usually involves divisions,
so you need to use a field type with
Cartesian
representation or have to switch to homogeneous representation.
*double* is a possible, but imprecise field type. You can also
put any ring type into *Quotient* to get a field type and put it
into *Cartesian*, but you better put the ring type into
*Homogeneous*. *leda_rational* and *leda_real* are valid
field types, too.

Still other people will prefer the built-in type `double`, because
they need speed and can live with approximate results, or even
algorithms that, from time to time, crash or compute incorrect results
due to accumulated rounding errors.

You need just to include a representation class to obtain the the
geometric objects of the kernel that you would like to use with the
representation class, i.e., *CGAL/Cartesian_d.h* or
*CGAL/Homogeneous_d.h*

Next chapter: Kernel Geometry

The CGAL Project .
Tue, December 21, 2004 .