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An object of class Approximate_min_ellipsoid_d<Traits> is an approximation to the
ellipsoid of smallest volume enclosing a finite multiset of points
in d-dimensional Euclidean space d, d ≥ 2.
An ellipsoid in d is a Cartesian pointset of the form {
x ∈
d | xT E x + xT e + η ≤ 0 }, where E is some
positive definite matrix from the set ℝd × d, e is some
real d-vector, and η ∈ ℝ. A pointset P ⊆
d is
called full-dimensional if its affine hull has dimension d.
For a finite, full-dimensional pointset P we denote by (P) the
smallest ellipsoid that contains all points of P; this ellipsoid
exists and is unique.
For a given finite and full-dimensional pointset P ⊂ d and a
real number ε ≥ 0, we say that an ellipsoid E ⊂
d is an (1+ε)-appoximation to (P) if
P ⊂ E and (E) ≤ (1+ε)
((P)). In other words, an (1+ε)-approximation to
(P) is an enclosing ellipsoid whose volume is by at most a
factor of 1+ε larger than the volume of the smallest
enclosing ellipsoid of P.
Given this notation, an object of class Approximate_min_ellipsoid_d<Traits> represents an
(1+ε)-approximation to (P) for a given finite and
full-dimensional multiset of points P ⊂ d and a real constant
ε>0.1 When an
Approximate_min_ellipsoid_d<Traits> object is constructed, an
iterator over the points P and the number ε have to be
specified; the number ε defines the desired
approximation ratio 1+ε. The underlying algorithm will then
try to compute an (1+ε)-approximation to (P), and one of
the following two cases takes place.
Important note: due to rounding errors, the algorithm cannot
in all cases decide correctly whether P is full-dimensional or
not. If is_full_dimensional() returns false, the points
lie in such a ``thin'' subspace of d that the algorithm is
incapable of computing an approximation to (P). More
precisely, if is_full_dimensional() returns false, there
exist two parallel hyperplanes in
d with the points P in
between so that the distance δ between the hyperplanes is
very small, possible zero. (If δ=0 then P is not
full-dimensional.)
If P is not full-dimensional, linear algebra techniques should be
used to determine an affine subspace S of d that contains the
points P as a (w.r.t. S) full-dimensional pointset; once S is
determined, the algorithm can be invoked again to compute an
approximation to (the lower-dimensional) (P) in S. Since
is_full_dimensional() might (due to rounding errors, see
above) return false even though P is full-dimensional, the
lower-dimensional subspace S containing P need not exist.
Therefore, it might be more advisable to fit a hyperplane H
through the pointset P, project P onto this affine subspace H,
and compute an approximation to the minimum-volume enclosing
ellipsoid of the projected points within H; the fitting can be
done for instance using the linear_least_squares_fitting()
function from the Cgal package Principal_component_analysis.
The ellipsoid E computed by the algorithm satisfies the inclusions
| E ⊆ (P) ⊆ E |
The underlying algorithm can cope with all kinds of inputs (multisets P, ε ∈ [0,∞)) and terminates in all cases. There is, however, no guarantee that any desired approximation ratio is actually achieved; the performance of the algorithm in this respect highly depends on the input pointset. Values of at least 0.01 for ε are usually handled without problems.
Internally, the algorithm represents the input points' Cartesian coordinates as double's. For this conversion to work, the input point coordinates must be convertible to double. Also, in order to compute the achieved epsilon ε' mentioned above, the algorithm requires a number type ET that provides exact arithmetic. (Both these aspects are discussed in the documentation of the concept ApproximateMinEllipsoid_d_Traits_d.)
#include <CGAL/Approximate_min_ellipsoid_d.h>
The template parameter Traits is a model for ApproximateMinEllipsoid_d_Traits_d.
We provide the model CGAL::Approximate_min_ellipsoid_d_traits_d<K> using the d-dimensional Cgal kernel; the models CGAL::Approximate_min_ellipsoid_d_traits_2<K> and CGAL::Approximate_min_ellipsoid_d_traits_3<K> are for use with the 2- and 3-dimensional Cgal kernel, respectively.
Approximate_min_ellipsoid_d<Traits>::FT | |
typedef Traits::FT FT (which is always a
typedef to double).
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Approximate_min_ellipsoid_d<Traits>::ET | |
typedef Traits::ET ET (which is an exact number type used for exact computation like for example in achieved_epsilon()).
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Approximate_min_ellipsoid_d<Traits>::Point | |
typedef Traits::Point Point
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Approximate_min_ellipsoid_d<Traits>::Cartesian_const_iterator | |
typedef Traits::Cartesian_const_iterator Cartesian_const_iterator
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Approximate_min_ellipsoid_d<Traits>::Center_coordinate_iterator | |
A model of STL concept
RandomAccessIterator with value type double that is used
to iterate over the Cartesian center coordinates of the computed
ellipsoid, see center_cartesian_begin().
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Approximate_min_ellipsoid_d<Traits>::Axes_lengths_iterator | |
A model of STL concept
RandomAccessIterator with value type double that is used
to iterate over the lengths of the semiaxes of the computed ellipsoid,
see axes_lengths_begin().
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Approximate_min_ellipsoid_d<Traits>::Axis_direction_iterator | |
A model of STL concept
RandomAccessIterator with value type double that is used
to iterate over the Cartesian coordinates of the direction of a fixed
axis of the computed ellipsoid, see
axis_direction_cartesian_begin().
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An object of type Approximate_min_ellipsoid_d<Traits> can be created from an arbitrary point set P and some nonnegative double value eps.
template < class Iterator > | |||||
Approximate_min_ellipsoid_d<Traits> ame ( double eps, Iterator first, Iterator last, Traits traits = Traits()); | |||||
initializes ame to an (1+ε)-approximation of
(P) with P being the set of points in the range
[first,last). The number ε in this will
be at most eps, if possible. However, due to the
limited precision in the algorithm's underlying arithmetic, it
can happen that the computed approximation ellipsoid has a
worse approximation ratio (and ε can thus be larger
than eps in general). In any case, the number
ε (and with this, the achived approximation
1+ε) can be queried by calling the routine
achieved_epsilon() discussed below.
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The following methods can be used to query the achieved approximation
ratio 1+ε' and the computed ellipsoid E = { x ∈ d
| xT E x + xT e + η ≤ 0 }. The methods
defining_matrix(), defining_vector(), and
defining_scalar() do not return E, e, and η directly
but yield multiples of these quantities that are exactly representable
using the double type. (This is necessary because the
parameters E, e, and η of the computed approximation
ellipsoid E might not be exactly representable as
double numbers.)
unsigned int | ame.number_of_points () const | returns the number of points of ame, i.e., |P|. | ||||
double | ame.achieved_epsilon () const |
returns a number
ε' such that the computed approximation is (under exact
arithmetic) guaranteed to be an (1+ε')-approximation to
(P).
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double | ame.defining_matrix ( int i, int j) const | |||||
gives access to the (i,j)th entry of the matrix E in the
representation { x ∈ ![]()
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double | ame.defining_vector ( int i) const | |||||
gives access to the ith entry of the vector e in the
representation { x ∈ ![]()
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double | ame.defining_scalar () const |
gives access to the scalar η from the
representation { x ∈ ![]()
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Traits | ame.traits () const | returns a const reference to the traits class object. | ||||
int | ame.dimension () const | returns the dimension of the ambient space, i.e., the dimension of the points P. |
In order to access the center and semiaxes of the computed approximation ellipsoid, the functions center_cartesian_begin(), axes_lengths_begin(), and axis_direction_cartesian_begin() can be used. In constrast to the above access functions achieved_epsilon(), defining_matrix(), defining_vector(), and defining_scalar(), which return the described quantities exactly, the routines below return numerical approximations to the real center and real semiaxes of the computed ellipsoid; the comprised relative error may be larger than zero, and there are no guarantees for the returned quantities.
Center_coordinate_iterator | ame.center_cartesian_begin () |
returns an iterator pointing to the first of the d Cartesian
coordinates of the computed ellipsoid's center. The returned point is a floating-point approximation to the ellipsoid's exact center; no guarantee is given w.r.t. the involved relative error.
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Center_coordinate_iterator | ame.center_cartesian_end () |
returns the past-the-end iterator corresponding to
center_cartesian_begin().
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Axes_lengths_iterator | ame.axes_lengths_begin () |
returns an iterator pointing to the first of the d descendantly
sorted lengths of the computed ellipsoid's axes. The d returned
numbers are floating-point approximations to the exact
axes-lengths of the computed ellipsoid; no guarantee is given
w.r.t. the involved relative error. (See also method
axes_direction_cartesian_begin().)
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Axes_lengths_iterator | ame.axes_lengths_end () |
returns the past-the-end iterator corresponding to
axes_lengths_begin().
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Axes_direction_coordinate_iterator | ||||
ame.axis_direction_cartesian_begin ( int i) | ||||
returns an iterator
pointing to the first of the d Cartesian coordinates of the
computed ellipsoid's ith axis direction (i.e., unit vector in
direction of the ellipsoid's ith axis). The direction described
by this iterator is a floating-point approximation to the exact
axis direction of the computed ellipsoid; no guarantee is given
w.r.t. the involved relative error. An approximation to the
length of axis i is given by the ith entry of
axes_lengths_begin().
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Axes_direction_coordinate_iterator | ||||
ame.axis_direction_cartesian_end ( int i) | ||||
returns the past-the-end
iterator corresponding to
axis_direction_cartesian_begin().
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An object ame is valid iff
We implement Khachyian's algorithm for rounding polytopes [Kha96]. Internally, we use double-arithmetic and (initially a single) Cholesky-decomposition. The algorithm's running time is O(nd2(ε-1+ln d + ln ln (n))), where n=|P| and 1+ε is the desired approximation ratio.
To illustrate the usage of Approximate_min_ellipsoid_d<Traits> we give two examples in 2D. The
first program generates a random set P ⊂ 2 and outputs the
points and a 1.01-approximation of (P) as an EPS-file, which
you can view using gv, for instance. (In both examples you can
change the variables n and d to experiment with the code.)
File: examples/Approximate_min_ellipsoid_d/ellipsoid.cpp
#include <CGAL/Cartesian_d.h> #include <CGAL/MP_Float.h> #include <CGAL/point_generators_d.h> #include <CGAL/Approximate_min_ellipsoid_d.h> #include <CGAL/Approximate_min_ellipsoid_d_traits_d.h> #include <vector> #include <iostream> typedef CGAL::Cartesian_d<double> Kernel; typedef CGAL::MP_Float ET; typedef CGAL::Approximate_min_ellipsoid_d_traits_d<Kernel, ET> Traits; typedef Traits::Point Point; typedef std::vector<Point> Point_list; typedef CGAL::Approximate_min_ellipsoid_d<Traits> AME; int main() { const int n = 1000; // number of points const int d = 2; // dimension const double eps = 0.01; // approximation ratio is (1+eps) // create a set of random points: Point_list P; CGAL::Random_points_in_cube_d<Point> rpg(d,100.0); for (int i = 0; i < n; ++i) { P.push_back(*rpg); ++rpg; } // compute approximation: Traits traits; AME ame(eps, P.begin(), P.end(), traits); // write EPS file: if (ame.is_full_dimensional() && d == 2) ame.write_eps("example.eps"); // output center coordinates: std::cout << "Cartesian center coordinates: "; for (AME::Center_coordinate_iterator c_it = ame.center_cartesian_begin(); c_it != ame.center_cartesian_end(); ++c_it) std::cout << *c_it << ' '; std::cout << ".\n"; if (d == 2 || d == 3) { // output axes: AME::Axes_lengths_iterator axes = ame.axes_lengths_begin(); for (int i = 0; i < d; ++i) { std::cout << "Semiaxis " << i << " has length " << *axes++ << "\n" << "and Cartesian coordinates "; for (AME::Axes_direction_coordinate_iterator d_it = ame.axis_direction_cartesian_begin(i); d_it != ame.axis_direction_cartesian_end(i); ++d_it) std::cout << *d_it << ' '; std::cout << ".\n"; } } }
The second program outputs the approximation in a format suitable for display in Maplesoft's Maple.
File: examples/Approximate_min_ellipsoid_d/ellipsoid_for_maple.cpp
#include <CGAL/Cartesian_d.h> #include <CGAL/MP_Float.h> #include <CGAL/point_generators_d.h> #include <CGAL/Approximate_min_ellipsoid_d.h> #include <CGAL/Approximate_min_ellipsoid_d_traits_d.h> #include <vector> #include <iostream> #include <iomanip> typedef CGAL::Cartesian_d<double> Kernel; typedef CGAL::MP_Float ET; typedef CGAL::Approximate_min_ellipsoid_d_traits_d<Kernel, ET> Traits; typedef Traits::Point Point; typedef std::vector<Point> Point_list; typedef CGAL::Approximate_min_ellipsoid_d<Traits> AME; int main() { const int n = 100; // number of points const int d = 2; // dimension const double eps = 0.01; // approximation ratio is (1+eps) // create a set of random points: Point_list P; CGAL::Random_points_in_cube_d<Point> rpg(d,1.0); for (int i = 0; i < n; ++i) { P.push_back(*rpg); ++rpg; } // compute approximation: Traits traits; AME mel(eps, P.begin(), P.end(), traits); // output for Maple: if (mel.is_full_dimensional() && d == 2) { const double alpha = (1+mel.achieved_epsilon())*(d+1); // output points: using std::cout; cout << "restart;\n" << "with(LinearAlgebra):\n" << "with(plottools):\n" << "n:= " << n << ":\n" << "P:= Matrix(" << d << "," << n << "):\n"; for (int i=0; i<n; ++i) for (int j=0; j<d; ++j) cout << "P[" << j+1 << "," << i+1 << "] := " << std::setiosflags(std::ios::scientific) << std::setprecision(20) << P[i][j] << ":\n"; cout << "\n"; // output defining equation: cout << "Mp:= Matrix([\n"; for (int i=0; i<d; ++i) { cout << " ["; for (int j=0; j<d; ++j) { cout << mel.defining_matrix(i,j)/alpha; if (j<d-1) cout << ","; } cout << "]"; if (i<d-1) cout << ","; cout << "\n"; } cout << "]);\n" << "mp:= Vector(["; for (int i=0; i<d; ++i) { cout << mel.defining_vector(i)/alpha; if (i<d-1) cout << ","; } cout << "]);\n" << "eta:= " << (mel.defining_scalar()/alpha-1.0) << ";\n" << "v:= Vector([x,y]):\n" << "e:= Transpose(v).Mp.v+Transpose(v).mp+eta;\n" << "plots[display]({seq(point([P[1,i],P[2,i]]),i=1..n),\n" << " plots[implicitplot](e,x=-5..5,y=-5..5,numpoints=10000)},\n" << " scaling=CONSTRAINED);\n"; } }
1 | A multiset is a set where elements may have multiplicity greater than 1. |