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CGAL 5.3.2 - dD Geometry Kernel
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CGAL 5.3.2 - dD Geometry Kernel
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#include <CGAL/Kernel_d/Point_d.h>
An instance of data type Point_d<Kernel> is a point of Euclidean space in dimension \( d\).
A point \( p = (p_0,\ldots,p_{ d - 1 })\) in \( d\)-dimensional space can be represented by homogeneous coordinates \( (h_0,h_1,\ldots,h_d)\) of number type RT such that \( p_i = h_i/h_d\), which is of type FT. The homogenizing coordinate \( h_d\) is positive.
We call \( p_i\), \( 0 \leq i < d\) the \( i\)-th Cartesian coordinate and \( h_i\), \( 0 \le i \le d\), the \( i\)-th homogeneous coordinate. We call \( d\) the dimension of the point.
Downward compatibility
We provide operations of the lower dimensional interface x(), y(), z(), hx(), hy(), hz(), hw().
Implementation
Points are implemented by arrays of RT items. All operations like creation, initialization, tests, point - vector arithmetic, input and output on a point \( p\) take time \( O(p.dimension())\). dimension(), coordinate access and conversions take constant time. The space requirement for points is \( O(p.dimension())\).
Types | |
| typedef unspecified_type | LA |
| the linear algebra layer. | |
| typedef unspecified_type | Cartesian_const_iterator |
| a read-only iterator for the Cartesian coordinates. | |
| typedef unspecified_type | Homogeneous_const_iterator |
| a read-only iterator for the homogeneous coordinates. | |
Creation | |
| Point_d () | |
introduces a variable p of type Point_d<Kernel>. | |
| Point_d (int d, Origin) | |
introduces a variable p of type Point_d<Kernel> in \( d\)-dimensional space, initialized to the origin. | |
| template<class InputIterator > | |
| Point_d (int d, InputIterator first, InputIterator last) | |
introduces a variable p of type Point_d<Kernel> in dimension d. More... | |
| template<class InputIterator > | |
| Point_d (int d, InputIterator first, InputIterator last, RT D) | |
introduces a variable p of type Point_d<Kernel> in dimension d initialized to the point with homogeneous coordinates as defined by H = set [first,last) and D: \( (\pm H[0], \pm H[1], \ldots, \pm H[d-1], \pm D)\). More... | |
| Point_d (RT x, RT y, RT w=1) | |
introduces a variable p of type Point_d<Kernel> in \( 2\)-dimensional space. More... | |
| Point_d (RT x, RT y, RT z, RT w) | |
introduces a variable p of type Point_d<Kernel> in \( 3\)-dimensional space. More... | |
Operations | |
| int | dimension () |
returns the dimension of p. | |
| FT | cartesian (int i) |
returns the \( i\)-th Cartesian coordinate of p. More... | |
| FT | operator[] (int i) |
returns the \( i\)-th Cartesian coordinate of p. More... | |
| RT | homogeneous (int i) |
returns the \( i\)-th homogeneous coordinate of p. More... | |
| Cartesian_const_iterator | cartesian_begin () |
returns an iterator pointing to the zeroth Cartesian coordinate \( p_0\) of p. | |
| Cartesian_const_iterator | cartesian_end () |
returns an iterator pointing beyond the last Cartesian coordinate of p. | |
| Homogeneous_const_iterator | homogeneous_begin () |
returns an iterator pointing to the zeroth homogeneous coordinate \( h_0\) of p. | |
| Homogeneous_const_iterator | homogeneous_end () |
returns an iterator pointing beyond the last homogeneous coordinate of p. | |
| Point_d< Kernel > | transform (const Aff_transformation_d< Kernel > &t) |
| returns \( t(p)\). | |
Arithmetic Operators, Tests and IO | |
| Vector_d< Kernel > | operator- (const Origin &o) |
| returns the vector \( p-O\). | |
| Vector_d< Kernel > | operator- (const Point_d< Kernel > &q) |
| returns \( p - q\). More... | |
| Point_d< Kernel > | operator+ (const Vector_d< Kernel > &v) |
| returns \( p + v\). More... | |
| Point_d< Kernel > | operator- (const Vector_d< Kernel > &v) |
| returns \( p - v\). More... | |
| Point_d< Kernel > & | operator+= (const Vector_d< Kernel > &v) |
adds v to p. More... | |
| Point_d< Kernel > & | operator-= (const Vector_d< Kernel > &v) |
subtracts v from p. More... | |
| bool | operator== (const Origin &) |
returns true if p is the origin. | |
| bool | operator< (const Point_d< Kernel > &q) |
returns true iff p is lexicographically smaller than q with respect to Cartesian lexicographic order of points. More... | |
| bool | operator> (const Point_d< Kernel > &q) |
returns true iff p is lexicographically greater than q with respect to Cartesian lexicographic order of points. More... | |
| bool | operator<= (const Point_d< Kernel > &q) |
returns true iff p is lexicographically smaller than or equal to q with respect to Cartesian lexicographic order of points. More... | |
| bool | operator>= (const Point_d< Kernel > &q) |
returns true iff p is lexicographically greater than or equal to q with respect to Cartesian lexicographic order of points. More... | |
| CGAL::Point_d< Kernel >::Point_d | ( | int | d, |
| InputIterator | first, | ||
| InputIterator | last | ||
| ) |
introduces a variable p of type Point_d<Kernel> in dimension d.
If size [first,last) == d this creates a point with Cartesian coordinates set [first,last). If size [first,last) == d+1 the range specifies the homogeneous coordinates \( H = set [first,last) = (\pm h_0, \pm h_1, \ldots, \pm h_d)\) where the sign chosen is the sign of \( h_d\).
d is nonnegative, [first,last) has d or d+1 elements where the last has to be non-zero. | InputIterator | has RT as value type. |
| CGAL::Point_d< Kernel >::Point_d | ( | int | d, |
| InputIterator | first, | ||
| InputIterator | last, | ||
| RT | D | ||
| ) |
introduces a variable p of type Point_d<Kernel> in dimension d initialized to the point with homogeneous coordinates as defined by H = set [first,last) and D: \( (\pm H[0], \pm H[1], \ldots, \pm H[d-1], \pm D)\).
The sign chosen is the sign of \( D\).
D is non-zero, the iterator range defines a \( d\)-tuple of RT. | InputIterator | has RT as value type. |
| CGAL::Point_d< Kernel >::Point_d | ( | RT | x, |
| RT | y, | ||
| RT | w = 1 |
||
| ) |
| CGAL::Point_d< Kernel >::Point_d | ( | RT | x, |
| RT | y, | ||
| RT | z, | ||
| RT | w | ||
| ) |
| FT CGAL::Point_d< Kernel >::cartesian | ( | int | i | ) |
returns the \( i\)-th Cartesian coordinate of p.
| RT CGAL::Point_d< Kernel >::homogeneous | ( | int | i | ) |
returns the \( i\)-th homogeneous coordinate of p.
| Point_d<Kernel> CGAL::Point_d< Kernel >::operator+ | ( | const Vector_d< Kernel > & | v | ) |
returns \( p + v\).
p.dimension() == v.dimension(). | Point_d<Kernel>& CGAL::Point_d< Kernel >::operator+= | ( | const Vector_d< Kernel > & | v | ) |
adds v to p.
p.dimension() == v.dimension(). | Vector_d<Kernel> CGAL::Point_d< Kernel >::operator- | ( | const Point_d< Kernel > & | q | ) |
returns \( p - q\).
p.dimension() == q.dimension(). | Point_d<Kernel> CGAL::Point_d< Kernel >::operator- | ( | const Vector_d< Kernel > & | v | ) |
returns \( p - v\).
p.dimension() == v.dimension(). | Point_d<Kernel>& CGAL::Point_d< Kernel >::operator-= | ( | const Vector_d< Kernel > & | v | ) |
subtracts v from p.
p.dimension() == v.dimension(). | bool CGAL::Point_d< Kernel >::operator< | ( | const Point_d< Kernel > & | q | ) |
returns true iff p is lexicographically smaller than q with respect to Cartesian lexicographic order of points.
p.dimension() == q.dimension(). | bool CGAL::Point_d< Kernel >::operator<= | ( | const Point_d< Kernel > & | q | ) |
returns true iff p is lexicographically smaller than or equal to q with respect to Cartesian lexicographic order of points.
p.dimension() == q.dimension(). | bool CGAL::Point_d< Kernel >::operator> | ( | const Point_d< Kernel > & | q | ) |
returns true iff p is lexicographically greater than q with respect to Cartesian lexicographic order of points.
p.dimension() == q.dimension(). | bool CGAL::Point_d< Kernel >::operator>= | ( | const Point_d< Kernel > & | q | ) |
returns true iff p is lexicographically greater than or equal to q with respect to Cartesian lexicographic order of points.
p.dimension() == q.dimension(). | FT CGAL::Point_d< Kernel >::operator[] | ( | int | i | ) |
returns the \( i\)-th Cartesian coordinate of p.
1.8.13