Points in 3D-Space ( d3_point )

Definition

An instance of the data type d3_point is a point in the three-dimensional space R³. We use (x, y, z) to denote a point with first (or x-) coordinate x, second (or y-) coordinate y, and third (or z-) coordinate z.

#include < LEDA/geo/d3_point.h >

Creation

d3_point	p	introduces a variable p of type d3_point initialized to the point (0, 0, 0).
d3_point	p(double x, double y, double z)
		introduces a variable p of type d3_point initialized to the point (x, y, z).
d3_point	p(vector v)	introduces a variable p of type d3_point initialized to the point (v[0], v[1], v[2]). Precondition v.dim() = 3.

Operations

double	p.xcoord()	returns the first coordinate of p.
double	p.ycoord()	returns the second coordinate of p.
double	p.zcoord()	returns the third coordinate of p.
vector	p.to_vector()	returns the vector $\vec{{xyz}}\,$.
point	p.project_xy()	returns p projected into the xy-plane.
point	p.project_yz()	returns p projected into the yz-plane.
point	p.project_xz()	returns p projected into the xz-plane.
double	p.sqr_dist(const d3_point& q)
		returns the square of the Euclidean distance between p and q.
double	p.xdist(const d3_point& q)
		returns the x-distance between p and q.
double	p.ydist(const d3_point& q)
		returns the y-distance between p and q.
double	p.zdist(const d3_point& q)
		returns the z-distance between p and q.
double	p.distance(const d3_point& q)
		returns the Euclidean distance between p and q.
double	p.distance()	returns the Euclidean distance between p and the origin.
d3_point	p.translate(double dx, double dy, double dz)
		returns p translated by vector (dx, dy, dz).
d3_point	p.translate(const vector& v)
		returns p+ v, i.e., p translated by vector v. Precondition v.dim() = 3.
d3_point	p + const vector& v	returns p translated by vector v.
d3_point	p - const vector& v	returns p translated by vector - v.
d3_point	p.reflect(const d3_point& q, const d3_point& r, const d3_point& s)
		returns p reflected across the plane passing through q, r and s.
d3_point	p.reflect(const d3_point& q)
		returns p reflected across point q.
d3_point	p.rotate_around_axis(int a, double phi)
		returns p rotated by angle phi around the x-axis if a = 1, aournd the y-axis if a = 1, or around the z-axis if a = 2.
d3_point	p.rotate_around_vector(const vector& u, double phi)
		returns p rotated by angle phi around the axis defined by vector u.
d3_point	p.cartesian_to_polar()	returns p converted to polar coordinates.
d3_point	p.polar_to_cartesian()	returns p converted to cartesian coordinates.
vector	p - const d3_point& q	returns the difference vector of the coordinates.
ostream&	ostream& O « const d3_point& p
		writes p to output stream O.
istream&	istream& I » d3_point& p	reads the coordinates of p (three double numbers) from input stream I.

Non-Member Functions

int	cmp_distances(const d3_point& p1, const d3_point& p2, const d3_point& p3, const d3_point& p4)
		compares the distances (p1,p2) and (p3,p4). Returns +1 (-1) if distance (p1,p2) is larger (smaller) than distance (p3,p4), otherwise 0.
d3_point	center(const d3_point& a, const d3_point& b)
		returns the center of a and b, i.e. a + $\vec{{ab}}\,$/2.
d3_point	midpoint(const d3_point& a, const d3_point& b)
		returns the center of a and b.
int	orientation(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d)
		computes the orientation of points a, b, c, and d as the sign of the determinant $\left\vert\begin{array}{cccc} 1 & 1 & 1 & 1\\ a_x & b_x & c_x & d_x\\ a_y & b_y & c_y & d_y\\ a_z & b_z & c_z & d_z \end{array} \right\vert$
int	orientation_xy(const d3_point& a, const d3_point& b, const d3_point& c)
		returns the orientation of the projections of a, b and c into the xy-plane.
int	orientation_yz(const d3_point& a, const d3_point& b, const d3_point& c)
		returns the orientation of the projections of a, b and c into the yz-plane.
int	orientation_xz(const d3_point& a, const d3_point& b, const d3_point& c)
		returns the orientation of the projections of a, b and c into the xz-plane.
double	volume(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d)
		computes the signed volume of the simplex determined by a,b, c, and d, positive if orientation(a, b, c, d ) > 0 and negative otherwise.
bool	collinear(const d3_point& a, const d3_point& b, const d3_point& c)
		returns true if points a, b, c are collinear and false otherwise.
bool	coplanar(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d)
		returns true if points a, b, c, d are coplanar and false otherwise.
int	side_of_sphere(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d, const d3_point& x)
		returns +1 (-1) if point x lies on the positive (negative) side of the oriented sphere through points a, b, c, and d, and 0 if x is contained in this sphere.
int	region_of_sphere(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d, const d3_point& x)
		determines whether the point x lies inside (= + 1), on (= 0), or outside (= - 1) the sphere through points a, b, c, d, (equivalent to orientation(a,b,c,d) * side_of_sphere(a,b,c,d,x)) Precondition orientation(A)! = 0
bool	contained_in_simplex(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d, const d3_point& x)
		determines whether x is contained in the simplex spanned by the points a, b, c, d. Precondition a, b, c, d are affinely independent.
bool	contained_in_simplex(const array<d3_point>& A, const d3_point& x)
		determines whether x is contained in the simplex spanned by the points in A. Precondition A must have size < = 4 and the points in A must be affinely independent.
bool	contained_in_affine_hull(const list<d3_point>& L, const d3_point& x)
		determines whether x is contained in the affine hull of the points in L.
bool	contained_in_affine_hull(const array<d3_point>& A, const d3_point& x)
		determines whether x is contained in the affine hull of the points in A.
int	affine_rank(const array<d3_point>& L)
		computes the affine rank of the points in L.
int	affine_rank(const array<d3_point>& A)
		computes the affine rank of the points in A.
bool	affinely_independent(const list<d3_point>& L)
		decides whether the points in A are affinely independent.
bool	affinely_independent(const array<d3_point>& A)
		decides whether the points in A are affinely independent.
bool	inside_sphere(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d, const d3_point& e)
		returns true if point e lies in the interior of the sphere through points a, b, c, and d, and false otherwise.
bool	outside_sphere(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d, const d3_point& e)
		returns true if point e lies in the exterior of the sphere through points a, b, c, and d, and false otherwise.
bool	on_sphere(const d3_point& a, const d3_point& b, const d3_point& c, const d3_point& d, const d3_point& e)
		returns true if a, b, c, d, and e lie on a common sphere.
d3_point	point_on_positive_side(const d3_point& a, const d3_point& b, const d3_point& c)
		returns a point d with orientation(a, b, c, d ) > 0.