Points ( point )

Definition

An instance of the data type point is a point in the two-dimensional plane  R ². We use (x, y) to denote a point with first (or x-) coordinate x and second (or y-) coordinate y.

#include < LEDA/point.h >

Creation

point	p	introduces a variable p of type point initialized to the point (0, 0).
point	p(double x, double y)	introduces a variable p of type point initialized to the point (x, y).
point	p(vector v)	introduces a variable p of type point initialized to the point (v[0], v[1]). Precondition: : v.dim() = 2.
point	p(point p, int prec)	introduces a variable p of type point initialized to the point with coordinates ($\lfloor$P*x$\rfloor$/P,$\lfloor$P*x$\rfloor$/P), where p = (x, y) and P = 2prec. If prec is non-positive, the new point has coordinates x and y.

Operations

double	p.xcoord()	returns the first coordinate of p.
double	p.ycoord()	returns the second coordinate of p.
vector	p.to_vector()	returns the vector $\vec{xy}\,$.
double	p.sqr_dist(point q)	returns the square of the Euclidean distance between p and q.
int	p.cmp_dist(point q, point r)
		returns compare(p.sqrdist(q), p.sqrdist(r)).
double	p.xdist(point q)	returns the horizontal distance between p and q.
double	p.ydist(point q)	returns the vertical distance between p and q.
double	p.distance(point q)	returns the Euclidean distance between p and q.
double	p.distance()	returns the Euclidean distance between p and (0, 0).
double	p.angle(point q, point r)	returns the angle between $\vec{p q}\,$ and $\vec{p r}\,$.
point	p.translate_by_angle(double alpha, double d)
		returns p translated in direction alpha by distance d. The direction is given by its angle with a right oriented horizontal ray.
point	p.translate(double dx, double dy)
		returns p translated by vector (dx, dy).
point	p.translate(vector v)	returns p+ v, i.e., p translated by vector v. Precondition: v.dim() = 2.
point	p + vector v	returns p translated by vector v.
point	p - vector v	returns p translated by vector - v.
point	p.rotate(point q, double a)
		returns p rotated about q by angle a.
point	p.rotate(double a)	returns p.rotate( point(0, 0), a).
point	p.rotate90(point q)	returns p rotated about q by an angle of 90 degrees.
point	p.rotate90()	returns p.rotate90( point(0, 0)).
point	p.reflect(point q, point r)
		returns p reflected across the straight line passing through q and r.
point	p.reflect(point q)	returns p reflected across point q.
vector	p - q	returns the difference vector of the coordinates.
ostream&	ostream& O << p	writes p to output stream O.
istream&	istream& I >> point& p	reads the coordinates of p (two double numbers) from input stream I.

Non-Member Functions

point	center(point a, point b)	returns the center of a and b, i.e. a + $\vec{ab}\,$/2.
point	midpoint(point a, point b)
		returns the center of a and b.
int	orientation(point a, point b, point c)
		computes the orientation of points a, b, and c as the sign of the determinant $\left\Vert \begin{array}{ccc} a_x & a_y & 1\\ b_x & b_y & 1\\ c_x & c_y & 1 \end{array} \right\Vert$ i.e., it returns +1 if point c lies left of the directed line through a and b, 0 if a,b, and c are collinear, and -1 otherwise.
int	cmp_signed_dist(point a, point b, point c, point d)
		compares (signed) distances of c and d to the straight line passing through a and b (directed from a to b). Returns +1 (-1)if c has larger (smaller) distance than d and 0 if distances are equal.
double	area(point a, point b, point c)
		computes the signed area of the triangle determined by a,b,c, positive if orientation(a, b, c) > 0 and negative otherwise.
bool	collinear(point a, point b, point c)
		returns true if points a, b, c are collinear, i.e., orientation(a, b, c) = 0, and false otherwise.
bool	right_turn(point a, point b, point c)
		returns true if points a, b, c form a righ turn, i.e., orientation(a, b, c) > 0, and false otherwise.
bool	left_turn(point a, point b, point c)
		returns true if points a, b, c form a left turn, i.e., orientation(a, b, c) < 0, and false otherwise.
int	side_of_halfspace(point a, point b, point c)
		returns the sign of the scalar product (b - a)*(c - a). If b not  = a this amounts to: Let h be the open halfspace orthogonal to the vector b - a, containing b, and having a in its boundary. Returns +1 if c is contained in h, returns 0 is c lies on the the boundary of h, and returns -1 is c is contained in the interior of the complement of h.
int	side_of_circle(point a, point b, point c, point d)
		returns +1 if point d lies left of the directed circle through points a, b, and c, 0 if a,b,c,and d are cocircular, and -1 otherwise.
bool	incircle(point a, point b, point c, point d)
		returns true if point d lies in the interior of the circle through points a, b, and c, and false otherwise.
bool	outcircle(point a, point b, point c, point d)
		returns true if point d lies outside of the circle through points a, b, and c, and false otherwise.
bool	cocircular(point a, point b, point c, point d)
		returns true if points a, b, c, and d are corcircular.
bool	affinely_independent(array<point> A)
		decides whether the points in A are affinely independent.
bool	contained_in_simplex(array<point> A, point p)
		determines whether p is contained in the simplex spanned by the points in A. A may consists of up to 3 points. Precondition: The points in A are affinely independent.
bool	contained_in_affine_hull(array<point> A, point p)
		determines whether p is contained in the affine hull of the points in A.