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Points ( point )

Definition

An instance of the data type point is a point in the two-dimensional plane R2. We use (x, y) to denote a point with first (or x-) coordinate x and second (or y-) coordinate y.

#include < LEDA/geo/point.h >

Types

point::coord_type the coordinate type (double).

point::point_type the point type (point).

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(const 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}}\,$.

int p.orientation(const point& q, const point& r)
    returns orientation(p, q, r) (see below).

double p.area(const point& q, const point& r)
    returns area(p, q, r) (see below).

double p.sqr_dist(const point& q)
    returns the square of the Euclidean distance between p and q.

int p.cmp_dist(const point& q, const point& r)
    returns compare(p.sqr$ \_$dist(q), p.sqr$ \_$dist(r)).

double p.xdist(const point& q) returns the horizontal distance between p and q.

double p.ydist(const point& q) returns the vertical distance between p and q.

double p.distance(const 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(const point& q, const 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(const vector& v)
    returns p+ v, i.e., p translated by vector v.
Precondition v.dim() = 2.

point p + const vector& v returns p translated by vector v.

point p - const vector& v returns p translated by vector - v.

point p.rotate(const 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(const point& q, int i=1)
    returns p rotated about q by an angle of i x 90 degrees. If i > 0 the rotation is counter-clockwise otherwise it is clockwise.

point p.rotate90(int i=1) returns p.rotate90( point(0, 0), i).

point p.reflect(const point& q, const point& r)
    returns p reflected across the straight line passing through q and r.

point p.reflect(const point& q) returns p reflected across point q.

vector p - const point& q returns the difference vector of the coordinates.

Non-Member Functions

int cmp_distances(const point& p1, const point& p2, const point& p3, const 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.

point center(const point& a, const point& b)
    returns the center of a and b, i.e. a + $ \vec{{ab}}\,$/2.

point midpoint(const point& a, const point& b)
    returns the center of a and b.

int orientation(const point& a, const point& b, const 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(const point& a, const point& b, const point& c, const 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(const point& a, const point& b, const 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(const point& a, const point& b, const point& c)
    returns true if points a, b, c are collinear, i.e., orientation(a, b, c) = 0, and false otherwise.

bool right_turn(const point& a, const point& b, const 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(const point& a, const point& b, const 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(const point& a, const point& b, const 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(const point& a, const point& b, const point& c, const 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 inside_circle(const point& a, const point& b, const point& c, const point& d)
    returns true if point d lies in the interior of the circle through points a, b, and c, and false otherwise.

bool outside_circle(const point& a, const point& b, const point& c, const point& d)
    returns true if point d lies outside of the circle through points a, b, and c, and false otherwise.

bool on_circle(const point& a, const point& b, const point& c, const point& d)
    returns true if points a, b, c, and d are cocircular.

bool cocircular(const point& a, const point& b, const point& c, const point& d)
    returns true if points a, b, c, and d are cocircular.

int compare_by_angle(const point& a, const point& b, const point& c, const point& d)
    compares vectors b-a and d-c by angle (more efficient than calling compare_by_angle(b-a,d-x) on vectors).

bool affinely_independent(const array<point>& A)
    decides whether the points in A are affinely independent.

bool contained_in_simplex(const array<point>& A, const point& p)
    determines whether p is contained in the simplex spanned by the points in A. A may consist of up to 3 points.
Precondition The points in A are affinely independent.

bool contained_in_affine_hull(const array<point>& A, const point& p)
    determines whether p is contained in the affine hull of the points in A.


next up previous contents index
Next: Segments ( segment ) Up: Basic Data Types for Previous: Basic Data Types for   Contents   Index
root 2008-01-09