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Rational Points ( rat_point )

Definition

An instance of data type rat_point is a point with rational coordinates in the two-dimensional plane. A point with cartesian coordinates (a, b) is represented by homogeneous coordinates (x, y, w) of arbitrary length integers (see Integers of Arbitrary Length) such that a = x/w and b = y/w and w > 0 .

#include < LEDA/geo/rat_point.h >

Types

rat_point::coord_type the coordinate type (rational).

rat_point::point_type the point type (rat_point).

rat_point::float_type the corresponding floating-point type (point).

Creation

rat_point p introduces a variable p of type rat_point initialized to the point (0, 0) .

rat_point p(const rational& a, const rational& b)
    introduces a variable p of type rat_point initialized to the point (a, b) .

rat_point p(integer a, integer b) introduces a variable p of type rat_point initialized to the point (a, b) .

rat_point p(integer x, integer y, integer w)
    introduces a variable p of type rat_point initialized to the point with homogeneous coordinates (x, y, w) if w > 0 and to point (- x, - y, - w) if w < 0 .
Precondition w $ \not=$ 0 .

rat_point p(const rat_vector& v) introduces a variable p of type rat_point initialized to the point (v[0], v[1]) .
Precondition: v.dim() = 2.

rat_point p(const point& p1, int prec = rat_point::default_precision)
    introduces a variable p of type rat_point initialized to the point with homogeneous coordinates ($ \lfloor$P*x$ \rfloor$,$ \lfloor$P*y$ \rfloor$, P) , where p1 = (x, y) and P = 2prec . If prec is non-positive, the conversion is without loss of precision, i.e., P is chosen as a sufficiently large power of two such that P*x and P*y are integers.

rat_point p(double x, double y, int prec = rat_point::default_precision)
    see constructor above with p = (x, y) .

Operations

point p.to_float() returns a floating point approximation of p.

rat_vector p.to_vector() returns the vector extending from the origin to p.

void p.normalize() simplifies the homogenous representation by dividing all coordinates by gcd (X, Y, W) .

integer p.X() returns the first homogeneous coordinate of p.

integer p.Y() returns the second homogeneous coordinate of p.

integer p.W() returns the third homogeneous coordinate of p.

double p.XD() returns a floating point approximation of p.X().

double p.YD() returns a floating point approximation of p.Y().

double p.WD() returns a floating point approximation of p.W().

rational p.xcoord() returns the x -coordinate of p.

rational p.ycoord() returns the y -coordinate of p.

double p.xcoordD() returns a floating point approximation of p.xcoord().

double p.ycoordD() returns a floating point approximation of p.ycoord().

rat_point p.rotate90(const rat_point& q, int i = 1)
    returns p rotated by i x 90 degrees about q . If i > 0 the rotation is counter-clockwise otherwise it is clockwise.

rat_point p.rotate90(int i = 1) returns p rotated by i x 90 degrees about the origin. If i > 0 the rotation is counter-clockwise otherwise it is clockwise.

rat_point p.reflect(const rat_point& p, const rat_point& q)
    returns p reflected across the straight line passing through p and q .
Precondition p $ \not=$q .

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

rat_point p.translate(const rational& dx, const rational& dy)
    returns p translated by vector (dx, dy) .

rat_point p.translate(integer dx, integer dy, integer dw)
    returns p translated by vector (dx/dw, dy/dw) .

rat_point p.translate(const rat_vector& v)
    returns p + v , i.e., p translated by vector v .
Precondition v .dim() = 2.

rat_point p + const rat_vector& v returns p translated by vector v .

rat_point p - const rat_vector& v returns p translated by vector - v .

rational p.sqr_dist(const rat_point& q)
    returns the squared distance between p and q .

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

rational p.xdist(const rat_point& q)
    returns the horizontal distance between p and q .

rational p.ydist(const rat_point& q)
    returns the vertical distance between p and q .

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

rational p.area(const rat_point& q, const rat_point& r)
    returns area(p, q, r) (see below).

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

Non-Member Functions

int cmp_signed_dist(const rat_point& a, const rat_point& b, const rat_point& c, const rat_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.

int orientation(const rat_point& a, const rat_point& b, const rat_point& c)
    computes the orientation of points a , b , c as the sign of the determinant

$ \left\vert\begin{array}{ccc} a_x & a_y & a_w\\
b_x & b_y & b_w\\
c_x & c_y & c_w
\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_distances(const rat_point& p1, const rat_point& p2, const rat_point& p3, const rat_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 .

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

rational area(const rat_point& a, const rat_point& b, const rat_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 rat_point& a, const rat_point& b, const rat_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 rat_point& a, const rat_point& b, const rat_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 rat_point& a, const rat_point& b, const rat_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 rat_point& a, const rat_point& b, const rat_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 rat_point& a, const rat_point& b, const rat_point& c, const rat_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(const rat_point& a, const rat_point& b, const rat_point& c, const rat_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(const rat_point& a, const rat_point& b, const rat_point& c, const rat_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 rat_point& a, const rat_point& b, const rat_point& c, const rat_point& d)
    returns true if points a , b , c , and d are cocircular.

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

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

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

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


next up previous contents index
Next: Rational Segments ( rat_segment Up: Basic Data Types for Previous: Iso-oriented Rectangles ( rectangle   Contents   Index
Christian Uhrig 2017-04-07