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Rational Circles ( rat_circle )

Definition

An instance C of data type rat_circle is an oriented circle in the plane. A circle is defined by three points p1, p2, p3 with rational coordinates (rat_points). The orientation of C is equal to the orientation of the three defining points, i.e., orientation(p1, p2, p3). Positive orientation corresponds to counterclockwise orientation and negative orientation corresponds to clockwise orientation.

Some triples of points are unsuitable for defining a circle. A triple is admissable if |{p1, p2, p3}| $ \not=$2. Assume now that p1, p2, p3 are admissable. If |{p1, p2, p3}| = 1 they define the circle with center p1 and radius zero. If p1, p2, and p3 are collinear C is a straight line passing through p1, p2 and p3 in this order and the center of C is undefined. If p1, p2, and p3 are not collinear, C is the circle passing through them.

#include < LEDA/geo/rat_circle.h >

Types

rat_circle::coord_type the coordinate type (rational).

rat_circle::point_type the point type (rat_point).

rat_circle::float_type the corresponding floatin-point type (circle).

Creation

rat_circle C(const rat_point& a, const rat_point& b, const rat_point& c)
    introduces a variable C of type rat$ \_$circle. C is initialized to the circle through points a, b, and c.
Precondition a, b, and c are admissable.

rat_circle C(const rat_point& a, const rat_point& b)
    introduces a variable C of type circle. C is initialized to the counter-clockwise oriented circle with center a passing through b.

rat_circle C(const rat_point& a) introduces a variable C of type circle. C is initialized to the trivial circle with center a.

rat_circle C introduces a variable C of type rat$ \_$circle. C is initialized to the trivial circle centered at (0, 0).

rat_circle C(const circle& c, int prec = rat_point::default_precision)
    introduces a variable C of type rat_circle. C is initialized to the circle obtained by approximating three defining points of c.

Operations

circle C.to_float() returns a floating point approximation of C.

void C.normalize() simplifies the homogenous representation by normalizing p1, p2, and p3.

int C.orientation() returns the orientation of C.

rat_point C.center() returns the center of C.
Precondition C has a center, i.e., is not a line.

rat_point C.point1() returns p1.

rat_point C.point2() returns p2.

rat_point C.point3() returns p3.

rational C.sqr_radius() returns the square of the radius of C.

rat_point C.point_on_circle(double alpha, double epsilon)
    returns a point p on C such that the angle of p differs from alpha by at most epsilon.

bool C.is_degenerate() returns true if the defining points are collinear.

bool C.is_trivial() returns true if C has radius zero.

bool C.is_line() returns true if C is a line.

rat_line C.to_line() returns line(point1(), point3()).

int C.side_of(const rat_point& p)
    returns -1, +1, or 0 if p lies right of, left of, or on C respectively.

bool C.inside(const rat_point& p)
    returns true iff p lies inside of C.

bool C.outside(const rat_point& p)
    returns true iff p lies outside of C.

bool C.contains(const rat_point& p)
    returns true iff p lies on C.

rat_circle C.translate(const rational& dx, const rational& dy)
    returns C translated by vector (dx, dy).

rat_circle C.translate(integer dx, integer dy, integer dw)
    returns C translated by vector (dx/dw, dy/dw).

rat_circle C.translate(const rat_vector& v)
    returns C translated by vector v.

rat_circle C + const rat_vector& v returns C translated by vector v.

rat_circle C - const rat_vector& v returns C translated by vector - v.

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

rat_circle C.reflect(const rat_point& p, const rat_point& q)
    returns C reflected across the straight line passing through p and q.

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

rat_circle C.reverse() returns C reversed.

bool C == const rat_circle& D returns true if C and D are equal as oriented circles.

bool equal_as_sets(const rat_circle& C1, const rat_circle& C2)
    returns true if C1 and C2 are equal as unoriented circles.

bool radical_axis(const rat_circle& C1, const rat_circle& C2, rat_line& rad_axis)
    if the radical axis for C1 and C2 exists, it is assigned to rad_axis and true is returned; otherwise the result is false.

ostream& ostream& out « const rat_circle& c
    writes the three defining points.

istream& istream& in » rat_circle& c
    reads three points and assigns the circle defined by them to c.


next up previous contents index
Next: Rational Triangles ( rat_triangle Up: Basic Data Types for Previous: Straight Rational Lines (   Contents   Index
root 2008-01-09