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.