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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}| 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 ratcircle . 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 ratcircle . 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: Rational Triangles ( rat_triangle Up: Basic Data Types for Previous: Straight Rational Lines (   Contents   Index
Christian Uhrig 2017-04-07