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 p_1, p_2, p_3 with rational coordinates (rat_points). The orientation of C is equal to the orientation of the three defining points, i.e., orientation(p_1,p_2,p_3). 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 $\hbox{$\vert\,\{p_1,p_2,p_3\}\,\vert$ } \not= 2$. Assume now that p_1, p_2, p_3 are admissable. If $\hbox{$\vert\,\{p_1,p_2,p_3\}\,\vert$ } = 1$ they define the circle with center p_1 and radius zero. If p_1, p_2, and p_3 are collinear C is a straight line passing through p_1, p_2 and p_3 in this order and the center of C is undefined. If p_1, p_2, and p_3 are not collinear, C is the circle passing through them.

Creation

rat_circle	C(rat_point a, rat_point b, 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(rat_point a, 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(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).

Operations

circle	C.to_circle()	returns a floating point approximation of C.
int	C.orientation()	returns the orientation of C.
rat_point	C.center()	returns the center of C. Precondition: C has a center, i.e., it not a line.
rat_point	C.point1()	returns p_1.
rat_point	C.point2()	returns p_2.
rat_point	C.point3()	returns p_3.
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.
int	C.side_of(rat_point p)	returns -1, +1, or 0 if p lies right of, left of, or on C respectively.
bool	C.inside(rat_point p)	returns true if p lies inside C, false otherwise.
bool	C.outside(rat_point p)	returns true if p lies outside C.
bool	C.contains(rat_point p)	returns true if p lies on C, false otherwise.
rat_circle	C.translate(rational dx, 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(rat_vector v)	returns C translated by vector v.
rat_circle	C + rat_vector v	returns C translated by vector v.
rat_circle	C - rat_vector v	returns C translated by vector -v.
rat_circle	C.rotate90(rat_point q)	returns C rotated about q by an angle of 90 degrees.
rat_circle	C.reflect(rat_point p, rat_point q)
		returns C reflected across the straight line passing through p and q.
rat_circle	C.reflect(rat_point p)	returns C reflected across point p.
rat_circle	C.reverse()	returns C reversed.
bool	C == D	returns true if C and D are equal as oriented circles
bool	C.equal_as_sets(rat_circle C1, rat_circle C2)
		returns true if C1 and C2 are equal as unoriented circles
ostream&	ostream& out << c	writes the three defining points
istream&	istream& in >> rat_circle& c
		reads three points and assigns the circle defined by them to c.