/*
 * Some useful declarations for complex arithmetic.
 * A point in the plane may be considered complex once
 * we define arithmetic on it.
 */

/*
 * cpoint represents a point in the plane with 
 * x and y coordinates
 *
 */
struct cpoint {
  double x, y;
};

/*
 * cplus implements complex addition
 * 
 * parameters:
 *    z1 and z2 are complex points
 *
 * return:
 *    a new complex point which is the vector
 *    sum of the parameters
 */
extern struct cpoint cplus(struct cpoint z1, struct cpoint z2);

/*
 * ctimes implements complex multiplication
 *
 * parameters:
 *    z1 and z2 are complex points
 *
 * return:
 *    a new complex point which the the complex product of
 *    complex points z1 and z2. Defined by the formula:
 *    z1 * z1 == z1.x * z2.x - z1.y * z2.y (new x component)
 *               z1.x * z2.y + z2.y * z1.x (new y component)
 *
 */
extern struct cpoint ctimes(struct cpoint z1, struct cpoint z2);

/*
 * mag returns the magnitude (modulus) of a complex point
 *
 * parameter: z is a complex point
 *
 * return: The magnitude of z is define by:
 *         |z| == sqrt(z.x**2 + z.y**2)
 */
extern double mag(struct cpoint z);

/*
 * conj returns the complex conjugate of a complex point
 *
 * parameter: z is a complex point
 * 
 * return: The conjugate of z has the same x component, and
 *         the negation of the y component
 */
extern struct cpoint conj(struct cpoint z);

/*
 * cdiv implements complex multiplication
 *
 * parameters:
 *    z1 and z2 are complex points
 *
 * return:
 *    a new complex number z1/z2 which is defined by the
 *    formula:
 *    z1/z2 == ctimes(z1 , conj(z2)) / mag(z2)**2
 *    (where / is floating point division).
 *    Not defined when mag(z2) == 0
 */
extern struct cpoint cdiv(struct cpoint z1, struct cpoint z2);