Version 5.0 The LEDA User Manual

5.14 Rational Vectors ( rat vector )
1. Definition
An instance of data type rat vector is a vector of rational numbers. A d-dimensional
vector r = (r 0 , . . . , r d−1 ) is represented in homogeneous coordinates (h 0 , . . . , h d ), where
r i = h i /h d and the h i ’s are of type integer. We call the r i ’s the cartesian coordinates of
the vector. The homogenizing coordinate h d is positive.
This data type is meant for use in computational geometry. It realizes free vectors as
opposed to position vectors (type rat point). The main difference between position vectors
and free vectors is their behavior under affine transformations, e.g., free vectors are
invariant under translations.
rat vector is an item type.
#include < LEDA/numbers/rat vector.h >
2. Creation
rat vector v(int d = 2); introduces a variable v of type rat vector initialized to
the zero vector of dimension d.
rat vector
v(integer a, integer b, integer D);
introduces a variable v of type rat vector initialized to
the two-dimensional vector with homogeneous representation
(a, b, D) if D is positive and representation
(−a, −b, −D) if D is negative.
Precondition: D is non-zero.
rat vector
rat vector
rat vector
v(rational x, rational y);
introduces a variable v of type rat vector initialized to
the two-dimensional vector with homogeneous representation
(a, b, D), where x = a/D and y = b/D.
v(integer a, integer b, integer c, integer D);
v(rational x, rational y, rational z);
introduces a variable v of type rat vector initialized to
the three-dimensional vector with homogeneous representation
(a, b, c, D) if D is positive and representation
(−a, −b, −c, −D) if D is negative.
Precondition: D is non-zero.
introduces a variable v of type rat vector initialized to
the three-dimensional vector with homogeneous representation
(a, b, c, D), where x = a/D, y = b/D and
z = c/D.