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# Rational Vectors ( rat_vector )

Definition

An instance of data type rat_vector is a vector of rational numbers. A d-dimensional vector r = (r0,..., rd - 1) is represented in homogeneous coordinates (h0,..., hd), where ri = hi/hd and the hi's are of type integer. We call the ri's the cartesian coordinates of the vector. The homogenizing coordinate hd 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/rat_vector.h >

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 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. rat_vector v(integer a, integer b, integer c, integer D) introduces a variable v of type rat_vector initialized to the two-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. rat_vector v(rational x, rational y, rational z) introduces a variable v of type rat_vector initialized to the two-dimensional vector with homogeneous representation (a, b, c, D), where x = a/D, y = b/D and z = c/D. rat_vector v(integer a, integer b) introduces a variable v of type rat_vector initialized to the two-dimensional vector with homogeneous representation (a, b, 1). rat_vector v(integer_vector c, integer D) introduces a variable v of type rat_vector initialized to the vector with homogeneous coordinates (c0,...,cd - 1,D), where d is the dimension of c and the sign chosen is the sign of D. Precondition D is non-zero. rat_vector v(integer_vector c) introduces a variable v of type rat_vector initialized to the direction with homogeneous coordinate vector c, where the sign chosen is the sign of the last component of c. Precondition The last component of c is non-zero. rat_vector v(vector w, int prec) introduces a variable v of type rat_vector initialized to (P*w0,...,P*wd - 1, P), where d is the dimension of w and P = 2prec.

Operations

3.1 Initialization, Access and Conversions

 rat_vector rat_vector::d2(integer a, integer b, integer D) returns a rat_vector of dimension 2 initialized to a 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::d3(integer a, integer b, integer c, integer D) returns a rat_vector of dimension 3 initialized to a 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. rat_vector rat_vector::unit(int i, int d=2) returns a rat_vector of dimension d initialized to the i-th unit vector. Precondition 0 < = i < d. rat_vector rat_vector::zero(int d=2) returns the zero vector in d-dimensional space. int v.dim() returns the dimension of v. integer v.hcoord(int i) returns the i-th homogeneous coordinate of v. rational v.coord(int i) returns the i-th cartesian coordinate of v. rational v[int i] returns the i-th cartesian coordinate of v. rational v.sqr_length() returns the square of the length of v. vector v.to_float() returns a floating point approximation of v.

Additional Operations for vectors in two and three-dimensional space

 rational v.xcoord() returns the zero-th cartesian coordinate of v. rational v.ycoord() returns the first cartesian coordinate of v. rational v.zcoord() returns the second cartesian coordinate of v. integer v.X() returns the zero-th homogeneous coordinate of v. integer v.Y() returns the first homogeneous coordinate of v. integer v.Z() returns the second homogeneous coordinate of v. integer v.W() returns the homogenizing coordinate of v. rat_vector v.rotate90() returns the v rotated by 90 degrees. Precondition v.dim() == 2. int compare_by_angle(rat_vector v1, rat_vector v2) For a non-zero vector v let (v) be the angle by which the positive x-axis has to be turned counter-clockwise until it aligns with v. The function compares the angles defined by v1 and v2, respectively. The zero-vector precedes all non-zero vectors in the angle-order.

3.2 Tests

 bool v == w Test for equality. bool v != w Test for inequality.

3.3 Arithmetical Operators

 rat_vector integer n * v multiplies all cartesian coordinates by n. rat_vector rational r * v multiplies all cartesian coordinates by r. rat_vector& v *= integer n multiplies all cartesian coordinates by n. rat_vector& v *= rational r multiplies all cartesian coordinates by r. rat_vector v / integer n divides all cartesian coordinates by n. rat_vector v / rational r divides all cartesian coordinates by r. rat_vector& v /= integer n divides all cartesian coordinates by n. rat_vector& v /= rational r divides all cartesian coordinates by r. rational const v * w scalar product, i.e., viwi, where vi and wi are the cartesian coordinates of v and w respectively. rat_vector v + w adds cartesian coordinates. rat_vector& v += w addition plus assignment. rat_vector v - w subtracts cartesian coordinates. rat_vector& v -= w subtraction plus assignment. rat_vector -v returns -v.

3.4 Input and Output

 ostream& ostream& O << v writes v's homogeneous coordinates componentwise to the output stream O. istream& istream& I >> rat_vector& v reads v's homogeneous coordinates componentwise from the input stream I. The operator uses the current dimension of v.

3.5 Linear Hull, Dependence and Rank

 bool contained_in_linear_hull(array A, rat_vector x) determines whether x is contained in the linear hull of the vectors in A. int linear_rank(array A) computes the linear rank of the vectors in A. bool linearly_independent(array A) decides whether the vectors in A are linearly independent. array linear_base(array A) computes a basis of the linear space spanned by the vectors in A.

Implementation

Vectors are implemented by arrays of integers as an item type. All operations like creation, initialization, tests, vector arithmetic, input and output on an vector v take time O(v.dim()). dim(), coordinate access and conversions take constant time. The operations for linear hull, rank and independence have the cubic costs of the used matrix operations. The space requirement is O(v.dim()).

Next: Functions of Numerical Analysis Up: Number Types and Linear Previous: Matrices with Integer Entries   Contents   Index
LEDA research project
2000-10-12