/**
* \file
* \brief Infinite straight line
*//*
* Authors:
* Marco Cecchetti <mrcekets at gmail.com>
* Krzysztof Kosiński <tweenk.pl@gmail.com>
* Copyright 2008-2011 Authors
*
* This library is free software; you can redistribute it and/or
* modify it either under the terms of the GNU Lesser General Public
* License version 2.1 as published by the Free Software Foundation
* (the "LGPL") or, at your option, under the terms of the Mozilla
* Public License Version 1.1 (the "MPL"). If you do not alter this
* notice, a recipient may use your version of this file under either
* the MPL or the LGPL.
*
* You should have received a copy of the LGPL along with this library
* in the file COPYING-LGPL-2.1; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
* You should have received a copy of the MPL along with this library
* in the file COPYING-MPL-1.1
*
* The contents of this file are subject to the Mozilla Public License
* Version 1.1 (the "License"); you may not use this file except in
* compliance with the License. You may obtain a copy of the License at
* http://www.mozilla.org/MPL/
*
* This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
* OF ANY KIND, either express or implied. See the LGPL or the MPL for
* the specific language governing rights and limitations.
*/
#ifndef LIB2GEOM_SEEN_LINE_H
#define LIB2GEOM_SEEN_LINE_H
#include <cmath>
#include <boost/optional.hpp>
#include <2geom/bezier-curve.h> // for LineSegment
#include <2geom/rect.h>
#include <2geom/crossing.h>
#include <2geom/exception.h>
#include <2geom/ray.h>
#include <2geom/angle.h>
#include <2geom/intersection.h>
namespace Geom
{
// class docs in cpp file
class Line
: boost::equality_comparable< Line >
{
private:
Point _initial;
Point _final;
public:
/// @name Creating lines.
/// @{
/** @brief Create a default horizontal line.
* Creates a line with unit speed going in +X direction. */
Line()
: _initial(0,0), _final(1,0)
{}
/** @brief Create a line with the specified inclination.
* @param origin One of the points on the line
* @param angle Angle of the line in mathematical convention */
Line(Point const &origin, Coord angle)
: _initial(origin)
{
Point v;
sincos(angle, v[Y], v[X]);
_final = _initial + v;
}
/** @brief Create a line going through two points.
* The first point will be at time 0, while the second one
* will be at time 1.
* @param a Initial point
* @param b First point */
Line(Point const &a, Point const &b)
: _initial(a)
, _final(b)
{}
/** @brief Create a line based on the coefficients of its equation.
@see Line::setCoefficients() */
Line(double a, double b, double c) {
setCoefficients(a, b, c);
}
/// Create a line by extending a line segment.
explicit Line(LineSegment const &seg)
: _initial(seg.initialPoint())
, _final(seg.finalPoint())
{}
/// Create a line by extending a ray.
explicit Line(Ray const &r)
: _initial(r.origin())
, _final(r.origin() + r.vector())
{}
/// Create a line normal to a vector at a specified distance from origin.
static Line from_normal_distance(Point const &n, Coord c) {
Point start = c * n.normalized();
Line l(start, start + rot90(n));
return l;
}
/** @brief Create a line from origin and unit vector.
* Note that each line direction has two possible unit vectors.
* @param o Point through which the line will pass
* @param v Unit vector of the line's direction */
static Line from_origin_and_vector(Point const &o, Point const &v) {
Line l(o, o + v);
return l;
}
Line* duplicate() const {
return new Line(*this);
}
/// @}
/// @name Retrieve and set the line's parameters.
/// @{
/// Get the line's origin point.
Point origin() const { return _initial; }
/** @brief Get the line's raw direction vector.
* The retrieved vector is normalized to unit length. */
Point vector() const { return _final - _initial; }
/** @brief Get the line's normalized direction vector.
* The retrieved vector is normalized to unit length. */
Point versor() const { return (_final - _initial).normalized(); }
/// Angle the line makes with the X axis, in mathematical convention.
Coord angle() const {
Point d = _final - _initial;
double a = std::atan2(d[Y], d[X]);
if (a < 0) a += M_PI;
if (a == M_PI) a = 0;
return a;
}
/** @brief Set the point at zero time.
* The orientation remains unchanged, modulo numeric errors during addition. */
void setOrigin(Point const &p) {
Point d = p - _initial;
_initial = p;
_final += d;
}
/** @brief Set the speed of the line.
* Origin remains unchanged. */
void setVector(Point const &v) {
_final = _initial + v;
}
/** @brief Set the angle the line makes with the X axis.
* Origin remains unchanged. */
void setAngle(Coord angle) {
Point v;
sincos(angle, v[Y], v[X]);
v *= distance(_initial, _final);
_final = _initial + v;
}
/// Set a line based on two points it should pass through.
void setPoints(Point const &a, Point const &b) {
_initial = a;
_final = b;
}
/** @brief Set the coefficients of the line equation.
* The line equation is: \f$ax + by = c\f$. Points that satisfy the equation
* are on the line. */
void setCoefficients(double a, double b, double c);
/** @brief Get the coefficients of the line equation as a vector.
* @return STL vector @a v such that @a v[0] contains \f$a\f$, @a v[1] contains \f$b\f$,
* and @a v[2] contains \f$c\f$. */
std::vector<double> coefficients() const;
/// Get the coefficients of the line equation by reference.
void coefficients(Coord &a, Coord &b, Coord &c) const;
/** @brief Check if the line has more than one point.
* A degenerate line can be created if the line is created from a line equation
* that has no solutions.
* @return True if the line has no points or exactly one point */
bool isDegenerate() const {
return _initial == _final;
}
/// Check if the line is horizontal (y is constant).
bool isHorizontal() const {
return _initial[Y] == _final[Y];
}
/// Check if the line is vertical (x is constant).
bool isVertical() const {
return _initial[X] == _final[X];
}
/** @brief Reparametrize the line so that it has unit speed.
* Note that the direction of the line may also change. */
void normalize() {
// this helps with the nasty case of a line that starts somewhere far
// and ends very close to the origin
if (L2sq(_final) < L2sq(_initial)) {
std::swap(_initial, _final);
}
Point v = _final - _initial;
v.normalize();
_final = _initial + v;
}
/** @brief Return a new line reparametrized for unit speed. */
Line normalized() const {
Point v = _final - _initial;
v.normalize();
Line ret(_initial, _initial + v);
return ret;
}
/// @}
/// @name Evaluate the line as a function.
///@{
Point initialPoint() const {
return _initial;
}
Point finalPoint() const {
return _final;
}
Point pointAt(Coord t) const {
return lerp(t, _initial, _final);;
}
Coord valueAt(Coord t, Dim2 d) const {
return lerp(t, _initial[d], _final[d]);
}
Coord timeAt(Point const &p) const;
/** @brief Get a time value corresponding to a projection of a point on the line.
* @param p Arbitrary point.
* @return Time value corresponding to a point closest to @c p. */
Coord timeAtProjection(Point const& p) const {
if ( isDegenerate() ) return 0;
Point v = vector();
return dot(p - _initial, v) / dot(v, v);
}
/** @brief Find a point on the line closest to the query point.
* This is an alias for timeAtProjection(). */
Coord nearestTime(Point const &p) const {
return timeAtProjection(p);
}
std::vector<Coord> roots(Coord v, Dim2 d) const;
Coord root(Coord v, Dim2 d) const;
/// @}
/// @name Create other objects based on this line.
/// @{
void reverse() {
std::swap(_final, _initial);
}
/** @brief Create a line containing the same points, but in opposite direction.
* @return Line \f$g\f$ such that \f$g(t) = f(1-t)\f$ */
Line reversed() const {
Line result(_final, _initial);
return result;
}
/** @brief Same as segment(), but allocate the line segment dynamically. */
// TODO remove this?
Curve* portion(Coord f, Coord t) const {
LineSegment* seg = new LineSegment(pointAt(f), pointAt(t));
return seg;
}
/** @brief Create a segment of this line.
* @param f Time value for the initial point of the segment
* @param t Time value for the final point of the segment
* @return Created line segment */
LineSegment segment(Coord f, Coord t) const {
return LineSegment(pointAt(f), pointAt(t));
}
/// Return the portion of the line that is inside the given rectangle
boost::optional<LineSegment> clip(Rect const &r) const;
/** @brief Create a ray starting at the specified time value.
* The created ray will go in the direction of the line's vector (in the direction
* of increasing time values).
* @param t Time value where the ray should start
* @return Ray starting at t and going in the direction of the vector */
Ray ray(Coord t) {
Ray result;
result.setOrigin(pointAt(t));
result.setVector(vector());
return result;
}
/** @brief Create a derivative of the line.
* The new line will always be degenerate. Its origin will be equal to this
* line's vector. */
Line derivative() const {
Point v = vector();
Line result(v, v);
return result;
}
/// Create a line transformed by an affine transformation.
Line transformed(Affine const& m) const {
Line l(_initial * m, _final * m);
return l;
}
/** @brief Get a unit vector normal to the line.
* If Y grows upwards, then this is the left normal. If Y grows downwards,
* then this is the right normal. */
Point normal() const {
return rot90(vector()).normalized();
}
// what does this do?
Point normalAndDist(double & dist) const {
Point n = normal();
dist = -dot(n, _initial);
return n;
}
/// Compute an affine matrix representing a reflection about the line.
Affine reflection() const {
Point v = versor();
Coord x2 = v[X]*v[X], y2 = v[Y]*v[Y], xy = v[X]*v[Y];
Affine m(x2-y2, 2.*xy,
2.*xy, y2-x2,
_initial[X], _initial[Y]);
m = Translate(-_initial) * m;
return m;
}
/** @brief Compute an affine which transforms all points on the line to zero X or Y coordinate.
* This operation is useful in reducing intersection problems to root-finding problems.
* There are many affines which do this transformation. This function returns one that
* preserves angles, areas and distances - a rotation combined with a translation, and
* additionaly moves the initial point of the line to (0,0). This way it works without
* problems even for lines perpendicular to the target, though may in some cases have
* lower precision than e.g. a shear transform.
* @param d Which coordinate of points on the line should be zero after the transformation */
Affine rotationToZero(Dim2 d) const {
Point v = vector();
if (d == X) {
std::swap(v[X], v[Y]);
} else {
v[Y] = -v[Y];
}
Affine m = Translate(-_initial) * Rotate(v);
return m;
}
/** @brief Compute a rotation affine which transforms the line to one of the axes.
* @param d Which line should be the axis */
Affine rotationToAxis(Dim2 d) const {
Affine m = rotationToZero(other_dimension(d));
return m;
}
Affine transformTo(Line const &other) const;
/// @}
std::vector<ShapeIntersection> intersect(Line const &other) const;
std::vector<ShapeIntersection> intersect(Ray const &r) const;
std::vector<ShapeIntersection> intersect(LineSegment const &ls) const;
template <typename T>
Line &operator*=(T const &tr) {
BOOST_CONCEPT_ASSERT((TransformConcept<T>));
_initial *= tr;
_final *= tr;
return *this;
}
bool operator==(Line const &other) const {
if (distance(pointAt(nearestTime(other._initial)), other._initial) != 0) return false;
if (distance(pointAt(nearestTime(other._final)), other._final) != 0) return false;
return true;
}
template <typename T>
friend Line operator*(Line const &l, T const &tr) {
BOOST_CONCEPT_ASSERT((TransformConcept<T>));
Line result(l);
result *= tr;
return result;
}
}; // end class Line
/** @brief Removes intersections outside of the unit interval.
* A helper used to implement line segment intersections.
* @param xs Line intersections
* @param a Whether the first time value has to be in the unit interval
* @param b Whether the second time value has to be in the unit interval
* @return Appropriately filtered intersections */
void filter_line_segment_intersections(std::vector<ShapeIntersection> &xs, bool a=false, bool b=true);
void filter_ray_intersections(std::vector<ShapeIntersection> &xs, bool a=false, bool b=true);
/// @brief Compute distance from point to line.
/// @relates Line
inline
double distance(Point const &p, Line const &line)
{
if (line.isDegenerate()) {
return ::Geom::distance(p, line.initialPoint());
} else {
Coord t = line.nearestTime(p);
return ::Geom::distance(line.pointAt(t), p);
}
}
inline
bool are_near(Point const &p, Line const &line, double eps = EPSILON)
{
return are_near(distance(p, line), 0, eps);
}
inline
bool are_parallel(Line const &l1, Line const &l2, double eps = EPSILON)
{
return are_near(cross(l1.vector(), l2.vector()), 0, eps);
}
/** @brief Test whether two lines are approximately the same.
* This tests for being parallel and the origin of one line being close to the other,
* so it tests whether the images of the lines are similar, not whether the same time values
* correspond to similar points. For example a line from (1,1) to (2,2) and a line from
* (-1,-1) to (0,0) will the the same, because their images match, even though there is
* no time value for which the lines give similar points.
* @relates Line */
inline
bool are_same(Line const &l1, Line const &l2, double eps = EPSILON)
{
return are_parallel(l1, l2, eps) && are_near(l1.origin(), l2, eps);
}
/// Test whether two lines are perpendicular.
/// @relates Line
inline
bool are_orthogonal(Line const &l1, Line const &l2, double eps = EPSILON)
{
return are_near(dot(l1.vector(), l2.vector()), 0, eps);
}
// evaluate the angle between l1 and l2 rotating l1 in cw direction
// until it overlaps l2
// the returned value is an angle in the interval [0, PI[
inline
double angle_between(Line const& l1, Line const& l2)
{
double angle = angle_between(l1.vector(), l2.vector());
if (angle < 0) angle += M_PI;
if (angle == M_PI) angle = 0;
return angle;
}
inline
double distance(Point const &p, LineSegment const &seg)
{
double t = seg.nearestTime(p);
return distance(p, seg.pointAt(t));
}
inline
bool are_near(Point const &p, LineSegment const &seg, double eps = EPSILON)
{
return are_near(distance(p, seg), 0, eps);
}
// build a line passing by _point and orthogonal to _line
inline
Line make_orthogonal_line(Point const &p, Line const &line)
{
Point d = line.vector().cw();
Line l(p, p + d);
return l;
}
// build a line passing by _point and parallel to _line
inline
Line make_parallel_line(Point const &p, Line const &line)
{
Line result(line);
result.setOrigin(p);
return result;
}
// build a line passing by the middle point of _segment and orthogonal to it.
inline
Line make_bisector_line(LineSegment const& _segment)
{
return make_orthogonal_line( middle_point(_segment), Line(_segment) );
}
// build the bisector line of the angle between ray(O,A) and ray(O,B)
inline
Line make_angle_bisector_line(Point const &A, Point const &O, Point const &B)
{
AngleInterval ival(Angle(A-O), Angle(B-O));
Angle bisect = ival.angleAt(0.5);
return Line(O, bisect);
}
// prj(P) = rot(v, Point( rot(-v, P-O)[X], 0 )) + O
inline
Point projection(Point const &p, Line const &line)
{
return line.pointAt(line.nearestTime(p));
}
inline
LineSegment projection(LineSegment const &seg, Line const &line)
{
return line.segment(line.nearestTime(seg.initialPoint()),
line.nearestTime(seg.finalPoint()));
}
inline
boost::optional<LineSegment> clip(Line const &l, Rect const &r) {
return l.clip(r);
}
namespace detail
{
OptCrossing intersection_impl(Ray const& r1, Line const& l2, unsigned int i);
OptCrossing intersection_impl( LineSegment const& ls1,
Line const& l2,
unsigned int i );
OptCrossing intersection_impl( LineSegment const& ls1,
Ray const& r2,
unsigned int i );
}
inline
OptCrossing intersection(Ray const& r1, Line const& l2)
{
return detail::intersection_impl(r1, l2, 0);
}
inline
OptCrossing intersection(Line const& l1, Ray const& r2)
{
return detail::intersection_impl(r2, l1, 1);
}
inline
OptCrossing intersection(LineSegment const& ls1, Line const& l2)
{
return detail::intersection_impl(ls1, l2, 0);
}
inline
OptCrossing intersection(Line const& l1, LineSegment const& ls2)
{
return detail::intersection_impl(ls2, l1, 1);
}
inline
OptCrossing intersection(LineSegment const& ls1, Ray const& r2)
{
return detail::intersection_impl(ls1, r2, 0);
}
inline
OptCrossing intersection(Ray const& r1, LineSegment const& ls2)
{
return detail::intersection_impl(ls2, r1, 1);
}
OptCrossing intersection(Line const& l1, Line const& l2);
OptCrossing intersection(Ray const& r1, Ray const& r2);
OptCrossing intersection(LineSegment const& ls1, LineSegment const& ls2);
} // end namespace Geom
#endif // LIB2GEOM_SEEN_LINE_H
/*
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*/
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