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+/**
+ * \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 <optional>
+#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
+ std::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
+ * additionally 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 be 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
+std::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
+
+
+/*
+ Local Variables:
+ mode:c++
+ c-file-style:"stroustrup"
+ c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
+ indent-tabs-mode:nil
+ fill-column:99
+ End:
+*/
+// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 :
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