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/**
* \file
* \brief Symmetric power basis curve
*//*
* Authors:
* MenTaLguY <mental@rydia.net>
* Marco Cecchetti <mrcekets at gmail.com>
* Krzysztof Kosiński <tweenk.pl@gmail.com>
*
* Copyright 2007-2009 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_SBASIS_CURVE_H
#define LIB2GEOM_SEEN_SBASIS_CURVE_H
#include <2geom/curve.h>
#include <2geom/nearest-point.h>
#include <2geom/sbasis-geometric.h>
namespace Geom
{
/** @brief Symmetric power basis curve.
*
* Symmetric power basis (S-basis for short) polynomials are a versatile numeric
* representation of arbitrary continuous curves. They are the main representation of curves
* in 2Geom.
*
* S-basis is defined for odd degrees and composed of the following polynomials:
* \f{align*}{
P_k^0(t) &= t^k (1-t)^{k+1} \\
P_k^1(t) &= t^{k+1} (1-t)^k \f}
* This can be understood more easily with the help of the chart below. Each square
* represents a product of a specific number of \f$t\f$ and \f$(1-t)\f$ terms. Red dots
* are the canonical (monomial) basis, the green dots are the Bezier basis, and the blue
* dots are the S-basis, all of them of degree 7.
*
* @image html sbasis.png "Illustration of the monomial, Bezier and symmetric power bases"
*
* The S-Basis has several important properties:
* - S-basis polynomials are closed under multiplication.
* - Evaluation is fast, using a modified Horner scheme.
* - Degree change is as trivial as in the monomial basis. To elevate, just add extra
* zero coefficients. To reduce the degree, truncate the terms in the highest powers.
* Compare this with Bezier curves, where degree change is complicated.
* - Conversion between S-basis and Bezier basis is numerically stable.
*
* More in-depth information can be found in the following paper:
* J Sanchez-Reyes, "The symmetric analogue of the polynomial power basis".
* ACM Transactions on Graphics, Vol. 16, No. 3, July 1997, pages 319--357.
* http://portal.acm.org/citation.cfm?id=256162
*
* @ingroup Curves
*/
class SBasisCurve : public Curve {
private:
D2<SBasis> inner;
public:
explicit SBasisCurve(D2<SBasis> const &sb) : inner(sb) {}
explicit SBasisCurve(Curve const &other) : inner(other.toSBasis()) {}
#ifndef DOXYGEN_SHOULD_SKIP_THIS
virtual Curve *duplicate() const { return new SBasisCurve(*this); }
virtual Point initialPoint() const { return inner.at0(); }
virtual Point finalPoint() const { return inner.at1(); }
virtual bool isDegenerate() const { return inner.isConstant(); }
virtual Point pointAt(Coord t) const { return inner.valueAt(t); }
virtual std::vector<Point> pointAndDerivatives(Coord t, unsigned n) const {
return inner.valueAndDerivatives(t, n);
}
virtual Coord valueAt(Coord t, Dim2 d) const { return inner[d].valueAt(t); }
virtual void setInitial(Point const &v) {
for (unsigned d = 0; d < 2; d++) { inner[d][0][0] = v[d]; }
}
virtual void setFinal(Point const &v) {
for (unsigned d = 0; d < 2; d++) { inner[d][0][1] = v[d]; }
}
virtual Rect boundsFast() const { return *bounds_fast(inner); }
virtual Rect boundsExact() const { return *bounds_exact(inner); }
virtual OptRect boundsLocal(OptInterval const &i, unsigned deg) const {
return bounds_local(inner, i, deg);
}
virtual std::vector<Coord> roots(Coord v, Dim2 d) const { return Geom::roots(inner[d] - v); }
virtual Coord nearestPoint( Point const& p, Coord from = 0, Coord to = 1 ) const {
return nearest_point(p, inner, from, to);
}
virtual std::vector<Coord> allNearestPoints( Point const& p, Coord from = 0,
Coord to = 1 ) const
{
return all_nearest_points(p, inner, from, to);
}
virtual Coord length(Coord tolerance) const { return ::Geom::length(inner, tolerance); }
virtual Curve *portion(Coord f, Coord t) const {
return new SBasisCurve(Geom::portion(inner, f, t));
}
virtual Curve *transformed(Affine const &m) const { return new SBasisCurve(inner * m); }
virtual Curve *derivative() const {
return new SBasisCurve(Geom::derivative(inner));
}
virtual D2<SBasis> toSBasis() const { return inner; }
virtual int degreesOfFreedom() const {
return inner[0].degreesOfFreedom() + inner[1].degreesOfFreedom();
}
#endif
};
} // end namespace Geom
#endif // LIB2GEOM_SEEN_SBASIS_CURVE_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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