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+/**
+ * \file
+ * \brief Symmetric power basis curve
+ *//*
+ * Authors:
+ * MenTaLguY <mental@rydia.net>
+ * Marco Cecchetti <mrcekets at gmail.com>
+ * Krzysztof KosiƄski <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/exception.h>
+#include <2geom/nearest-time.h>
+#include <2geom/sbasis-geometric.h>
+#include <2geom/transforms.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()) {}
+
+ 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(0); }
+ virtual bool isLineSegment() const { return inner[X].size() == 1; }
+ 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 nearestTime( Point const& p, Coord from = 0, Coord to = 1 ) const {
+ return nearest_time(p, inner, from, to);
+ }
+ virtual std::vector<Coord> allNearestTimes( Point const& p, Coord from = 0,
+ Coord to = 1 ) const
+ {
+ return all_nearest_times(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));
+ }
+
+ using Curve::operator*=;
+ virtual void operator*=(Affine const &m) { inner = inner * m; }
+
+ virtual Curve *derivative() const {
+ return new SBasisCurve(Geom::derivative(inner));
+ }
+ virtual D2<SBasis> toSBasis() const { return inner; }
+ virtual bool operator==(Curve const &c) const {
+ SBasisCurve const *other = dynamic_cast<SBasisCurve const *>(&c);
+ if (!other) return false;
+ return inner == other->inner;
+ }
+ virtual bool isNear(Curve const &/*c*/, Coord /*eps*/) const {
+ THROW_NOTIMPLEMENTED();
+ return false;
+ }
+ virtual int degreesOfFreedom() const {
+ return inner[0].degreesOfFreedom() + inner[1].degreesOfFreedom();
+ }
+};
+
+} // 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 :