curve.h revision 76addc201c409e81eaaa73fe27cc0f79c4db097c
/**
* \file
* \brief Abstract curve type
*
*//*
* Authors:
* MenTaLguY <mental@rydia.net>
* Marco Cecchetti <mrcekets at gmail.com>
* Krzysztof Kosiński <tweenk.pl@gmail.com>
*
* Copyright 2007-2009 Authors
*
* 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
*
* 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_CURVE_H
#define LIB2GEOM_SEEN_CURVE_H
#include <vector>
#include <boost/operators.hpp>
typedef Intersection<> CurveIntersection;
/**
* @brief Abstract continuous curve on a plane defined on [0,1].
*
* Formally, a curve in 2Geom is defined as a function
* \f$\mathbf{C}: [0, 1] \to \mathbb{R}^2\f$
* (a function that maps the unit interval to points on a 2D plane). Its image (the set of points
* the curve passes through) will be denoted \f$\mathcal{C} = \mathbf{C}[ [0, 1] ]\f$.
* All curve types available in 2Geom are continuous and differentiable on their
* interior, e.g. \f$(0, 1)\f$. Sometimes the curve's image (value set) is referred to as the curve
* itself for simplicity, but keep in mind that it's not strictly correct.
*
* It is common to think of the parameter as time. The curve can then be interpreted as
* describing the position of some moving object from time \f$t=0\f$ to \f$t=1\f$.
* Because of this, the parameter is frequently called the time value.
*
* Some methods return pointers to newly allocated curves. They are expected to be freed
* by the caller when no longer used. Default implementations are provided for some methods.
*
* @ingroup Curves
*/
{
/// @name Evaluate the curve
/// @{
/** @brief Retrieve the start of the curve.
* @return The point corresponding to \f$\mathbf{C}(0)\f$. */
/** Retrieve the end of the curve.
* @return The point corresponding to \f$\mathbf{C}(1)\f$. */
/** @brief Check whether the curve has exactly zero length.
* @return True if the curve's initial point is exactly the same as its final point, and it contains
* no other points (its value set contains only one element). */
virtual bool isDegenerate() const = 0;
/** @brief Get the interval of allowed time values.
* @return \f$[0, 1]\f$ */
return tr;
}
/** @brief Evaluate the curve at a specified time value.
* @param t Time value
* @return \f$\mathbf{C}(t)\f$ */
/** @brief Evaluate one of the coordinates at the specified time value.
* @param t Time value
* @param d The dimension to evaluate
* @return The specified coordinate of \f$\mathbf{C}(t)\f$ */
/** @brief Evaluate the function at the specified time value. Allows curves to be used
* as functors. */
/** @brief Evaluate the curve and its derivatives.
* This will return a vector that contains the value of the curve and the specified number
* of derivatives. However, the returned vector might contain less elements than specified
* if some derivatives do not exist.
* @param t Time value
* @param n The number of derivatives to compute
* @return Vector of at most \f$n+1\f$ elements of the form \f$[\mathbf{C}(t),
\mathbf{C}'(t), \mathbf{C}''(t), \ldots]\f$ */
/// @}
/// @name Change the curve's endpoints
/// @{
/** @brief Change the starting point of the curve.
* After calling this method, it is guaranteed that \f$\mathbf{C}(0) = \mathbf{p}\f$,
* and the curve is still continuous. The precise new shape of the curve varies with curve
* type.
* @param p New starting point of the curve */
/** @brief Change the ending point of the curve.
* After calling this method, it is guaranteed that \f$\mathbf{C}(0) = \mathbf{p}\f$,
* and the curve is still continuous. The precise new shape of the curve varies
* with curve type.
* @param p New ending point of the curve */
/// @}
/// @name Compute the bounding box
/// @{
/** @brief Quickly compute the curve's approximate bounding box.
* The resulting rectangle is guaranteed to contain all points belonging to the curve,
* but it might not be the smallest such rectangle. This method is usually fast.
* @return A rectangle that contains all points belonging to the curve. */
/** @brief Compute the curve's exact bounding box.
* This method can be dramatically slower than boundsExact() depending on the curve type.
* @return The smallest possible rectangle containing all of the curve's points. */
// I have no idea what the 'deg' parameter is for, so this is undocumented for now.
/** @brief Compute the bounding box of a part of the curve.
* Since this method returns the smallest possible bounding rectangle of the specified portion,
* it can also be rather slow.
* @param a An interval specifying a part of the curve, or nothing.
* If \f$[0, 1] \subseteq a\f$, then the bounding box for the entire curve
* is calculated.
* @return The smallest possible rectangle containing all points in \f$\mathbf{C}[a]\f$,
* or nothing if the supplied interval is empty. */
/// @}
/// @name Create new curves based on this one
/// @{
/** @brief Create an exact copy of this curve.
* @return Pointer to a newly allocated curve, identical to the original */
/** @brief Transform this curve by an affine transformation.
* Because of this method, all curve types must be closed under affine
* transformations.
* @param m Affine describing the affine transformation */
/** @brief Create a curve transformed by an affine transformation.
* This method returns a new curve instead modifying the existing one.
* @param m Affine describing the affine transformation
* @return Pointer to a new, transformed curve */
return ret;
}
/** @brief Create a curve that corresponds to a part of this curve.
* For \f$a > b\f$, the returned portion will be reversed with respect to the original.
* The returned curve will always be of the same type.
* @param a Beginning of the interval specifying the portion of the curve
* @param b End of the interval
* @return New curve \f$\mathbf{D}\f$ such that:
* - \f$\mathbf{D}(0) = \mathbf{C}(a)\f$
* - \f$\mathbf{D}(1) = \mathbf{C}(b)\f$
* - \f$\mathbf{D}[ [0, 1] ] = \mathbf{C}[ [a?b] ]\f$,
* where \f$[a?b] = [\min(a, b), \max(a, b)]\f$ */
/** @brief A version of that accepts an Interval. */
/** @brief Create a reversed version of this curve.
* The result corresponds to <code>portion(1, 0)</code>, but this method might be faster.
* @return Pointer to a new curve \f$\mathbf{D}\f$ such that
* \f$\forall_{x \in [0, 1]} \mathbf{D}(x) = \mathbf{C}(1-x)\f$ */
/** @brief Create a derivative of this curve.
* It's best to think of the derivative in physical terms: if the curve describes
* the position of some object on the plane from time \f$t=0\f$ to \f$t=1\f$ as said in the
* introduction, then the curve's derivative describes that object's speed at the same times.
* The second derivative refers to its acceleration, the third to jerk, etc.
* @return New curve \f$\mathbf{D} = \mathbf{C}'\f$. */
/// @}
/// @name Advanced operations
/// @{
/** @brief Compute a time value at which the curve comes closest to a specified point.
* The first value with the smallest distance is returned if there are multiple such points.
* @param p Query point
* @param a Minimum time value to consider
* @param b Maximum time value to consider; \f$a < b\f$
* @return \f$q \in [a, b]: ||\mathbf{C}(q) - \mathbf{p}|| =
\inf(\{r \in \mathbb{R} : ||\mathbf{C}(r) - \mathbf{p}||\})\f$ */
/** @brief A version that takes an Interval. */
}
/** @brief Compute time values at which the curve comes closest to a specified point.
* @param p Query point
* @param a Minimum time value to consider
* @param b Maximum time value to consider; \f$a < b\f$
* @return Vector of points closest and equally far away from the query point */
/** @brief A version that takes an Interval. */
}
/** @brief Compute the arc length of this curve.
* For a curve \f$\mathbf{C}(t) = (C_x(t), C_y(t))\f$, arc length is defined for 2D curves as
* \f[ \ell = \int_{0}^{1} \sqrt { [C_x'(t)]^2 + [C_y'(t)]^2 }\, \text{d}t \f]
* In other words, we divide the curve into infinitely small linear segments
* and add together their lengths. Of course we can't subdivide the curve into
* infinitely many segments on a computer, so this method returns an approximation.
* Not that there is usually no closed form solution to such integrals, so this
* method might be slow.
* @param tolerance Maximum allowed error
* @return Total distance the curve's value travels on the plane when going from 0 to 1 */
/** @brief Computes time values at which the curve intersects an axis-aligned line.
* @param v The coordinate of the line
* @param d Which axis the coordinate is on. X means a vertical line, Y a horizontal line. */
/** @brief Compute the partial winding number of this curve.
* The partial winding number is equal to the difference between the number
* of roots at which the curve goes in the +Y direction and the number of roots
* at which the curve goes in the -Y direction. This method is mainly useful
* for implementing path winding calculation. It will ignore roots which
* are local maxima on the Y axis.
* @param p Point where the winding number should be determined
* @return Winding number contribution at p */
/// Compute intersections with another curve.
/// Compute intersections of this curve with itself.
/** @brief Compute a vector tangent to the curve.
* This will return an unit vector (a Point with length() equal to 1) that denotes a vector
* tangent to the curve. This vector is defined as
* \f$ \mathbf{v}(t) = \frac{\mathbf{C}'(t)}{||\mathbf{C}'(t)||} \f$. It is pointed
* in the direction of increasing \f$t\f$, at the specfied time value. The method uses
* l'Hopital's rule when the derivative is zero. A zero vector is returned if no non-zero
* derivative could be found.
* @param t Time value
* @param n The maximum order of derivative to consider
* @return Unit tangent vector \f$\mathbf{v}(t)\f$
* @bug This method might currently break for the case of t being exactly 1. A workaround
* is to reverse the curve and use the negated unit tangent at 0 like this:
* @code
Curve *c_reverse = c.reverse();
Point tangent = - c_reverse->unitTangentAt(0);
delete c_reverse; @endcode */
/** @brief Convert the curve to a symmetric power basis polynomial.
* Symmetric power basis polynomials (S-basis for short) are numerical representations
* of curves with excellent numerical properties. Most high level operations provided by 2Geom
* are implemented in terms of S-basis operations, so every curve has to provide a method
* to convert it to an S-basis polynomial on two variables. See SBasis class reference
* for more information. */
/// @}
/// @name Miscellaneous
/// @{
/** Return the number of independent parameters required to represent all variations
* of this curve. For example, for Bezier curves it returns the curve's order
* multiplied by 2. */
virtual int degreesOfFreedom() const { return 0;}
/** @brief Test equality of two curves.
* Equality means that for any time value, the evaluation of either curve will yield
* the same value. This means non-degenerate curves are not equal to their reverses.
* Note that this tests for exact equality.
* @return True if the curves are identical, false otherwise */
/** @brief Feed the curve to a PathSink */
/// @}
};
inline
return c.nearestTime(p);
}
// for make benefit glorious library of Boost Pointer Container
inline
return c.duplicate();
}
} // end namespace Geom
#endif // _2GEOM_CURVE_H_
/*
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*/
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