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309 lines
10 KiB
309 lines
10 KiB
10 months ago
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// David Eberly, Geometric Tools, Redmond WA 98052
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// Copyright (c) 1998-2021
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// Distributed under the Boost Software License, Version 1.0.
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// https://www.boost.org/LICENSE_1_0.txt
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// https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
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// Version: 4.0.2021.03.08
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#pragma once
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#include <Mathematics/Integration.h>
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#include <Mathematics/RootsBisection.h>
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#include <Mathematics/Vector.h>
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namespace gte
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{
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template <int N, typename Real>
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class ParametricCurve
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{
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protected:
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// Abstract base class for a parameterized curve X(t), where t is the
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// parameter in [tmin,tmax] and X is an N-tuple position. The first
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// constructor is for single-segment curves. The second constructor is
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// for multiple-segment curves. The times must be strictly increasing.
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ParametricCurve(Real tmin, Real tmax)
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:
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mTime(2),
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mSegmentLength(1, (Real)0),
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mAccumulatedLength(1, (Real)0),
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mRombergOrder(DEFAULT_ROMBERG_ORDER),
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mMaxBisections(DEFAULT_MAX_BISECTIONS),
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mConstructed(false)
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{
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mTime[0] = tmin;
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mTime[1] = tmax;
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}
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ParametricCurve(int numSegments, Real const* times)
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:
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mTime(numSegments + 1),
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mSegmentLength(numSegments, (Real)0),
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mAccumulatedLength(numSegments, (Real)0),
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mRombergOrder(DEFAULT_ROMBERG_ORDER),
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mMaxBisections(DEFAULT_MAX_BISECTIONS),
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mConstructed(false)
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{
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std::copy(times, times + numSegments + 1, mTime.begin());
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}
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public:
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virtual ~ParametricCurve()
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{
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}
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// To validate construction, create an object as shown:
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// DerivedClassCurve<N, Real> curve(parameters);
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// if (!curve) { <constructor failed, handle accordingly>; }
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inline operator bool() const
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{
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return mConstructed;
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}
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// Member access.
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inline Real GetTMin() const
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{
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return mTime.front();
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}
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inline Real GetTMax() const
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{
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return mTime.back();
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}
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inline int GetNumSegments() const
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{
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return static_cast<int>(mSegmentLength.size());
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}
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inline Real const* GetTimes() const
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{
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return &mTime[0];
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}
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// This function applies only when the first constructor is used (two
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// times rather than a sequence of three or more times).
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void SetTimeInterval(Real tmin, Real tmax)
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{
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if (mTime.size() == 2)
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{
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mTime[0] = tmin;
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mTime[1] = tmax;
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}
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}
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// Parameters used in GetLength(...), GetTotalLength() and
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// GetTime(...).
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// The default value is 8.
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inline void SetRombergOrder(int order)
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{
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mRombergOrder = std::max(order, 1);
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}
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// The default value is 1024.
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inline void SetMaxBisections(unsigned int maxBisections)
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{
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mMaxBisections = std::max(maxBisections, 1u);
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}
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// Evaluation of the curve. The function supports derivative
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// calculation through order 3; that is, order <= 3 is required. If
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// you want/ only the position, pass in order of 0. If you want the
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// position and first derivative, pass in order of 1, and so on. The
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// output array 'jet' must have enough storage to support the maximum
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// order. The values are ordered as: position, first derivative,
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// second derivative, third derivative.
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enum { SUP_ORDER = 4 };
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virtual void Evaluate(Real t, unsigned int order, Vector<N, Real>* jet) const = 0;
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void Evaluate(Real t, unsigned int order, Real* values) const
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{
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Evaluate(t, order, reinterpret_cast<Vector<N, Real>*>(values));
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}
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// Differential geometric quantities.
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Vector<N, Real> GetPosition(Real t) const
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{
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Vector<N, Real> position;
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Evaluate(t, 0, &position);
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return position;
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}
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Vector<N, Real> GetTangent(Real t) const
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{
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std::array<Vector<N, Real>, 2> jet; // (position, tangent)
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Evaluate(t, 1, jet.data());
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Normalize(jet[1]);
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return jet[1];
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}
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Real GetSpeed(Real t) const
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{
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std::array<Vector<N, Real>, 2> jet; // (position, tangent)
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Evaluate(t, 1, jet.data());
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return Length(jet[1]);
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}
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Real GetLength(Real t0, Real t1) const
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{
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std::function<Real(Real)> speed = [this](Real t)
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{
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return GetSpeed(t);
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};
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if (mSegmentLength[0] == (Real)0)
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{
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// Lazy initialization of lengths of segments.
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int const numSegments = static_cast<int>(mSegmentLength.size());
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Real accumulated = (Real)0;
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for (int i = 0; i < numSegments; ++i)
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{
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mSegmentLength[i] = Integration<Real>::Romberg(mRombergOrder,
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mTime[i], mTime[i + 1], speed);
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accumulated += mSegmentLength[i];
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mAccumulatedLength[i] = accumulated;
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}
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}
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t0 = std::max(t0, GetTMin());
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t1 = std::min(t1, GetTMax());
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auto iter0 = std::lower_bound(mTime.begin(), mTime.end(), t0);
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int index0 = static_cast<int>(iter0 - mTime.begin());
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auto iter1 = std::lower_bound(mTime.begin(), mTime.end(), t1);
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int index1 = static_cast<int>(iter1 - mTime.begin());
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Real length;
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if (index0 < index1)
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{
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length = (Real)0;
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if (t0 < *iter0)
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{
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length += Integration<Real>::Romberg(mRombergOrder, t0,
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mTime[index0], speed);
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}
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int isup;
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if (t1 < *iter1)
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{
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length += Integration<Real>::Romberg(mRombergOrder,
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mTime[index1 - 1], t1, speed);
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isup = index1 - 1;
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}
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else
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{
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isup = index1;
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}
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for (int i = index0; i < isup; ++i)
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{
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length += mSegmentLength[i];
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}
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}
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else
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{
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length = Integration<Real>::Romberg(mRombergOrder, t0, t1, speed);
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}
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return length;
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}
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Real GetTotalLength() const
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{
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if (mAccumulatedLength.back() == (Real)0)
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{
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// Lazy evaluation of the accumulated length array.
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return GetLength(mTime.front(), mTime.back());
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}
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return mAccumulatedLength.back();
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}
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// Inverse mapping of s = Length(t) given by t = Length^{-1}(s). The
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// inverse length function generally cannot be written in closed form,
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// in which case it is not directly computable. Instead, we can
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// specify s and estimate the root t for F(t) = Length(t) - s. The
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// derivative is F'(t) = Speed(t) >= 0, so F(t) is nondecreasing. To
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// be robust, we use bisection to locate the root, although it is
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// possible to use a hybrid of Newton's method and bisection. For
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// details, see the document
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// https://www.geometrictools.com/Documentation/MovingAlongCurveSpecifiedSpeed.pdf
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Real GetTime(Real length) const
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{
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if (length > (Real)0)
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{
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if (length < GetTotalLength())
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{
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std::function<Real(Real)> F = [this, &length](Real t)
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{
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return Integration<Real>::Romberg(mRombergOrder,
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mTime.front(), t, [this](Real z) { return GetSpeed(z); })
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- length;
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};
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// We know that F(tmin) < 0 and F(tmax) > 0, which allows us to
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// use bisection. Rather than bisect the entire interval, let's
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// narrow it down with a reasonable initial guess.
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Real ratio = length / GetTotalLength();
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Real omratio = (Real)1 - ratio;
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Real tmid = omratio * mTime.front() + ratio * mTime.back();
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Real fmid = F(tmid);
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if (fmid > (Real)0)
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{
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RootsBisection<Real>::Find(F, mTime.front(), tmid, (Real)-1,
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(Real)1, mMaxBisections, tmid);
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}
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else if (fmid < (Real)0)
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{
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RootsBisection<Real>::Find(F, tmid, mTime.back(), (Real)-1,
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(Real)1, mMaxBisections, tmid);
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}
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return tmid;
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}
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else
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{
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return mTime.back();
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}
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}
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else
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{
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return mTime.front();
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}
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}
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// Compute a subset of curve points according to the specified attribute.
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// The input 'numPoints' must be two or larger.
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void SubdivideByTime(int numPoints, Vector<N, Real>* points) const
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{
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Real delta = (mTime.back() - mTime.front()) / (Real)(numPoints - 1);
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for (int i = 0; i < numPoints; ++i)
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{
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Real t = mTime.front() + delta * i;
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points[i] = GetPosition(t);
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}
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}
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void SubdivideByLength(int numPoints, Vector<N, Real>* points) const
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{
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Real delta = GetTotalLength() / (Real)(numPoints - 1);
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for (int i = 0; i < numPoints; ++i)
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{
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Real length = delta * i;
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Real t = GetTime(length);
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points[i] = GetPosition(t);
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}
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}
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protected:
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enum
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{
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DEFAULT_ROMBERG_ORDER = 8,
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DEFAULT_MAX_BISECTIONS = 1024
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};
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std::vector<Real> mTime;
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mutable std::vector<Real> mSegmentLength;
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mutable std::vector<Real> mAccumulatedLength;
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int mRombergOrder;
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unsigned int mMaxBisections;
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bool mConstructed;
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};
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}
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