libcruft-util/bezier3.cpp

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/*
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* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
* file, You can obtain one at http://mozilla.org/MPL/2.0/.
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*
* Copyright 2015-2016 Danny Robson <danny@nerdcruft.net>
*/
#include "bezier.hpp"
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#include "coord/iostream.hpp"
#include <array>
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///////////////////////////////////////////////////////////////////////////////
namespace cruft {
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template <>
point2f
bezier<3>::eval (float t) const
{
CHECK_GE (t, 0.f);
CHECK_LE (t, 1.f);
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auto v0 = pow (1 - t, 3u) * m_points[0];
auto v1 = 3 * pow (1 - t, 2u) * t * m_points[1];
auto v2 = 3 * pow (1 - t, 2u) * t * m_points[2];
auto v3 = pow (t, 3u) * m_points[3];
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return {
v0.x + v1.x + v2.x + v3.x,
v0.y + v1.y + v2.y + v3.y
};
}
}
//-----------------------------------------------------------------------------
namespace cruft {
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template <>
std::array<cruft::vector2f,4>
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bezier<3>::coeffs (void) const
{
const auto &v = m_coeffs;
return {
-1.f * v[0] +3.f * v[1] -3.f * v[2] +1.f * v[3],
3.f * v[0] -6.f * v[1] +3.f * v[2],
-3.f * v[0] +3.f * v[1],
1.f * v[0]
};
}
}
//-----------------------------------------------------------------------------
float refine_cubic (cruft::bezier<3> b,
cruft::point2f target,
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float t,
float d,
float p)
{
// TODO: use an iteration of newton before handing back
if (p < 0.00001) {
return t;
}
float t_l = std::max (0.f, t - p);
float t_r = std::min (1.f, t + p);
cruft::point2f p_l = b.eval (t_l);
cruft::point2f p_r = b.eval (t_r);
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float d_l = cruft::distance (target, p_l);
float d_r = cruft::distance (target, p_r);
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if (d_l < d) { return refine_cubic (b, target, t_l, d_l, p); }
if (d_r < d) { return refine_cubic (b, target, t_r, d_r, p); }
return refine_cubic (b, target, t, d, p / 2);
}
//-----------------------------------------------------------------------------
namespace cruft {
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// TODO: use a more reliable method like [Xiao-Dia Chen 2010]
template <>
float
bezier<3>::distance (cruft::point2f target) const noexcept
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{
static constexpr int SUBDIV = 32;
std::array<cruft::point2f, SUBDIV> lookup;
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for (int i = 0; i < SUBDIV; ++i)
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lookup[i] = eval (float (i) / (SUBDIV - 1.f));
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size_t best = 0;
for (size_t i = 1; i < lookup.size (); ++i) {
auto d_i = cruft::distance2 (target, lookup[i]);
auto d_b = cruft::distance2 (target, lookup[best]);
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if (d_i < d_b)
best = i;
}
return refine_cubic (*this,
target,
best / float (SUBDIV - 1),
cruft::distance (target, lookup[best]),
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1.f / SUBDIV);
}
}
///////////////////////////////////////////////////////////////////////////////
namespace cruft {
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template <>
cruft::vector2f
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bezier<3>::tangent (const float t) const
{
CHECK_INCLUSIVE (t, 0.f, 1.f);
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return mix (
mix (m_coeffs[1] - m_coeffs[0], m_coeffs[2] - m_coeffs[1], t),
mix (m_coeffs[2] - m_coeffs[1], m_coeffs[3] - m_coeffs[2], t),
t
);
}
}
//-----------------------------------------------------------------------------
namespace cruft {
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template <>
cruft::vector2f
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bezier<3>::d1 (const float t) const noexcept
{
const auto &P0 = m_points[0];
const auto &P1 = m_points[1];
const auto &P2 = m_points[2];
const auto &P3 = m_points[3];
return 3 * (1 - t) * (1 - t) * (P1 - P0) +
6 * (1 - t) * t * (P2 - P1) +
3 * t * t * (P3 - P2);
}
}
//-----------------------------------------------------------------------------
namespace cruft {
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template <>
cruft::vector2f
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bezier<3>::d2 (const float t) const noexcept
{
const auto &P0 = m_points[0];
const auto &P1 = m_points[1];
const auto &P2 = m_points[2];
const auto &P3 = m_points[3];
return 6 * (1 - t) * (P2 - P1 + P0 - P1) +
6 * t * (P3 - P2 + P1 - P2);
}
}
///////////////////////////////////////////////////////////////////////////////
namespace cruft {
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template <>
sdot_t
bezier<3>::sdot (point2f src) const noexcept
{
const auto ab = m_points[1] - m_points[0];
const auto cd = m_points[3] - m_points[2];
const auto qa = m_points[0] - src;
const auto qd = m_points[3] - src;
// setup variables for minimisation
float d = std::numeric_limits<float>::infinity ();
float t = std::numeric_limits<float>::quiet_NaN ();
// distance from A
const auto d_a = cruft::sign (cross (ab, qa)) * norm (qa);
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if (abs (d_a) < abs (d)) {
d = d_a;
t = -dot (ab, qa) / norm2 (ab);
}
// distance from D
const auto d_d = cruft::sign (cross (cd, qd)) * norm (qd);
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if (abs (d_d) < abs (d)) {
d = d_d;
t = -dot (cd, qd) / norm2 (cd);
}
// Iterative minimum distance search
static constexpr int SUBDIVISIONS = 4;
static constexpr int REFINEMENTS = 4;
for (int i = 0; i <= SUBDIVISIONS; ++i) {
auto r = float (i) / SUBDIVISIONS;
for (int step = 0; ; ++step) {
const auto qp = eval (r) - src;
const auto d_p = sign (cross (tangent (r), qp)) * norm (qp);
if (abs (d_p) < abs (d)) {
d = d_p;
t = r;
}
if (step == REFINEMENTS)
break;
// converge a little using newton's method
const auto d1_ = d1 (r);
const auto d2_ = d2 (r);
r -= dot (qp, d1_) / (dot (d1_, d1_) + dot (qp, d2_));
// bail if it looks like we're going to escape the curve
if (r < 0 || r > 1)
break;
}
}
if (t >= 0 && t <= 1)
return { d, 0 };
if (t < 0)
return { d, abs (dot (normalised (ab), normalised (qa))) };
else
return { d, abs (dot (normalised (cd), normalised (qd))) };
}
}