nerdcruft/libcruft-util
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libcruft-util/bezier.cpp

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/*
* Licensed under the Apache License, Version 2.0 (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.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
* Copyright 2015 Danny Robson <danny@nerdcruft.net>
*/
#include "bezier.hpp"
#include "debug.hpp"
#include "polynomial.hpp"
#include <algorithm>
#include <iterator>
//-----------------------------------------------------------------------------
template <size_t S>
util::bezier<S>::bezier (const util::point2f (&_points)[S+1])
{
std::copy (_points, _points + S + 1, m_points);
}
//-----------------------------------------------------------------------------
namespace util {
template <>
point2f
bezier<1>::eval (float t) const
{
CHECK_GE (t, 0);
CHECK_LE (t, 1);
auto v0 = (1 - t) * m_points[0];
auto v1 = t * m_points[1];
return {
v0.x + v1.x,
v0.y + v1.y
};
}
}
//-----------------------------------------------------------------------------
namespace util {
template <>
point2f
bezier<2>::eval (float t) const
{
CHECK_GE (t, 0);
CHECK_LE (t, 1);
auto v0 = pow2 (1 - t) * m_points[0];
auto v1 = 2 * (1 - t) * t * m_points[1];
auto v2 = pow2 (t) * m_points[2];
return {
v0.x + v1.x + v2.x,
v0.y + v1.y + v2.y
};
}
}
//-----------------------------------------------------------------------------
namespace util {
template <>
point2f
bezier<3>::eval (float t) const
{
CHECK_GE (t, 0);
CHECK_LE (t, 1);
auto v0 = pow (1 - t, 3) * m_points[0];
auto v1 = 3 * pow2 (1 - t) * t * m_points[1];
auto v2 = 3 * pow2 (1 - t) * t * m_points[2];
auto v3 = pow (t, 3) * m_points[3];
return {
v0.x + v1.x + v2.x + v3.x,
v0.y + v1.y + v2.y + v3.y
};
}
}
//-----------------------------------------------------------------------------
namespace util {
template <>
float
bezier<1>::distance (util::point2f target) const
{
auto v = m_points[1] - m_points[0];
auto w = target - m_points[0];
auto c1 = dot (w, v);
if (c1 <= 0)
return m_points[0].distance (target);
auto c2 = dot (v, v);
if (c2 <= c1)
return m_points[1].distance (target);
auto b = c1 / c2;
auto p = m_points[0] + b * v;
return p.distance (target);
}
}
//-----------------------------------------------------------------------------
namespace util {
template <>
std::array<util::vector2f,4>
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]
};
}
}
//-----------------------------------------------------------------------------
namespace util {
template <>
std::array<util::vector2f,3>
bezier<2>::coeffs (void) const
{
auto &v = m_coeffs;
return {
+1.f * v[2] -2.f * v[1] + 1.f * v[0],
-2.f * v[2] +2.f * v[1],
+1.f * v[2]
};
}
}
//-----------------------------------------------------------------------------
namespace util {
template <>
std::array<util::vector2f,2>
bezier<1>::coeffs (void) const
{
auto &v = m_coeffs;
return {
-1.f * v[1] + 1.f * v[0],
+1.f * v[1],
};
}
}
//-----------------------------------------------------------------------------
// XXX: If the line is co-linear we'll have no solutions. But we return 1
// anyway as this function is used to find any point that intersects as part
// of other more comprehensive tests.
template <size_t S>
size_t
util::bezier<S>::intersections (point2f p0, point2f p1) const
{
float A = p1.y - p0.y; // A = y2 - y1
float B = p0.x - p1.x; // B = x1 - x2
float C = p0.x * (p0.y - p1.y) + // C = x1 (y1 - y2) + y1 (x2 - x1)
p0.y * (p1.x - p0.x);
// Build the intersection polynomial
const std::array<vector2f,S+1> bcoeff = coeffs ();
std::array<float,S+1> pcoeff;
for (size_t i = 0; i < pcoeff.size (); ++i)
pcoeff[i] = A * bcoeff[i].x + B * bcoeff[i].y;
pcoeff.back () += C;
const auto r = polynomial::solve<S> (pcoeff);
// The curve and line are colinear
if (std::all_of (r.begin (), r.end (), [] (auto i) { return std::isnan (i); }))
return 1;
size_t count = 0;
for (size_t i = 0; i < S; ++i) {
// Ensure the solutions are on the curve
const auto t = r[i];
if (std::isnan (t))
break;
if (t < 0.f || t > 1.f)
continue;
// Find the line's intersection point
const util::vector2f q = polynomial::eval (bcoeff, t);
const auto s = almost_equal (p0.x, p1.x) ?
(q.y-p0.y) / (p1.y-p0.y) :
(q.x-p0.x) / (p1.x-p0.x) ; // vertical
// Check if the point is on the line
if (s >= 0.f && s <= 1.f)
++count;
}
return count;
}
//-----------------------------------------------------------------------------
namespace util {
template <>
float
bezier<2>::distance (util::point2f target) const
{
// Using procedure from: http://blog.gludion.com/2009/08/distance-to-quadratic-bezier-curve.html
auto p0 = m_points[0];
auto p1 = m_points[1];
auto p2 = m_points[2];
// Parametric form: P(t) = (1-t)^2*P0 + 2t(1-t)P1 + t^2*P2
//
// Derivative: dP/dt = -2(1-t)P0 + 2(1-2t)P1 + 2tP2
// = 2(A+Bt), A=(P1-P0), B=(P2-P1-A)
//
auto A = p1 - p0;
auto B = p2 - p1 - A;
// Make: dot(target, dP/dt) == 0
// dot (M - P(t), A+Bt) == 0
//
// Solve: at^3 + bt^2 + ct + d,
// a = B^2,
// b = 3A.B,
// c = 2A^2+M'.B,
// d = M'.A,
// M' = P0-M
const auto M = target;
const auto M_ = p0 - M;
float a = dot (B, B);
float b = 3.f * dot (A, B);
float c = 2.f * dot (A, A) + dot (M_, B);
float d = dot (M_, A);
// We have our cubic, so pass off to the solver
auto solutions = util::polynomial::solve<3> ({a, b, c, d});
// Find the smallest distance and return
float dist = std::numeric_limits<float>::infinity ();
for (auto t: solutions) {
if (std::isnan (t))
continue;
if (t <= 0)
dist = min (dist, p0.distance (target));
else if (t > 1)
dist = min (p2.distance (target));
else {
auto p = eval (t);
dist = min (dist, p.distance (target));
}
}
return dist;
}
}
//-----------------------------------------------------------------------------
float refine_cubic (util::bezier<3> b,
util::point2f target,
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);
util::point2f p_l = b.eval (t_l);
util::point2f p_r = b.eval (t_r);
float d_l = p_l.distance (target);
float d_r = p_r.distance (target);
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 util {
// TODO: use a more reliable method like [Xiao-Dia Chen 2010]
template <>
float
bezier<3>::distance (util::point2f target) const
{
static constexpr size_t SUBDIV = 32;
std::array<util::point2f, SUBDIV> lookup;
for (size_t i = 0; i < SUBDIV; ++i)
lookup[i] = eval (i / float (SUBDIV - 1));
size_t best = 0;
for (size_t i = 1; i < lookup.size (); ++i)
if (lookup[i].distance2 (target) < lookup[best].distance2 (target))
best = i;
return refine_cubic (*this,
target,
best / float (SUBDIV - 1),
lookup[best].distance (target),
1.f / SUBDIV);
}
}
//-----------------------------------------------------------------------------
template <size_t S>
util::region2f
util::bezier<S>::region (void) const
{
float x0 = m_points[0].x;
float y0 = m_points[0].y;
float x1 = x0;
float y1 = y0;
for (size_t i = 1; i < S+1; ++i) {
x0 = min (x0, m_points[i].x);
y0 = min (y0, m_points[i].y);
x1 = max (x1, m_points[i].x);
y1 = max (y1, m_points[i].y);
}
util::point2f p0 { x0, y0 };
util::point2f p1 { x1, y1 };
return { p0, p1 };
}
//-----------------------------------------------------------------------------
template <size_t S>
util::point2f&
util::bezier<S>::operator[] (size_t idx)
{
CHECK_LE (idx, S);
return m_points[idx];
}
//-----------------------------------------------------------------------------
template <size_t S>
const util::point2f&
util::bezier<S>::operator[] (size_t idx) const
{
CHECK_LE (idx, S);
return m_points[idx];
}
//-----------------------------------------------------------------------------
template <size_t S>
std::ostream&
util::operator<< (std::ostream &os, const bezier<S> &b)
{
os << b[0];
for (size_t i = 1; i < S+1; ++i)
os << ", " << b[i];
return os;
}
//-----------------------------------------------------------------------------
#define INSTANTIATE(S) \
template class util::bezier<S>; \
template std::ostream& util::operator<< (std::ostream&, const bezier<S>&);
INSTANTIATE(1)
INSTANTIATE(2)
INSTANTIATE(3)
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