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Started thedmd/imgui-node-editor integration
2023-04-21 11:55:36 +02:00
//------------------------------------------------------------------------------
// VERSION 0.1
//
// LICENSE
// This software is dual-licensed to the public domain and under the following
// license: you are granted a perpetual, irrevocable license to copy, modify,
// publish, and distribute this file as you see fit.
//
// CREDITS
// Written by Michal Cichon
//------------------------------------------------------------------------------
# ifndef __IMGUI_BEZIER_MATH_INL__
# define __IMGUI_BEZIER_MATH_INL__
# pragma once
//------------------------------------------------------------------------------
# include "imgui_bezier_math.h"
# include <map> // used in ImCubicBezierFixedStep
//------------------------------------------------------------------------------
template <typename T>
inline T ImLinearBezier(const T& p0, const T& p1, float t)
{
return p0 + t * (p1 - p0);
}
template <typename T>
inline T ImLinearBezierDt(const T& p0, const T& p1, float t)
{
IM_UNUSED(t);
return p1 - p0;
}
template <typename T>
inline T ImQuadraticBezier(const T& p0, const T& p1, const T& p2, float t)
{
const auto a = 1 - t;
return a * a * p0 + 2 * t * a * p1 + t * t * p2;
}
template <typename T>
inline T ImQuadraticBezierDt(const T& p0, const T& p1, const T& p2, float t)
{
return 2 * (1 - t) * (p1 - p0) + 2 * t * (p2 - p1);
}
template <typename T>
inline T ImCubicBezier(const T& p0, const T& p1, const T& p2, const T& p3, float t)
{
const auto a = 1 - t;
const auto b = a * a * a;
const auto c = t * t * t;
return b * p0 + 3 * t * a * a * p1 + 3 * t * t * a * p2 + c * p3;
}
template <typename T>
inline T ImCubicBezierDt(const T& p0, const T& p1, const T& p2, const T& p3, float t)
{
const auto a = 1 - t;
const auto b = a * a;
const auto c = t * t;
const auto d = 2 * t * a;
return -3 * p0 * b + 3 * p1 * (b - d) + 3 * p2 * (d - c) + 3 * p3 * c;
}
template <typename T>
inline T ImCubicBezierSample(const T& p0, const T& p1, const T& p2, const T& p3, float t)
{
const auto cp0_zero = ImLengthSqr(p1 - p0) < 1e-5f;
const auto cp1_zero = ImLengthSqr(p3 - p2) < 1e-5f;
if (cp0_zero && cp1_zero)
return ImLinearBezier(p0, p3, t);
else if (cp0_zero)
return ImQuadraticBezier(p0, p2, p3, t);
else if (cp1_zero)
return ImQuadraticBezier(p0, p1, p3, t);
else
return ImCubicBezier(p0, p1, p2, p3, t);
}
template <typename T>
inline T ImCubicBezierSample(const ImCubicBezierPointsT<T>& curve, float t)
{
return ImCubicBezierSample(curve.P0, curve.P1, curve.P2, curve.P3, t);
}
template <typename T>
inline T ImCubicBezierTangent(const T& p0, const T& p1, const T& p2, const T& p3, float t)
{
const auto cp0_zero = ImLengthSqr(p1 - p0) < 1e-5f;
const auto cp1_zero = ImLengthSqr(p3 - p2) < 1e-5f;
if (cp0_zero && cp1_zero)
return ImLinearBezierDt(p0, p3, t);
else if (cp0_zero)
return ImQuadraticBezierDt(p0, p2, p3, t);
else if (cp1_zero)
return ImQuadraticBezierDt(p0, p1, p3, t);
else
return ImCubicBezierDt(p0, p1, p2, p3, t);
}
template <typename T>
inline T ImCubicBezierTangent(const ImCubicBezierPointsT<T>& curve, float t)
{
return ImCubicBezierTangent(curve.P0, curve.P1, curve.P2, curve.P3, t);
}
template <typename T>
inline float ImCubicBezierLength(const T& p0, const T& p1, const T& p2, const T& p3)
{
// Legendre-Gauss abscissae with n=24 (x_i values, defined at i=n as the roots of the nth order Legendre polynomial Pn(x))
static const float t_values[] =
{
-0.0640568928626056260850430826247450385909f,
0.0640568928626056260850430826247450385909f,
-0.1911188674736163091586398207570696318404f,
0.1911188674736163091586398207570696318404f,
-0.3150426796961633743867932913198102407864f,
0.3150426796961633743867932913198102407864f,
-0.4337935076260451384870842319133497124524f,
0.4337935076260451384870842319133497124524f,
-0.5454214713888395356583756172183723700107f,
0.5454214713888395356583756172183723700107f,
-0.6480936519369755692524957869107476266696f,
0.6480936519369755692524957869107476266696f,
-0.7401241915785543642438281030999784255232f,
0.7401241915785543642438281030999784255232f,
-0.8200019859739029219539498726697452080761f,
0.8200019859739029219539498726697452080761f,
-0.8864155270044010342131543419821967550873f,
0.8864155270044010342131543419821967550873f,
-0.9382745520027327585236490017087214496548f,
0.9382745520027327585236490017087214496548f,
-0.9747285559713094981983919930081690617411f,
0.9747285559713094981983919930081690617411f,
-0.9951872199970213601799974097007368118745f,
0.9951872199970213601799974097007368118745f
};
// Legendre-Gauss weights with n=24 (w_i values, defined by a function linked to in the Bezier primer article)
static const float c_values[] =
{
0.1279381953467521569740561652246953718517f,
0.1279381953467521569740561652246953718517f,
0.1258374563468282961213753825111836887264f,
0.1258374563468282961213753825111836887264f,
0.1216704729278033912044631534762624256070f,
0.1216704729278033912044631534762624256070f,
0.1155056680537256013533444839067835598622f,
0.1155056680537256013533444839067835598622f,
0.1074442701159656347825773424466062227946f,
0.1074442701159656347825773424466062227946f,
0.0976186521041138882698806644642471544279f,
0.0976186521041138882698806644642471544279f,
0.0861901615319532759171852029837426671850f,
0.0861901615319532759171852029837426671850f,
0.0733464814110803057340336152531165181193f,
0.0733464814110803057340336152531165181193f,
0.0592985849154367807463677585001085845412f,
0.0592985849154367807463677585001085845412f,
0.0442774388174198061686027482113382288593f,
0.0442774388174198061686027482113382288593f,
0.0285313886289336631813078159518782864491f,
0.0285313886289336631813078159518782864491f,
0.0123412297999871995468056670700372915759f,
0.0123412297999871995468056670700372915759f
};
static_assert(sizeof(t_values) / sizeof(*t_values) == sizeof(c_values) / sizeof(*c_values), "");
auto arc = [p0, p1, p2, p3](float t)
{
const auto p = ImCubicBezierDt(p0, p1, p2, p3, t);
const auto l = ImLength(p);
return l;
};
const auto z = 0.5f;
const auto n = sizeof(t_values) / sizeof(*t_values);
auto accumulator = 0.0f;
for (size_t i = 0; i < n; ++i)
{
const auto t = z * t_values[i] + z;
accumulator += c_values[i] * arc(t);
}
return z * accumulator;
}
template <typename T>
inline float ImCubicBezierLength(const ImCubicBezierPointsT<T>& curve)
{
return ImCubicBezierLength(curve.P0, curve.P1, curve.P2, curve.P3);
}
template <typename T>
inline ImCubicBezierSplitResultT<T> ImCubicBezierSplit(const T& p0, const T& p1, const T& p2, const T& p3, float t)
{
const auto z1 = t;
const auto z2 = z1 * z1;
const auto z3 = z1 * z1 * z1;
const auto s1 = z1 - 1;
const auto s2 = s1 * s1;
const auto s3 = s1 * s1 * s1;
return ImCubicBezierSplitResultT<T>
{
ImCubicBezierPointsT<T>
{
p0,
z1 * p1 - s1 * p0,
z2 * p2 - 2 * z1 * s1 * p1 + s2 * p0,
z3 * p3 - 3 * z2 * s1 * p2 + 3 * z1 * s2 * p1 - s3 * p0
},
ImCubicBezierPointsT<T>
{
z3 * p0 - 3 * z2 * s1 * p1 + 3 * z1 * s2 * p2 - s3 * p3,
z2 * p1 - 2 * z1 * s1 * p2 + s2 * p3,
z1 * p2 - s1 * p3,
p3,
}
};
}
template <typename T>
inline ImCubicBezierSplitResultT<T> ImCubicBezierSplit(const ImCubicBezierPointsT<T>& curve, float t)
{
return ImCubicBezierSplit(curve.P0, curve.P1, curve.P2, curve.P3, t);
}
inline ImRect ImCubicBezierBoundingRect(const ImVec2& p0, const ImVec2& p1, const ImVec2& p2, const ImVec2& p3)
{
auto a = 3 * p3 - 9 * p2 + 9 * p1 - 3 * p0;
auto b = 6 * p0 - 12 * p1 + 6 * p2;
auto c = 3 * p1 - 3 * p0;
auto delta_squared = ImMul(b, b) - 4 * ImMul(a, c);
auto tl = ImMin(p0, p3);
auto rb = ImMax(p0, p3);
# define IM_VEC2_INDEX(v, i) *(&v.x + i)
for (int i = 0; i < 2; ++i)
{
if (IM_VEC2_INDEX(a, i) == 0.0f)
continue;
if (IM_VEC2_INDEX(delta_squared, i) >= 0)
{
auto delta = ImSqrt(IM_VEC2_INDEX(delta_squared, i));
auto t0 = (-IM_VEC2_INDEX(b, i) + delta) / (2 * IM_VEC2_INDEX(a, i));
if (t0 > 0 && t0 < 1)
{
auto p = ImCubicBezier(IM_VEC2_INDEX(p0, i), IM_VEC2_INDEX(p1, i), IM_VEC2_INDEX(p2, i), IM_VEC2_INDEX(p3, i), t0);
IM_VEC2_INDEX(tl, i) = ImMin(IM_VEC2_INDEX(tl, i), p);
IM_VEC2_INDEX(rb, i) = ImMax(IM_VEC2_INDEX(rb, i), p);
}
auto t1 = (-IM_VEC2_INDEX(b, i) - delta) / (2 * IM_VEC2_INDEX(a, i));
if (t1 > 0 && t1 < 1)
{
auto p = ImCubicBezier(IM_VEC2_INDEX(p0, i), IM_VEC2_INDEX(p1, i), IM_VEC2_INDEX(p2, i), IM_VEC2_INDEX(p3, i), t1);
IM_VEC2_INDEX(tl, i) = ImMin(IM_VEC2_INDEX(tl, i), p);
IM_VEC2_INDEX(rb, i) = ImMax(IM_VEC2_INDEX(rb, i), p);
}
}
}
# undef IM_VEC2_INDEX
return ImRect(tl, rb);
}
inline ImRect ImCubicBezierBoundingRect(const ImCubicBezierPoints& curve)
{
return ImCubicBezierBoundingRect(curve.P0, curve.P1, curve.P2, curve.P3);
}
inline ImProjectResult ImProjectOnCubicBezier(const ImVec2& point, const ImVec2& p0, const ImVec2& p1, const ImVec2& p2, const ImVec2& p3, const int subdivisions)
{
// http://pomax.github.io/bezierinfo/#projections
const float epsilon = 1e-5f;
const float fixed_step = 1.0f / static_cast<float>(subdivisions - 1);
ImProjectResult result;
result.Point = point;
result.Time = 0.0f;
result.Distance = FLT_MAX;
// Step 1: Coarse check
for (int i = 0; i < subdivisions; ++i)
{
auto t = i * fixed_step;
auto p = ImCubicBezier(p0, p1, p2, p3, t);
auto s = point - p;
auto d = ImDot(s, s);
if (d < result.Distance)
{
result.Point = p;
result.Time = t;
result.Distance = d;
}
}
if (result.Time == 0.0f || ImFabs(result.Time - 1.0f) <= epsilon)
{
result.Distance = ImSqrt(result.Distance);
return result;
}
// Step 2: Fine check
auto left = result.Time - fixed_step;
auto right = result.Time + fixed_step;
auto step = fixed_step * 0.1f;
for (auto t = left; t < right + step; t += step)
{
auto p = ImCubicBezier(p0, p1, p2, p3, t);
auto s = point - p;
auto d = ImDot(s, s);
if (d < result.Distance)
{
result.Point = p;
result.Time = t;
result.Distance = d;
}
}
result.Distance = ImSqrt(result.Distance);
return result;
}
inline ImProjectResult ImProjectOnCubicBezier(const ImVec2& p, const ImCubicBezierPoints& curve, const int subdivisions)
{
return ImProjectOnCubicBezier(p, curve.P0, curve.P1, curve.P2, curve.P3, subdivisions);
}
inline ImCubicBezierIntersectResult ImCubicBezierLineIntersect(const ImVec2& p0, const ImVec2& p1, const ImVec2& p2, const ImVec2& p3, const ImVec2& a0, const ImVec2& a1)
{
auto cubic_roots = [](float a, float b, float c, float d, float* roots) -> int
{
int count = 0;
auto sign = [](float x) -> float { return x < 0 ? -1.0f : 1.0f; };
auto A = b / a;
auto B = c / a;
auto C = d / a;
auto Q = (3 * B - ImPow(A, 2)) / 9;
auto R = (9 * A * B - 27 * C - 2 * ImPow(A, 3)) / 54;
auto D = ImPow(Q, 3) + ImPow(R, 2); // polynomial discriminant
if (D >= 0) // complex or duplicate roots
{
auto S = sign(R + ImSqrt(D)) * ImPow(ImFabs(R + ImSqrt(D)), (1.0f / 3.0f));
auto T = sign(R - ImSqrt(D)) * ImPow(ImFabs(R - ImSqrt(D)), (1.0f / 3.0f));
roots[0] = -A / 3 + (S + T); // real root
roots[1] = -A / 3 - (S + T) / 2; // real part of complex root
roots[2] = -A / 3 - (S + T) / 2; // real part of complex root
auto Im = ImFabs(ImSqrt(3) * (S - T) / 2); // complex part of root pair
// discard complex roots
if (Im != 0)
count = 1;
else
count = 3;
}
else // distinct real roots
{
auto th = ImAcos(R / ImSqrt(-ImPow(Q, 3)));
roots[0] = 2 * ImSqrt(-Q) * ImCos(th / 3) - A / 3;
roots[1] = 2 * ImSqrt(-Q) * ImCos((th + 2 * IM_PI) / 3) - A / 3;
roots[2] = 2 * ImSqrt(-Q) * ImCos((th + 4 * IM_PI) / 3) - A / 3;
count = 3;
}
return count;
};
Updated imgui-node-editor to commit 32dc92e
2024-03-03 20:22:06 +01:00
// https://github.com/kaishiqi/Geometric-Bezier/blob/master/GeometricBezier/src/kaishiqi/geometric/intersection/Intersection.as
//
// Start with Bezier using Bernstein polynomials for weighting functions:
// (1-t^3)P0 + 3t(1-t)^2P1 + 3t^2(1-t)P2 + t^3P3
//
// Expand and collect terms to form linear combinations of original Bezier
// controls. This ends up with a vector cubic in t:
// (-P0+3P1-3P2+P3)t^3 + (3P0-6P1+3P2)t^2 + (-3P0+3P1)t + P0
// /\ /\ /\ /\
// || || || ||
// c3 c2 c1 c0
Started thedmd/imgui-node-editor integration
2023-04-21 11:55:36 +02:00
// Calculate the coefficients
auto c3 = -p0 + 3 * p1 - 3 * p2 + p3;
auto c2 = 3 * p0 - 6 * p1 + 3 * p2;
auto c1 = -3 * p0 + 3 * p1;
auto c0 = p0;
// Convert line to normal form: ax + by + c = 0
auto a = a1.y - a0.y;
auto b = a0.x - a1.x;
auto c = a0.x * (a0.y - a1.y) + a0.y * (a1.x - a0.x);
// Rotate each cubic coefficient using line for new coordinate system?
// Find roots of rotated cubic
float roots[3];
auto rootCount = cubic_roots(
a * c3.x + b * c3.y,
a * c2.x + b * c2.y,
a * c1.x + b * c1.y,
a * c0.x + b * c0.y + c,
roots);
// Any roots in closed interval [0,1] are intersections on Bezier, but
// might not be on the line segment.
// Find intersections and calculate point coordinates
auto min = ImMin(a0, a1);
auto max = ImMax(a0, a1);
ImCubicBezierIntersectResult result;
auto points = result.Points;
for (int i = 0; i < rootCount; ++i)
{
auto root = roots[i];
if (0 <= root && root <= 1)
{
// We're within the Bezier curve
// Find point on Bezier
auto p = ImCubicBezier(p0, p1, p2, p3, root);
// See if point is on line segment
// Had to make special cases for vertical and horizontal lines due
// to slight errors in calculation of p00
if (a0.x == a1.x)
{
if (min.y <= p.y && p.y <= max.y)
*points++ = p;
}
else if (a0.y == a1.y)
{
if (min.x <= p.x && p.x <= max.x)
*points++ = p;
}
else if (p.x >= min.x && p.y >= min.y && p.x <= max.x && p.y <= max.y)
{
*points++ = p;
}
}
}
result.Count = static_cast<int>(points - result.Points);
return result;
}
inline ImCubicBezierIntersectResult ImCubicBezierLineIntersect(const ImCubicBezierPoints& curve, const ImLine& line)
{
return ImCubicBezierLineIntersect(curve.P0, curve.P1, curve.P2, curve.P3, line.A, line.B);
}
inline void ImCubicBezierSubdivide(ImCubicBezierSubdivideCallback callback, void* user_pointer, const ImVec2& p0, const ImVec2& p1, const ImVec2& p2, const ImVec2& p3, float tess_tol, ImCubicBezierSubdivideFlags flags)
{
return ImCubicBezierSubdivide(callback, user_pointer, ImCubicBezierPoints{ p0, p1, p2, p3 }, tess_tol, flags);
}
inline void ImCubicBezierSubdivide(ImCubicBezierSubdivideCallback callback, void* user_pointer, const ImCubicBezierPoints& curve, float tess_tol, ImCubicBezierSubdivideFlags flags)
{
struct Tesselator
{
ImCubicBezierSubdivideCallback Callback;
void* UserPointer;
float TesselationTollerance;
ImCubicBezierSubdivideFlags Flags;
void Commit(const ImVec2& p, const ImVec2& t)
{
ImCubicBezierSubdivideSample sample;
sample.Point = p;
sample.Tangent = t;
Callback(sample, UserPointer);
}
void Subdivide(const ImCubicBezierPoints& curve, int level = 0)
{
float dx = curve.P3.x - curve.P0.x;
float dy = curve.P3.y - curve.P0.y;
float d2 = ((curve.P1.x - curve.P3.x) * dy - (curve.P1.y - curve.P3.y) * dx);
float d3 = ((curve.P2.x - curve.P3.x) * dy - (curve.P2.y - curve.P3.y) * dx);
d2 = (d2 >= 0) ? d2 : -d2;
d3 = (d3 >= 0) ? d3 : -d3;
if ((d2 + d3) * (d2 + d3) < TesselationTollerance * (dx * dx + dy * dy))
{
Commit(curve.P3, ImCubicBezierTangent(curve, 1.0f));
}
else if (level < 10)
{
const auto p12 = (curve.P0 + curve.P1) * 0.5f;
const auto p23 = (curve.P1 + curve.P2) * 0.5f;
const auto p34 = (curve.P2 + curve.P3) * 0.5f;
const auto p123 = (p12 + p23) * 0.5f;
const auto p234 = (p23 + p34) * 0.5f;
const auto p1234 = (p123 + p234) * 0.5f;
Subdivide(ImCubicBezierPoints { curve.P0, p12, p123, p1234 }, level + 1);
Subdivide(ImCubicBezierPoints { p1234, p234, p34, curve.P3 }, level + 1);
}
}
};
if (tess_tol < 0)
tess_tol = 1.118f; // sqrtf(1.25f)
Tesselator tesselator;
tesselator.Callback = callback;
tesselator.UserPointer = user_pointer;
tesselator.TesselationTollerance = tess_tol * tess_tol;
tesselator.Flags = flags;
if (!(tesselator.Flags & ImCubicBezierSubdivide_SkipFirst))
tesselator.Commit(curve.P0, ImCubicBezierTangent(curve, 0.0f));
tesselator.Subdivide(curve, 0);
}
template <typename F> inline void ImCubicBezierSubdivide(F& callback, const ImVec2& p0, const ImVec2& p1, const ImVec2& p2, const ImVec2& p3, float tess_tol, ImCubicBezierSubdivideFlags flags)
{
auto handler = [](const ImCubicBezierSubdivideSample& p, void* user_pointer)
{
auto& callback = *reinterpret_cast<F*>(user_pointer);
callback(p);
};
ImCubicBezierSubdivide(handler, &callback, ImCubicBezierPoints{ p0, p1, p2, p3 }, tess_tol, flags);
}
template <typename F> inline void ImCubicBezierSubdivide(F& callback, const ImCubicBezierPoints& curve, float tess_tol, ImCubicBezierSubdivideFlags flags)
{
auto handler = [](const ImCubicBezierSubdivideSample& p, void* user_pointer)
{
auto& callback = *reinterpret_cast<F*>(user_pointer);
callback(p);
};
ImCubicBezierSubdivide(handler, &callback, curve, tess_tol, flags);
}
inline void ImCubicBezierFixedStep(ImCubicBezierFixedStepCallback callback, void* user_pointer, const ImVec2& p0, const ImVec2& p1, const ImVec2& p2, const ImVec2& p3, float step, bool overshoot, float max_value_error, float max_t_error)
{
if (step <= 0.0f || !callback || max_value_error <= 0 || max_t_error <= 0)
return;
ImCubicBezierFixedStepSample sample;
sample.T = 0.0f;
sample.Length = 0.0f;
sample.Point = p0;
sample.BreakSearch = false;
callback(sample, user_pointer);
if (sample.BreakSearch)
return;
const auto total_length = ImCubicBezierLength(p0, p1, p2, p3);
const auto point_count = static_cast<int>(total_length / step) + (overshoot ? 2 : 1);
const auto t_min = 0.0f;
const auto t_max = step * point_count / total_length;
const auto t_0 = (t_min + t_max) * 0.5f;
// #todo: replace map with ImVector + binary search
std::map<float, float> cache;
for (int point_index = 1; point_index < point_count; ++point_index)
{
const auto targetLength = point_index * step;
float t_start = t_min;
float t_end = t_max;
float t = t_0;
float t_best = t;
float error_best = total_length;
while (true)
{
auto cacheIt = cache.find(t);
if (cacheIt == cache.end())
{
const auto front = ImCubicBezierSplit(p0, p1, p2, p3, t).Left;
const auto split_length = ImCubicBezierLength(front);
cacheIt = cache.emplace(t, split_length).first;
}
const auto length = cacheIt->second;
const auto error = targetLength - length;
if (error < error_best)
{
error_best = error;
t_best = t;
}
if (ImFabs(error) <= max_value_error || ImFabs(t_start - t_end) <= max_t_error)
{
sample.T = t;
sample.Length = length;
sample.Point = ImCubicBezier(p0, p1, p2, p3, t);
callback(sample, user_pointer);
if (sample.BreakSearch)
return;
break;
}
else if (error < 0.0f)
t_end = t;
else // if (error > 0.0f)
t_start = t;
t = (t_start + t_end) * 0.5f;
}
}
}
inline void ImCubicBezierFixedStep(ImCubicBezierFixedStepCallback callback, void* user_pointer, const ImCubicBezierPoints& curve, float step, bool overshoot, float max_value_error, float max_t_error)
{
ImCubicBezierFixedStep(callback, user_pointer, curve.P0, curve.P1, curve.P2, curve.P3, step, overshoot, max_value_error, max_t_error);
}
// F has signature void(const ImCubicBezierFixedStepSample& p)
template <typename F>
inline void ImCubicBezierFixedStep(F& callback, const ImVec2& p0, const ImVec2& p1, const ImVec2& p2, const ImVec2& p3, float step, bool overshoot, float max_value_error, float max_t_error)
{
auto handler = [](ImCubicBezierFixedStepSample& sample, void* user_pointer)
{
auto& callback = *reinterpret_cast<F*>(user_pointer);
callback(sample);
};
ImCubicBezierFixedStep(handler, &callback, p0, p1, p2, p3, step, overshoot, max_value_error, max_t_error);
}
template <typename F>
inline void ImCubicBezierFixedStep(F& callback, const ImCubicBezierPoints& curve, float step, bool overshoot, float max_value_error, float max_t_error)
{
auto handler = [](ImCubicBezierFixedStepSample& sample, void* user_pointer)
{
auto& callback = *reinterpret_cast<F*>(user_pointer);
callback(sample);
};
ImCubicBezierFixedStep(handler, &callback, curve.P0, curve.P1, curve.P2, curve.P3, step, overshoot, max_value_error, max_t_error);
}
//------------------------------------------------------------------------------
# endif // __IMGUI_BEZIER_MATH_INL__