/*
* Copyright (c) 2021 - 2026 ThorVG project. All rights reserved.
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
* The above copyright notice and this permission notice shall be included in all
* copies or substantial portions of the Software.
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
* SOFTWARE.
*/
#include "tvgMath.h"
#include "tvgArray.h"
#define BEZIER_EPSILON 1e-2f
/************************************************************************/
/* Internal Class Implementation */
/************************************************************************/
static float _lineLengthApprox(const Point& pt1, const Point& pt2)
{
/* approximate sqrt(x*x + y*y) using alpha max plus beta min algorithm.
With alpha = 1, beta = 3/8, giving results with the largest error less
than 7% compared to the exact value. */
Point diff = {pt2.x - pt1.x, pt2.y - pt1.y};
if (diff.x < 0) diff.x = -diff.x;
if (diff.y < 0) diff.y = -diff.y;
return (diff.x > diff.y) ? (diff.x + diff.y * 0.375f) : (diff.y + diff.x * 0.375f);
}
static float _lineLength(const Point& pt1, const Point& pt2)
{
Point diff = {pt2.x - pt1.x, pt2.y - pt1.y};
return sqrtf(diff.x * diff.x + diff.y * diff.y);
}
template<typename LengthFunc>
float _bezLength(const Bezier& cur, LengthFunc lineLengthFunc)
{
Bezier left, right;
auto len = lineLengthFunc(cur.start, cur.ctrl1) + lineLengthFunc(cur.ctrl1, cur.ctrl2) + lineLengthFunc(cur.ctrl2, cur.end);
auto chord = lineLengthFunc(cur.start, cur.end);
if (fabsf(len - chord) > BEZIER_EPSILON) {
cur.split(left, right);
return _bezLength(left, lineLengthFunc) + _bezLength(right, lineLengthFunc);
}
return len;
}
template<typename LengthFunc>
float _bezAt(const Bezier& bz, float at, float length, LengthFunc lineLengthFunc)
{
auto biggest = 1.0f;
auto smallest = 0.0f;
auto t = 0.5f;
//just in case to prevent an infinite loop
if (at <= 0) return 0.0f;
if (at >= length) return 1.0f;
while (true) {
auto right = bz;
Bezier left;
right.split(t, left);
length = _bezLength(left, lineLengthFunc);
if (fabsf(length - at) < BEZIER_EPSILON || fabsf(smallest - biggest) < 1e-3f) {
break;
}
if (length < at) {
smallest = t;
t = (t + biggest) * 0.5f;
} else {
biggest = t;
t = (smallest + t) * 0.5f;
}
}
return t;
}
/************************************************************************/
/* External Class Implementation */
/************************************************************************/
namespace tvg {
uint8_t lerp(const uint8_t &start, const uint8_t &end, float t)
{
return static_cast<uint8_t>(tvg::clamp(static_cast<int>(start + (end - start) * t), 0, 255));
}
float length(const PathCommand* cmds, uint32_t cmdsCnt, const Point* pts, uint32_t ptsCnt)
{
if (ptsCnt < 2) return 0.0f;
auto start = pts;
auto length = 0.0f;
while (cmdsCnt-- > 0) {
switch (*cmds) {
case PathCommand::Close: {
length += tvg::length(*(pts - 1), *start);
break;
}
case PathCommand::MoveTo: {
start = pts;
++pts;
break;
}
case PathCommand::LineTo: {
length += tvg::length(*(pts - 1), *pts);
++pts;
break;
}
case PathCommand::CubicTo: {
length += Bezier{*(pts - 1), *pts, *(pts + 1), *(pts + 2)}.length();
pts += 3;
break;
}
}
++cmds;
}
return length;
}
//https://en.wikipedia.org/wiki/Remez_algorithm
float atan2(float y, float x)
{
if (y == 0.0f && x == 0.0f) return 0.0f;
auto a = std::min(fabsf(x), fabsf(y)) / std::max(fabsf(x), fabsf(y));
auto s = a * a;
auto r = ((-0.0464964749f * s + 0.15931422f) * s - 0.327622764f) * s * a + a;
if (fabsf(y) > fabsf(x)) r = 1.57079637f - r;
if (x < 0) r = 3.14159274f - r;
if (y < 0) return -r;
return r;
}
bool inverse(const Matrix* m, Matrix* out)
{
auto det = m->e11 * (m->e22 * m->e33 - m->e32 * m->e23) -
m->e12 * (m->e21 * m->e33 - m->e23 * m->e31) +
m->e13 * (m->e21 * m->e32 - m->e22 * m->e31);
auto invDet = 1.0f / det;
if (std::isinf(invDet)) return false;
out->e11 = (m->e22 * m->e33 - m->e32 * m->e23) * invDet;
out->e12 = (m->e13 * m->e32 - m->e12 * m->e33) * invDet;
out->e13 = (m->e12 * m->e23 - m->e13 * m->e22) * invDet;
out->e21 = (m->e23 * m->e31 - m->e21 * m->e33) * invDet;
out->e22 = (m->e11 * m->e33 - m->e13 * m->e31) * invDet;
out->e23 = (m->e21 * m->e13 - m->e11 * m->e23) * invDet;
out->e31 = (m->e21 * m->e32 - m->e31 * m->e22) * invDet;
out->e32 = (m->e31 * m->e12 - m->e11 * m->e32) * invDet;
out->e33 = (m->e11 * m->e22 - m->e21 * m->e12) * invDet;
return true;
}
bool identity(const Matrix* m)
{
if (m->e11 != 1.0f || m->e12 != 0.0f || m->e13 != 0.0f ||
m->e21 != 0.0f || m->e22 != 1.0f || m->e23 != 0.0f ||
m->e31 != 0.0f || m->e32 != 0.0f || m->e33 != 1.0f) {
return false;
}
return true;
}
void rotate(Matrix* m, float degree)
{
if (degree == 0.0f) return;
auto radian = degree / 180.0f * MATH_PI;
auto cosVal = cosf(radian);
auto sinVal = sinf(radian);
m->e12 = m->e11 * -sinVal;
m->e11 *= cosVal;
m->e21 = m->e22 * sinVal;
m->e22 *= cosVal;
}
Matrix operator*(const Matrix& lhs, const Matrix& rhs)
{
Matrix m;
m.e11 = lhs.e11 * rhs.e11 + lhs.e12 * rhs.e21 + lhs.e13 * rhs.e31;
m.e12 = lhs.e11 * rhs.e12 + lhs.e12 * rhs.e22 + lhs.e13 * rhs.e32;
m.e13 = lhs.e11 * rhs.e13 + lhs.e12 * rhs.e23 + lhs.e13 * rhs.e33;
m.e21 = lhs.e21 * rhs.e11 + lhs.e22 * rhs.e21 + lhs.e23 * rhs.e31;
m.e22 = lhs.e21 * rhs.e12 + lhs.e22 * rhs.e22 + lhs.e23 * rhs.e32;
m.e23 = lhs.e21 * rhs.e13 + lhs.e22 * rhs.e23 + lhs.e23 * rhs.e33;
m.e31 = lhs.e31 * rhs.e11 + lhs.e32 * rhs.e21 + lhs.e33 * rhs.e31;
m.e32 = lhs.e31 * rhs.e12 + lhs.e32 * rhs.e22 + lhs.e33 * rhs.e32;
m.e33 = lhs.e31 * rhs.e13 + lhs.e32 * rhs.e23 + lhs.e33 * rhs.e33;
return m;
}
bool operator==(const Matrix& lhs, const Matrix& rhs)
{
if (!tvg::equal(lhs.e11, rhs.e11) || !tvg::equal(lhs.e12, rhs.e12) || !tvg::equal(lhs.e13, rhs.e13) ||
!tvg::equal(lhs.e21, rhs.e21) || !tvg::equal(lhs.e22, rhs.e22) || !tvg::equal(lhs.e23, rhs.e23) ||
!tvg::equal(lhs.e31, rhs.e31) || !tvg::equal(lhs.e32, rhs.e32) || !tvg::equal(lhs.e33, rhs.e33)) {
return false;
}
return true;
}
void operator*=(Point& pt, const Matrix& m)
{
auto tx = pt.x * m.e11 + pt.y * m.e12 + m.e13;
auto ty = pt.x * m.e21 + pt.y * m.e22 + m.e23;
pt.x = tx;
pt.y = ty;
}
Point operator*(const Point& pt, const Matrix& m)
{
auto tx = pt.x * m.e11 + pt.y * m.e12 + m.e13;
auto ty = pt.x * m.e21 + pt.y * m.e22 + m.e23;
return {tx, ty};
}
Point normal(const Point& p1, const Point& p2)
{
auto dir = p2 - p1;
auto len = length(dir);
if (tvg::zero(len)) return {};
auto unitDir = dir / len;
return {-unitDir.y, unitDir.x};
}
void normalize(Point& pt)
{
if (zero(pt)) return; //prevent zero division
auto ilength = 1.0f / sqrtf((pt.x * pt.x) + (pt.y * pt.y));
pt.x *= ilength;
pt.y *= ilength;
}
float Line::length() const
{
return _lineLength(pt1, pt2);
}
void Line::split(float at, Line& left, Line& right) const
{
auto len = length();
auto dx = ((pt2.x - pt1.x) / len) * at;
auto dy = ((pt2.y - pt1.y) / len) * at;
left.pt1 = pt1;
left.pt2.x = left.pt1.x + dx;
left.pt2.y = left.pt1.y + dy;
right.pt1 = left.pt2;
right.pt2 = pt2;
}
Bezier::Bezier(const Point& st, const Point& ed, float radius)
{
// Calculate the angle between the start and end points
auto angle = tvg::atan2(ed.y - st.y, ed.x - st.x);
// Calculate the control points of the cubic bezier curve
auto c = radius * PATH_KAPPA; // c = radius * (4/3) * tan(pi/8)
start = {st.x, st.y};
ctrl1 = {st.x + radius * cos(angle), st.y + radius * sin(angle)};
ctrl2 = {ed.x - c * cos(angle), ed.y - c * sin(angle)};
end = {ed.x, ed.y};
}
void Bezier::split(Bezier& left, Bezier& right) const
{
auto c = (ctrl1.x + ctrl2.x) * 0.5f;
left.ctrl1.x = (start.x + ctrl1.x) * 0.5f;
right.ctrl2.x = (ctrl2.x + end.x) * 0.5f;
left.start.x = start.x;
right.end.x = end.x;
left.ctrl2.x = (left.ctrl1.x + c) * 0.5f;
right.ctrl1.x = (right.ctrl2.x + c) * 0.5f;
left.end.x = right.start.x = (left.ctrl2.x + right.ctrl1.x) * 0.5f;
c = (ctrl1.y + ctrl2.y) * 0.5f;
left.ctrl1.y = (start.y + ctrl1.y) * 0.5f;
right.ctrl2.y = (ctrl2.y + end.y) * 0.5f;
left.start.y = start.y;
right.end.y = end.y;
left.ctrl2.y = (left.ctrl1.y + c) * 0.5f;
right.ctrl1.y = (right.ctrl2.y + c) * 0.5f;
left.end.y = right.start.y = (left.ctrl2.y + right.ctrl1.y) * 0.5f;
}
void Bezier::split(float at, Bezier& left, Bezier& right) const
{
right = *this;
auto t = right.at(at, right.length());
right.split(t, left);
}
float Bezier::length() const
{
return _bezLength(*this, _lineLength);
}
float Bezier::lengthApprox() const
{
return _bezLength(*this, _lineLengthApprox);
}
void Bezier::split(float t, Bezier& left)
{
left.start = start;
left.ctrl1.x = start.x + t * (ctrl1.x - start.x);
left.ctrl1.y = start.y + t * (ctrl1.y - start.y);
left.ctrl2.x = ctrl1.x + t * (ctrl2.x - ctrl1.x); //temporary holding spot
left.ctrl2.y = ctrl1.y + t * (ctrl2.y - ctrl1.y); //temporary holding spot
ctrl2.x = ctrl2.x + t * (end.x - ctrl2.x);
ctrl2.y = ctrl2.y + t * (end.y - ctrl2.y);
ctrl1.x = left.ctrl2.x + t * (ctrl2.x - left.ctrl2.x);
ctrl1.y = left.ctrl2.y + t * (ctrl2.y - left.ctrl2.y);
left.ctrl2.x = left.ctrl1.x + t * (left.ctrl2.x - left.ctrl1.x);
left.ctrl2.y = left.ctrl1.y + t * (left.ctrl2.y - left.ctrl1.y);
left.end.x = start.x = left.ctrl2.x + t * (ctrl1.x - left.ctrl2.x);
left.end.y = start.y = left.ctrl2.y + t * (ctrl1.y - left.ctrl2.y);
}
float Bezier::at(float at, float length) const
{
return _bezAt(*this, at, length, _lineLength);
}
float Bezier::atApprox(float at, float length) const
{
return _bezAt(*this, at, length, _lineLengthApprox);
}
Point Bezier::at(float t) const
{
Point cur;
auto it = 1.0f - t;
auto ax = start.x * it + ctrl1.x * t;
auto bx = ctrl1.x * it + ctrl2.x * t;
auto cx = ctrl2.x * it + end.x * t;
ax = ax * it + bx * t;
bx = bx * it + cx * t;
cur.x = ax * it + bx * t;
float ay = start.y * it + ctrl1.y * t;
float by = ctrl1.y * it + ctrl2.y * t;
float cy = ctrl2.y * it + end.y * t;
ay = ay * it + by * t;
by = by * it + cy * t;
cur.y = ay * it + by * t;
return cur;
}
float Bezier::angle(float t) const
{
if (t < 0 || t > 1) return 0;
//derivate
// p'(t) = 3 * (-(1-2t+t^2) * p0 + (1 - 4 * t + 3 * t^2) * p1 + (2 * t - 3 *
// t^2) * p2 + t^2 * p3)
float mt = 1.0f - t;
float d = t * t;
float a = -mt * mt;
float b = 1 - 4 * t + 3 * d;
float c = 2 * t - 3 * d;
Point pt ={a * start.x + b * ctrl1.x + c * ctrl2.x + d * end.x, a * start.y + b * ctrl1.y + c * ctrl2.y + d * end.y};
pt.x *= 3;
pt.y *= 3;
return rad2deg(tvg::atan2(pt.y, pt.x));
}
void Bezier::bounds(BBox& box, const Point& start, const Point& ctrl1, const Point& ctrl2, const Point& end)
{
if (box.min.x > start.x) box.min.x = start.x;
if (box.min.y > start.y) box.min.y = start.y;
if (box.min.x > end.x) box.min.x = end.x;
if (box.min.y > end.y) box.min.y = end.y;
if (box.max.x < start.x) box.max.x = start.x;
if (box.max.y < start.y) box.max.y = start.y;
if (box.max.x < end.x) box.max.x = end.x;
if (box.max.y < end.y) box.max.y = end.y;
//find x/y-direction extrema (solving derivative of Bezier curve)
auto findMinMax = [&](float start, float ctrl1, float ctrl2, float end, float& min, float& max) -> void {
auto a = -1.0f * start + 3.0f * ctrl1 - 3.0f * ctrl2 + end;
auto b = start - 2.0f * ctrl1 + ctrl2;
auto c = -1.0f * start + ctrl1;
auto h = b * b - a * c;
if (h <= 0.0f) return;
h = sqrtf(h);
float t[2] = {(-b - h) / a, (-b + h) / a};
for (int i = 0; i < 2; ++i) {
if (t[i] <= 0.0f || t[i] >= 1.0f) continue;
auto s = 1.0f - t[i];
auto q = s * s * s * start + 3.0f * s * s * t[i] * ctrl1 + 3.0f * s * t[i] * t[i] * ctrl2 + t[i] * t[i] * t[i] * end;
if (q < min) min = q;
if (q > max) max = q;
}
};
findMinMax(start.x, ctrl1.x, ctrl2.x, end.x, box.min.x, box.max.x);
findMinMax(start.y, ctrl1.y, ctrl2.y, end.y, box.min.y, box.max.y);
}
bool Bezier::flatten() const
{
float diff1_x = fabsf((ctrl1.x * 3.f) - (start.x * 2.f) - end.x);
float diff1_y = fabsf((ctrl1.y * 3.f) - (start.y * 2.f) - end.y);
float diff2_x = fabsf((ctrl2.x * 3.f) - (end.x * 2.f) - start.x);
float diff2_y = fabsf((ctrl2.y * 3.f) - (end.y * 2.f) - start.y);
if (diff1_x < diff2_x) diff1_x = diff2_x;
if (diff1_y < diff2_y) diff1_y = diff2_y;
return (diff1_x + diff1_y <= 0.5f);
}
uint32_t Bezier::segments() const
{
static constexpr uint32_t MaxSegments = 1u << 10; // 2^10 segments cap keeps runtime bounded
uint32_t count = 0;
Array<Bezier> stack;
stack.push(*this);
Bezier left, right;
while (!stack.empty()) {
auto current = stack.last();
stack.pop();
if (current.flatten()) {
++count;
continue;
}
if (count + stack.count + 2 >= MaxSegments) {
TVGERR("Common", "Bezier segments count exceeded the maximum limit.");
return MaxSegments;
}
current.split(left, right);
stack.push(left);
stack.push(right);
}
return count;
}
Bezier Bezier::operator*(const Matrix& m)
{
return Bezier{start * m, ctrl1* m, ctrl2 * m, end * m};
}
} // namespace tvg