godot/core/math/math_2d.cpp
Ferenc Arn 97d510531a Fix polar decomposition in 2D.
When performing polar decomposition in 2D as B = R.S, where R is rotation (with determinant +1) and S is scaling, use the convention that reflections are absorbed into S through a reflection around y axis.
In 3D, this is done by using a reflection along all three axes, but since the dimensionality is even in 2D, one axis needs to be chosen.

Fixes Matrix32::get_rotation and Matrix32::get_scale (which weren't properly fixed in #7445).
2017-03-29 12:04:49 -05:00

648 lines
15 KiB
C++

/*************************************************************************/
/* math_2d.cpp */
/*************************************************************************/
/* This file is part of: */
/* GODOT ENGINE */
/* http://www.godotengine.org */
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/* Copyright (c) 2007-2017 Juan Linietsky, Ariel Manzur. */
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#include "math_2d.h"
real_t Vector2::angle() const {
return Math::atan2(y, x);
}
real_t Vector2::length() const {
return Math::sqrt(x * x + y * y);
}
real_t Vector2::length_squared() const {
return x * x + y * y;
}
void Vector2::normalize() {
real_t l = x * x + y * y;
if (l != 0) {
l = Math::sqrt(l);
x /= l;
y /= l;
}
}
Vector2 Vector2::normalized() const {
Vector2 v = *this;
v.normalize();
return v;
}
real_t Vector2::distance_to(const Vector2 &p_vector2) const {
return Math::sqrt((x - p_vector2.x) * (x - p_vector2.x) + (y - p_vector2.y) * (y - p_vector2.y));
}
real_t Vector2::distance_squared_to(const Vector2 &p_vector2) const {
return (x - p_vector2.x) * (x - p_vector2.x) + (y - p_vector2.y) * (y - p_vector2.y);
}
real_t Vector2::angle_to(const Vector2 &p_vector2) const {
return Math::atan2(cross(p_vector2), dot(p_vector2));
}
real_t Vector2::angle_to_point(const Vector2 &p_vector2) const {
return Math::atan2(y - p_vector2.y, x - p_vector2.x);
}
real_t Vector2::dot(const Vector2 &p_other) const {
return x * p_other.x + y * p_other.y;
}
real_t Vector2::cross(const Vector2 &p_other) const {
return x * p_other.y - y * p_other.x;
}
Vector2 Vector2::cross(real_t p_other) const {
return Vector2(p_other * y, -p_other * x);
}
Vector2 Vector2::operator+(const Vector2 &p_v) const {
return Vector2(x + p_v.x, y + p_v.y);
}
void Vector2::operator+=(const Vector2 &p_v) {
x += p_v.x;
y += p_v.y;
}
Vector2 Vector2::operator-(const Vector2 &p_v) const {
return Vector2(x - p_v.x, y - p_v.y);
}
void Vector2::operator-=(const Vector2 &p_v) {
x -= p_v.x;
y -= p_v.y;
}
Vector2 Vector2::operator*(const Vector2 &p_v1) const {
return Vector2(x * p_v1.x, y * p_v1.y);
};
Vector2 Vector2::operator*(const real_t &rvalue) const {
return Vector2(x * rvalue, y * rvalue);
};
void Vector2::operator*=(const real_t &rvalue) {
x *= rvalue;
y *= rvalue;
};
Vector2 Vector2::operator/(const Vector2 &p_v1) const {
return Vector2(x / p_v1.x, y / p_v1.y);
};
Vector2 Vector2::operator/(const real_t &rvalue) const {
return Vector2(x / rvalue, y / rvalue);
};
void Vector2::operator/=(const real_t &rvalue) {
x /= rvalue;
y /= rvalue;
};
Vector2 Vector2::operator-() const {
return Vector2(-x, -y);
}
bool Vector2::operator==(const Vector2 &p_vec2) const {
return x == p_vec2.x && y == p_vec2.y;
}
bool Vector2::operator!=(const Vector2 &p_vec2) const {
return x != p_vec2.x || y != p_vec2.y;
}
Vector2 Vector2::floor() const {
return Vector2(Math::floor(x), Math::floor(y));
}
Vector2 Vector2::rotated(real_t p_by) const {
Vector2 v;
v.set_rotation(angle() + p_by);
v *= length();
return v;
}
Vector2 Vector2::project(const Vector2 &p_vec) const {
Vector2 v1 = p_vec;
Vector2 v2 = *this;
return v2 * (v1.dot(v2) / v2.dot(v2));
}
Vector2 Vector2::snapped(const Vector2 &p_by) const {
return Vector2(
Math::stepify(x, p_by.x),
Math::stepify(y, p_by.y));
}
Vector2 Vector2::clamped(real_t p_len) const {
real_t l = length();
Vector2 v = *this;
if (l > 0 && p_len < l) {
v /= l;
v *= p_len;
}
return v;
}
Vector2 Vector2::cubic_interpolate_soft(const Vector2 &p_b, const Vector2 &p_pre_a, const Vector2 &p_post_b, real_t p_t) const {
#if 0
k[0] = ((*this) (vi[0] + 1, vi[1], vi[2])) - ((*this) (vi[0],
vi[1],vi[2])); //fk = a0
k[1] = (((*this) (vi[0] + 1, vi[1], vi[2])) - ((*this) ((int) (v(0) -
1), vi[1],vi[2])))*0.5; //dk = a1
k[2] = (((*this) ((int) (v(0) + 2), vi[1], vi[2])) - ((*this) (vi[0],
vi[1],vi[2])))*0.5; //dk+1
k[3] = k[0]*3 - k[1]*2 - k[2];//a2
k[4] = k[1] + k[2] - k[0]*2;//a3
//ip = a3(t-tk)³ + a2(t-tk)² + a1(t-tk) + a0
//
//a3 = dk + dk+1 - Dk
//a2 = 3Dk - 2dk - dk+1
//a1 = dk
//a0 = fk
//
//dk = (fk+1 - fk-1)*0.5
//Dk = (fk+1 - fk)
real_t dk =
#endif
return Vector2();
}
Vector2 Vector2::cubic_interpolate(const Vector2 &p_b, const Vector2 &p_pre_a, const Vector2 &p_post_b, real_t p_t) const {
Vector2 p0 = p_pre_a;
Vector2 p1 = *this;
Vector2 p2 = p_b;
Vector2 p3 = p_post_b;
real_t t = p_t;
real_t t2 = t * t;
real_t t3 = t2 * t;
Vector2 out;
out = 0.5 * ((p1 * 2.0) +
(-p0 + p2) * t +
(2.0 * p0 - 5.0 * p1 + 4 * p2 - p3) * t2 +
(-p0 + 3.0 * p1 - 3.0 * p2 + p3) * t3);
return out;
/*
real_t mu = p_t;
real_t mu2 = mu*mu;
Vector2 a0 = p_post_b - p_b - p_pre_a + *this;
Vector2 a1 = p_pre_a - *this - a0;
Vector2 a2 = p_b - p_pre_a;
Vector2 a3 = *this;
return ( a0*mu*mu2 + a1*mu2 + a2*mu + a3 );
*/
/*
real_t t = p_t;
real_t t2 = t*t;
real_t t3 = t2*t;
real_t a = 2.0*t3- 3.0*t2 + 1;
real_t b = -2.0*t3+ 3.0*t2;
real_t c = t3- 2.0*t2 + t;
real_t d = t3- t2;
Vector2 p_a=*this;
return Vector2(
(a * p_a.x) + (b *p_b.x) + (c * p_pre_a.x) + (d * p_post_b.x),
(a * p_a.y) + (b *p_b.y) + (c * p_pre_a.y) + (d * p_post_b.y)
);
*/
}
Vector2 Vector2::slide(const Vector2 &p_vec) const {
return p_vec - *this * this->dot(p_vec);
}
Vector2 Vector2::reflect(const Vector2 &p_vec) const {
return p_vec - *this * this->dot(p_vec) * 2.0;
}
bool Rect2::intersects_segment(const Point2 &p_from, const Point2 &p_to, Point2 *r_pos, Point2 *r_normal) const {
real_t min = 0, max = 1;
int axis = 0;
real_t sign = 0;
for (int i = 0; i < 2; i++) {
real_t seg_from = p_from[i];
real_t seg_to = p_to[i];
real_t box_begin = pos[i];
real_t box_end = box_begin + size[i];
real_t cmin, cmax;
real_t csign;
if (seg_from < seg_to) {
if (seg_from > box_end || seg_to < box_begin)
return false;
real_t length = seg_to - seg_from;
cmin = (seg_from < box_begin) ? ((box_begin - seg_from) / length) : 0;
cmax = (seg_to > box_end) ? ((box_end - seg_from) / length) : 1;
csign = -1.0;
} else {
if (seg_to > box_end || seg_from < box_begin)
return false;
real_t length = seg_to - seg_from;
cmin = (seg_from > box_end) ? (box_end - seg_from) / length : 0;
cmax = (seg_to < box_begin) ? (box_begin - seg_from) / length : 1;
csign = 1.0;
}
if (cmin > min) {
min = cmin;
axis = i;
sign = csign;
}
if (cmax < max)
max = cmax;
if (max < min)
return false;
}
Vector2 rel = p_to - p_from;
if (r_normal) {
Vector2 normal;
normal[axis] = sign;
*r_normal = normal;
}
if (r_pos)
*r_pos = p_from + rel * min;
return true;
}
/* Point2i */
Point2i Point2i::operator+(const Point2i &p_v) const {
return Point2i(x + p_v.x, y + p_v.y);
}
void Point2i::operator+=(const Point2i &p_v) {
x += p_v.x;
y += p_v.y;
}
Point2i Point2i::operator-(const Point2i &p_v) const {
return Point2i(x - p_v.x, y - p_v.y);
}
void Point2i::operator-=(const Point2i &p_v) {
x -= p_v.x;
y -= p_v.y;
}
Point2i Point2i::operator*(const Point2i &p_v1) const {
return Point2i(x * p_v1.x, y * p_v1.y);
};
Point2i Point2i::operator*(const int &rvalue) const {
return Point2i(x * rvalue, y * rvalue);
};
void Point2i::operator*=(const int &rvalue) {
x *= rvalue;
y *= rvalue;
};
Point2i Point2i::operator/(const Point2i &p_v1) const {
return Point2i(x / p_v1.x, y / p_v1.y);
};
Point2i Point2i::operator/(const int &rvalue) const {
return Point2i(x / rvalue, y / rvalue);
};
void Point2i::operator/=(const int &rvalue) {
x /= rvalue;
y /= rvalue;
};
Point2i Point2i::operator-() const {
return Point2i(-x, -y);
}
bool Point2i::operator==(const Point2i &p_vec2) const {
return x == p_vec2.x && y == p_vec2.y;
}
bool Point2i::operator!=(const Point2i &p_vec2) const {
return x != p_vec2.x || y != p_vec2.y;
}
void Transform2D::invert() {
// FIXME: this function assumes the basis is a rotation matrix, with no scaling.
// Transform2D::affine_inverse can handle matrices with scaling, so GDScript should eventually use that.
SWAP(elements[0][1], elements[1][0]);
elements[2] = basis_xform(-elements[2]);
}
Transform2D Transform2D::inverse() const {
Transform2D inv = *this;
inv.invert();
return inv;
}
void Transform2D::affine_invert() {
real_t det = basis_determinant();
ERR_FAIL_COND(det == 0);
real_t idet = 1.0 / det;
SWAP(elements[0][0], elements[1][1]);
elements[0] *= Vector2(idet, -idet);
elements[1] *= Vector2(-idet, idet);
elements[2] = basis_xform(-elements[2]);
}
Transform2D Transform2D::affine_inverse() const {
Transform2D inv = *this;
inv.affine_invert();
return inv;
}
void Transform2D::rotate(real_t p_phi) {
*this = Transform2D(p_phi, Vector2()) * (*this);
}
real_t Transform2D::get_rotation() const {
real_t det = basis_determinant();
Transform2D m = orthonormalized();
if (det < 0) {
m.scale_basis(Size2(1, -1)); // convention to separate rotation and reflection for 2D is to absorb a flip along y into scaling.
}
return Math::atan2(m[0].y, m[0].x);
}
void Transform2D::set_rotation(real_t p_rot) {
real_t cr = Math::cos(p_rot);
real_t sr = Math::sin(p_rot);
elements[0][0] = cr;
elements[0][1] = sr;
elements[1][0] = -sr;
elements[1][1] = cr;
}
Transform2D::Transform2D(real_t p_rot, const Vector2 &p_pos) {
real_t cr = Math::cos(p_rot);
real_t sr = Math::sin(p_rot);
elements[0][0] = cr;
elements[0][1] = sr;
elements[1][0] = -sr;
elements[1][1] = cr;
elements[2] = p_pos;
}
Size2 Transform2D::get_scale() const {
real_t det_sign = basis_determinant() > 0 ? 1 : -1;
return Size2(elements[0].length(), det_sign * elements[1].length());
}
void Transform2D::scale(const Size2 &p_scale) {
scale_basis(p_scale);
elements[2] *= p_scale;
}
void Transform2D::scale_basis(const Size2 &p_scale) {
elements[0][0] *= p_scale.x;
elements[0][1] *= p_scale.y;
elements[1][0] *= p_scale.x;
elements[1][1] *= p_scale.y;
}
void Transform2D::translate(real_t p_tx, real_t p_ty) {
translate(Vector2(p_tx, p_ty));
}
void Transform2D::translate(const Vector2 &p_translation) {
elements[2] += basis_xform(p_translation);
}
void Transform2D::orthonormalize() {
// Gram-Schmidt Process
Vector2 x = elements[0];
Vector2 y = elements[1];
x.normalize();
y = (y - x * (x.dot(y)));
y.normalize();
elements[0] = x;
elements[1] = y;
}
Transform2D Transform2D::orthonormalized() const {
Transform2D on = *this;
on.orthonormalize();
return on;
}
bool Transform2D::operator==(const Transform2D &p_transform) const {
for (int i = 0; i < 3; i++) {
if (elements[i] != p_transform.elements[i])
return false;
}
return true;
}
bool Transform2D::operator!=(const Transform2D &p_transform) const {
for (int i = 0; i < 3; i++) {
if (elements[i] != p_transform.elements[i])
return true;
}
return false;
}
void Transform2D::operator*=(const Transform2D &p_transform) {
elements[2] = xform(p_transform.elements[2]);
real_t x0, x1, y0, y1;
x0 = tdotx(p_transform.elements[0]);
x1 = tdoty(p_transform.elements[0]);
y0 = tdotx(p_transform.elements[1]);
y1 = tdoty(p_transform.elements[1]);
elements[0][0] = x0;
elements[0][1] = x1;
elements[1][0] = y0;
elements[1][1] = y1;
}
Transform2D Transform2D::operator*(const Transform2D &p_transform) const {
Transform2D t = *this;
t *= p_transform;
return t;
}
Transform2D Transform2D::scaled(const Size2 &p_scale) const {
Transform2D copy = *this;
copy.scale(p_scale);
return copy;
}
Transform2D Transform2D::basis_scaled(const Size2 &p_scale) const {
Transform2D copy = *this;
copy.scale_basis(p_scale);
return copy;
}
Transform2D Transform2D::untranslated() const {
Transform2D copy = *this;
copy.elements[2] = Vector2();
return copy;
}
Transform2D Transform2D::translated(const Vector2 &p_offset) const {
Transform2D copy = *this;
copy.translate(p_offset);
return copy;
}
Transform2D Transform2D::rotated(real_t p_phi) const {
Transform2D copy = *this;
copy.rotate(p_phi);
return copy;
}
real_t Transform2D::basis_determinant() const {
return elements[0].x * elements[1].y - elements[0].y * elements[1].x;
}
Transform2D Transform2D::interpolate_with(const Transform2D &p_transform, real_t p_c) const {
//extract parameters
Vector2 p1 = get_origin();
Vector2 p2 = p_transform.get_origin();
real_t r1 = get_rotation();
real_t r2 = p_transform.get_rotation();
Size2 s1 = get_scale();
Size2 s2 = p_transform.get_scale();
//slerp rotation
Vector2 v1(Math::cos(r1), Math::sin(r1));
Vector2 v2(Math::cos(r2), Math::sin(r2));
real_t dot = v1.dot(v2);
dot = (dot < -1.0) ? -1.0 : ((dot > 1.0) ? 1.0 : dot); //clamp dot to [-1,1]
Vector2 v;
if (dot > 0.9995) {
v = Vector2::linear_interpolate(v1, v2, p_c).normalized(); //linearly interpolate to avoid numerical precision issues
} else {
real_t angle = p_c * Math::acos(dot);
Vector2 v3 = (v2 - v1 * dot).normalized();
v = v1 * Math::cos(angle) + v3 * Math::sin(angle);
}
//construct matrix
Transform2D res(Math::atan2(v.y, v.x), Vector2::linear_interpolate(p1, p2, p_c));
res.scale_basis(Vector2::linear_interpolate(s1, s2, p_c));
return res;
}
Transform2D::operator String() const {
return String(String() + elements[0] + ", " + elements[1] + ", " + elements[2]);
}