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#pragma once | ||
#include "Vec2.hpp" | ||
/*-------------------------------------------------------------------- | ||
±¾ÎļþÀ´×Ô | ||
http://www.flipcode.com/archives/2D_OBB_Intersection.shtml | ||
2D OBB Intersection | ||
For 2D graphics or movement of 3D objects on a 2D ground plane it is often | ||
useful to discover if the 2D oriented bounding boxes of two objects overlap | ||
(have a non - empty intersection).One motivating example is the placement of | ||
a new building in a Real - Time Strategy game.The UI needs to continuously | ||
check whether the footprint of the new building overlaps the footprint of | ||
any existing building.If there is any overlap, the UI should indicate that | ||
is an illegal placement. | ||
Stefan Gottschalk's thesis (Collision Queries using Oriented Bounding | ||
Boxes, Ph.D.Thesis, Department of Computer Science, University of North | ||
Carolina at Chapel Hill, 1999) introduces the separating - axis method | ||
for performing the equivalent test on 3D oriented bounding boxes. | ||
This method depends on the observation that | ||
for two boxes to be disjoint(i.e. * not* intersecting), there must be some axis | ||
along which their projections are disjoint.The 3D case considers each of 15 | ||
axes as a potential | ||
separating axis.These axes are the three edge axes of box 1, the three edge | ||
axes of box 2, and the nine cross products formed by taking some edge of box 1 | ||
and some edge of box 2. | ||
In 2D this simplifies dramatically and only four axes need be considered. | ||
These are | ||
the orthogonal edges of each bounding box.If a few values are precomputed | ||
every time a box moves, we end up performing only 16 dot products and some | ||
comparisons in the worst case for each overlap test.One nice property of the | ||
separating - axis method is that it can be | ||
structured in an early out fashion, so that many fewer operations are needed in | ||
the case where the boxes do not intersect.In general, the first test is | ||
extremely | ||
likely to fail(and return "no overlap") when there is no overlap.If it | ||
passes, | ||
the second test is even more likely to fail if there is no overlap, and so on. | ||
Only when the boxes are in extremely close proximity is there even a 50 % chance | ||
of | ||
executing more than 2 tests. | ||
The C++ code sample provided efficiently computes this fast 2D oriented | ||
bounding box | ||
overlap.I augmented the OBB2D class with some methods for rendering and | ||
construction to help visualize the result.OBB2D::overlaps1Way performs the | ||
real work.It tests to see whether the box passed as an argument overlaps the | ||
current box along either of the current box's dimensions. Note that this test | ||
must be performed for each box on the other to determine whether there is truly | ||
any overlap.To make the tests extremely efficient, OBB2D::origin stores the | ||
projection of corner number zero onto a box's axes and the axes are stored | ||
explicitly in OBB2D::axis.The magnitude of these stored axes is the inverse | ||
of the corresponding edge length so that all overlap tests can be performed on | ||
the interval[0, 1] without normalization, and square roots are avoided | ||
throughout the entire class. | ||
Morgan McGuire morgan@cs.brown.edu | ||
*/ | ||
class OBB2D { | ||
private: | ||
/** Corners of the box, where 0 is the lower left. */ | ||
Vec2d corner[4]; | ||
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/** Two edges of the box extended away from corner[0]. */ | ||
Vec2d axis[2]; | ||
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/** origin[a] = corner[0].dot(axis[a]); */ | ||
double origin[2]; | ||
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/** Returns true if other overlaps one dimension of this. */ | ||
bool overlaps1Way(const OBB2D& other) const { | ||
for (int a = 0; a < 2; ++a) { | ||
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double t = other.corner[0].dot(axis[a]); | ||
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// Find the extent of box 2 on axis a | ||
double tMin = t; | ||
double tMax = t; | ||
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for (int c = 1; c < 4; ++c) { | ||
t = other.corner[c].dot(axis[a]); | ||
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if (t < tMin) { | ||
tMin = t; | ||
} | ||
else if (t > tMax) { | ||
tMax = t; | ||
} | ||
} | ||
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// We have to subtract off the origin | ||
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// See if [tMin, tMax] intersects [0, 1] | ||
if ((tMin > 1 + origin[a]) || (tMax < origin[a])) { | ||
// There was no intersection along this dimension; | ||
// the boxes cannot possibly overlap. | ||
return false; | ||
} | ||
} | ||
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// There was no dimension along which there is no intersection. | ||
// Therefore the boxes overlap. | ||
return true; | ||
} | ||
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/** Updates the axes after the corners move. Assumes the | ||
corners actually form a rectangle. */ | ||
void computeAxes() { | ||
axis[0] = corner[1] - corner[0]; | ||
axis[1] = corner[3] - corner[0]; | ||
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// Make the length of each axis 1/edge length so we know any | ||
// dot product must be less than 1 to fall within the edge. | ||
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for (int a = 0; a < 2; ++a) { | ||
axis[a] /= axis[a].lengthSqr(); | ||
origin[a] = corner[0].dot(axis[a]); | ||
} | ||
} | ||
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class Obb2D { | ||
public: | ||
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OBB2D(const Vec2d& center, const double w, const double h, double angle) | ||
{ | ||
Vec2d X(cos(angle), sin(angle)); | ||
Vec2d Y(-sin(angle), cos(angle)); | ||
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X *= w / 2; | ||
Y *= h / 2; | ||
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corner[0] = center - X - Y; | ||
corner[1] = center + X - Y; | ||
corner[2] = center + X + Y; | ||
corner[3] = center - X + Y; | ||
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computeAxes(); | ||
} | ||
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/** For testing purposes. */ | ||
void moveTo(const Vec2d& center) { | ||
Vec2d centroid = (corner[0] + corner[1] + corner[2] + corner[3]) / 4; | ||
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Vec2d translation = center - centroid; | ||
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for (int c = 0; c < 4; ++c) { | ||
corner[c] += translation; | ||
} | ||
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computeAxes(); | ||
} | ||
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/** Returns true if the intersection of the boxes is non-empty. */ | ||
bool overlaps(const OBB2D& other) const { | ||
return overlaps1Way(other) && other.overlaps1Way(*this); | ||
} | ||
}; | ||
Obb2D(const Vec2d& center, const Vec2d& size, const double angle) noexcept { | ||
const auto x = Vec2d(cos(angle), sin(angle)) * size.x / 2, y = Vec2d(-sin(angle), cos(angle)) * size.y / 2; | ||
corner[0] = center - x - y; | ||
corner[1] = center + x - y; | ||
corner[2] = center + x + y; | ||
corner[3] = center - x + y; | ||
computeAxes(); | ||
} | ||
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void moveTo(const Vec2d& center) noexcept { | ||
const auto centroid = (corner[0] + corner[1] + corner[2] + corner[3]) / 4; | ||
const auto translation = center - centroid; | ||
for (auto c = 0; c < 4; ++c) { corner[c] += translation; } | ||
computeAxes(); | ||
} | ||
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bool overlaps(const Obb2D& other) const noexcept { return overlaps1Way(other) && other.overlaps1Way(*this); } | ||
private: | ||
Vec2d corner[4]{}, axis[2]{}; | ||
double origin[2]{}; | ||
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bool overlaps1Way(const Obb2D& other) const noexcept { | ||
for (auto a = 0; a < 2; ++a) { | ||
auto t = other.corner[0].dot(axis[a]); | ||
auto tMin = t, tMax = t; | ||
for (auto c = 1; c < 4; ++c) { | ||
t = other.corner[c].dot(axis[a]); | ||
if (t < tMin) tMin = t; | ||
else if (t > tMax) tMax = t; | ||
} | ||
if ((tMin > 1 + origin[a]) || (tMax < origin[a])) return false; | ||
} | ||
return true; | ||
} | ||
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void computeAxes() noexcept { | ||
axis[0] = corner[1] - corner[0]; | ||
axis[1] = corner[3] - corner[0]; | ||
for (auto a = 0; a < 2; ++a) { | ||
axis[a] /= axis[a].lengthSqr(); | ||
origin[a] = corner[0].dot(axis[a]); | ||
} | ||
} | ||
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}; |
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