// David Eberly, Geometric Tools, Redmond WA 98052
// Copyright (c) 1998-2021
// Distributed under the Boost Software License, Version 1.0.
// https://www.boost.org/LICENSE_1_0.txt
// https://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
// Version: 4.0.2019.08.13
#pragma once
#include <Mathematics/FIQuery.h>
#include <Mathematics/TIQuery.h>
#include <Mathematics/DistLineSegment.h>
#include <Mathematics/Capsule.h>
#include <Mathematics/Vector3.h>
// The queries consider the capsule to be a solid.
//
// The test-intersection queries are based on distance computations.
namespace gte
{
template <typename Real>
class TIQuery<Real, Line3<Real>, Capsule3<Real>>
{
public:
struct Result
{
bool intersect;
};
Result operator()(Line3<Real> const& line, Capsule3<Real> const& capsule)
{
Result result;
DCPQuery<Real, Line3<Real>, Segment3<Real>> lsQuery;
auto lsResult = lsQuery(line, capsule.segment);
result.intersect = (lsResult.distance <= capsule.radius);
return result;
}
};
template <typename Real>
class FIQuery<Real, Line3<Real>, Capsule3<Real>>
{
public:
struct Result
{
bool intersect;
int numIntersections;
std::array<Real, 2> parameter;
std::array<Vector3<Real>, 2> point;
};
Result operator()(Line3<Real> const& line, Capsule3<Real> const& capsule)
{
Result result;
DoQuery(line.origin, line.direction, capsule, result);
for (int i = 0; i < result.numIntersections; ++i)
{
result.point[i] = line.origin + result.parameter[i] * line.direction;
}
return result;
}
protected:
void DoQuery(Vector3<Real> const& lineOrigin,
Vector3<Real> const& lineDirection, Capsule3<Real> const& capsule,
Result& result)
{
// Initialize the result as if there is no intersection. If we
// discover an intersection, these values will be modified
// accordingly.
result.intersect = false;
result.numIntersections = 0;
// Create a coordinate system for the capsule. In this system,
// the capsule segment center C is the origin and the capsule axis
// direction W is the z-axis. U and V are the other coordinate
// axis directions. If P = x*U+y*V+z*W, the cylinder containing
// the capsule wall is x^2 + y^2 = r^2, where r is the capsule
// radius. The finite cylinder that makes up the capsule minus
// its hemispherical end caps has z-values |z| <= e, where e is
// the extent of the capsule segment. The top hemisphere cap is
// x^2+y^2+(z-e)^2 = r^2 for z >= e, and the bottom hemisphere cap
// is x^2+y^2+(z+e)^2 = r^2 for z <= -e.
Vector3<Real> segOrigin, segDirection;
Real segExtent;
capsule.segment.GetCenteredForm(segOrigin, segDirection, segExtent);
Vector3<Real> basis[3]; // {W, U, V}
basis[0] = segDirection;
ComputeOrthogonalComplement(1, basis);
Real rSqr = capsule.radius * capsule.radius;
// Convert incoming line origin to capsule coordinates.
Vector3<Real> diff = lineOrigin - segOrigin;
Vector3<Real> P{ Dot(basis[1], diff), Dot(basis[2], diff), Dot(basis[0], diff) };
// Get the z-value, in capsule coordinates, of the incoming line's
// unit-length direction.
Real dz = Dot(basis[0], lineDirection);
if (std::fabs(dz) == (Real)1)
{
// The line is parallel to the capsule axis. Determine
// whether the line intersects the capsule hemispheres.
Real radialSqrDist = rSqr - P[0] * P[0] - P[1] * P[1];
if (radialSqrDist >= (Real)0)
{
// The line intersects the hemispherical caps.
result.intersect = true;
result.numIntersections = 2;
Real zOffset = std::sqrt(radialSqrDist) + segExtent;
if (dz > (Real)0)
{
result.parameter[0] = -P[2] - zOffset;
result.parameter[1] = -P[2] + zOffset;
}
else
{
result.parameter[0] = P[2] - zOffset;
result.parameter[1] = P[2] + zOffset;
}
}
// else: The line outside the capsule's cylinder, no
// intersection.
return;
}
// Convert the incoming line unit-length direction to capsule
// coordinates.
Vector3<Real> D{ Dot(basis[1], lineDirection), Dot(basis[2], lineDirection), dz };
// Test intersection of line P+t*D with infinite cylinder
// x^2+y^2 = r^2. This reduces to computing the roots of a
// quadratic equation. If P = (px,py,pz) and D = (dx,dy,dz), then
// the quadratic equation is
// (dx^2+dy^2)*t^2 + 2*(px*dx+py*dy)*t + (px^2+py^2-r^2) = 0
Real a0 = P[0] * P[0] + P[1] * P[1] - rSqr;
Real a1 = P[0] * D[0] + P[1] * D[1];
Real a2 = D[0] * D[0] + D[1] * D[1];
Real discr = a1 * a1 - a0 * a2;
if (discr < (Real)0)
{
// The line does not intersect the infinite cylinder, so it
// cannot intersect the capsule.
return;
}
Real root, inv, tValue, zValue;
if (discr > (Real)0)
{
// The line intersects the infinite cylinder in two places.
root = std::sqrt(discr);
inv = (Real)1 / a2;
tValue = (-a1 - root) * inv;
zValue = P[2] + tValue * D[2];
if (std::fabs(zValue) <= segExtent)
{
result.intersect = true;
result.parameter[result.numIntersections++] = tValue;
}
tValue = (-a1 + root) * inv;
zValue = P[2] + tValue * D[2];
if (std::fabs(zValue) <= segExtent)
{
result.intersect = true;
result.parameter[result.numIntersections++] = tValue;
}
if (result.numIntersections == 2)
{
// The line intersects the capsule wall in two places.
return;
}
}
else
{
// The line is tangent to the infinite cylinder but intersects
// the cylinder in a single point.
tValue = -a1 / a2;
zValue = P[2] + tValue * D[2];
if (std::fabs(zValue) <= segExtent)
{
result.intersect = true;
result.numIntersections = 1;
result.parameter[0] = tValue;
// Used by derived classes.
result.parameter[1] = result.parameter[0];
return;
}
}
// Test intersection with bottom hemisphere. The quadratic
// equation is
// t^2 + 2*(px*dx+py*dy+(pz+e)*dz)*t
// + (px^2+py^2+(pz+e)^2-r^2) = 0
// Use the fact that currently a1 = px*dx+py*dy and
// a0 = px^2+py^2-r^2. The leading coefficient is a2 = 1, so no
// need to include in the construction.
Real PZpE = P[2] + segExtent;
a1 += PZpE * D[2];
a0 += PZpE * PZpE;
discr = a1 * a1 - a0;
if (discr > (Real)0)
{
root = std::sqrt(discr);
tValue = -a1 - root;
zValue = P[2] + tValue * D[2];
if (zValue <= -segExtent)
{
result.parameter[result.numIntersections++] = tValue;
if (result.numIntersections == 2)
{
result.intersect = true;
if (result.parameter[0] > result.parameter[1])
{
std::swap(result.parameter[0], result.parameter[1]);
}
return;
}
}
tValue = -a1 + root;
zValue = P[2] + tValue * D[2];
if (zValue <= -segExtent)
{
result.parameter[result.numIntersections++] = tValue;
if (result.numIntersections == 2)
{
result.intersect = true;
if (result.parameter[0] > result.parameter[1])
{
std::swap(result.parameter[0], result.parameter[1]);
}
return;
}
}
}
else if (discr == (Real)0)
{
tValue = -a1;
zValue = P[2] + tValue * D[2];
if (zValue <= -segExtent)
{
result.parameter[result.numIntersections++] = tValue;
if (result.numIntersections == 2)
{
result.intersect = true;
if (result.parameter[0] > result.parameter[1])
{
std::swap(result.parameter[0], result.parameter[1]);
}
return;
}
}
}
// Test intersection with top hemisphere. The quadratic equation
// is
// t^2 + 2*(px*dx+py*dy+(pz-e)*dz)*t
// + (px^2+py^2+(pz-e)^2-r^2) = 0
// Use the fact that currently a1 = px*dx+py*dy+(pz+e)*dz and
// a0 = px^2+py^2+(pz+e)^2-r^2. The leading coefficient is a2 = 1,
// so no need to include in the construction.
a1 -= ((Real)2) * segExtent * D[2];
a0 -= ((Real)4) * segExtent * P[2];
discr = a1 * a1 - a0;
if (discr > (Real)0)
{
root = std::sqrt(discr);
tValue = -a1 - root;
zValue = P[2] + tValue * D[2];
if (zValue >= segExtent)
{
result.parameter[result.numIntersections++] = tValue;
if (result.numIntersections == 2)
{
result.intersect = true;
if (result.parameter[0] > result.parameter[1])
{
std::swap(result.parameter[0], result.parameter[1]);
}
return;
}
}
tValue = -a1 + root;
zValue = P[2] + tValue * D[2];
if (zValue >= segExtent)
{
result.parameter[result.numIntersections++] = tValue;
if (result.numIntersections == 2)
{
result.intersect = true;
if (result.parameter[0] > result.parameter[1])
{
std::swap(result.parameter[0], result.parameter[1]);
}
return;
}
}
}
else if (discr == (Real)0)
{
tValue = -a1;
zValue = P[2] + tValue * D[2];
if (zValue >= segExtent)
{
result.parameter[result.numIntersections++] = tValue;
if (result.numIntersections == 2)
{
result.intersect = true;
if (result.parameter[0] > result.parameter[1])
{
std::swap(result.parameter[0], result.parameter[1]);
}
return;
}
}
}
if (result.numIntersections == 1)
{
// Used by derived classes.
result.parameter[1] = result.parameter[0];
}
}
};
}