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// Geometric Tools, LLC
// Copyright (c) 1998-2014
// Distributed under the Boost Software License, Version 1.0.
// http://www.boost.org/LICENSE_1_0.txt
// http://www.geometrictools.com/License/Boost/LICENSE_1_0.txt
//
// File Version: 5.0.1 (2010/10/01)
#include "Wm5MathematicsPCH.h"
#include "Wm5IntrLine3Cone3.h"
namespace Wm5
{
//----------------------------------------------------------------------------
template <typename Real>
IntrLine3Cone3<Real>::IntrLine3Cone3 (const Line3<Real>& line,
const Cone3<Real>& cone)
:
mLine(&line),
mCone(&cone)
{
}
//----------------------------------------------------------------------------
template <typename Real>
const Line3<Real>& IntrLine3Cone3<Real>::GetLine () const
{
return *mLine;
}
//----------------------------------------------------------------------------
template <typename Real>
const Cone3<Real>& IntrLine3Cone3<Real>::GetCone () const
{
return *mCone;
}
//----------------------------------------------------------------------------
template <typename Real>
bool IntrLine3Cone3<Real>::Find ()
{
// Set up the quadratic Q(t) = c2*t^2 + 2*c1*t + c0 that corresponds to
// the cone. Let the vertex be V, the unit-length direction vector be A,
// and the angle measured from the cone axis to the cone wall be Theta,
// and define g = cos(Theta). A point X is on the cone wall whenever
// Dot(A,(X-V)/|X-V|) = g. Square this equation and factor to obtain
// (X-V)^T * (A*A^T - g^2*I) * (X-V) = 0
// where the superscript T denotes the transpose operator. This defines
// a double-sided cone. The line is L(t) = P + t*D, where P is the line
// origin and D is a unit-length direction vector. Substituting
// X = L(t) into the cone equation above leads to Q(t) = 0. Since we
// want only intersection points on the single-sided cone that lives in
// the half-space pointed to by A, any point L(t) generated by a root of
// Q(t) = 0 must be tested for Dot(A,L(t)-V) >= 0.
Real AdD = mCone->Axis.Dot(mLine->Direction);
Real cosSqr = mCone->CosAngle*mCone->CosAngle;
Vector3<Real> E = mLine->Origin - mCone->Vertex;
Real AdE = mCone->Axis.Dot(E);
Real DdE = mLine->Direction.Dot(E);
Real EdE = E.Dot(E);
Real c2 = AdD*AdD - cosSqr;
Real c1 = AdD*AdE - cosSqr*DdE;
Real c0 = AdE*AdE - cosSqr*EdE;
Real dot;
// Solve the quadratic. Keep only those X for which Dot(A,X-V) >= 0.
if (Math<Real>::FAbs(c2) >= Math<Real>::ZERO_TOLERANCE)
{
// c2 != 0
Real discr = c1*c1 - c0*c2;
if (discr < (Real)0)
{
// Q(t) = 0 has no real-valued roots. The line does not
// intersect the double-sided cone.
mIntersectionType = IT_EMPTY;
mQuantity = 0;
}
else if (discr > Math<Real>::ZERO_TOLERANCE)
{
// Q(t) = 0 has two distinct real-valued roots. However, one or
// both of them might intersect the portion of the double-sided
// cone "behind" the vertex. We are interested only in those
// intersections "in front" of the vertex.
Real root = Math<Real>::Sqrt(discr);
Real invC2 = ((Real)1)/c2;
mQuantity = 0;
Real t = (-c1 - root)*invC2;
mPoint[mQuantity] = mLine->Origin + t*mLine->Direction;
E = mPoint[mQuantity] - mCone->Vertex;
dot = E.Dot(mCone->Axis);
if (dot > (Real)0)
{
mQuantity++;
}
t = (-c1 + root)*invC2;
mPoint[mQuantity] = mLine->Origin + t*mLine->Direction;
E = mPoint[mQuantity] - mCone->Vertex;
dot = E.Dot(mCone->Axis);
if (dot > (Real)0)
{
mQuantity++;
}
if (mQuantity == 2)
{
// The line intersects the single-sided cone in front of the
// vertex twice.
mIntersectionType = IT_SEGMENT;
}
else if (mQuantity == 1)
{
// The line intersects the single-sided cone in front of the
// vertex once. The other intersection is with the
// single-sided cone behind the vertex.
mIntersectionType = IT_RAY;
mPoint[mQuantity++] = mLine->Direction;
}
else
{
// The line intersects the single-sided cone behind the vertex
// twice.
mIntersectionType = IT_EMPTY;
}
}
else
{
// One repeated real root (line is tangent to the cone).
mPoint[0] = mLine->Origin - (c1/c2)*mLine->Direction;
E = mPoint[0] - mCone->Vertex;
if (E.Dot(mCone->Axis) > (Real)0)
{
mIntersectionType = IT_POINT;
mQuantity = 1;
}
else
{
mIntersectionType = IT_EMPTY;
mQuantity = 0;
}
}
}
else if (Math<Real>::FAbs(c1) >= Math<Real>::ZERO_TOLERANCE)
{
// c2 = 0, c1 != 0 (D is a direction vector on the cone boundary)
mPoint[0] = mLine->Origin - (((Real)0.5)*c0/c1)*mLine->Direction;
E = mPoint[0] - mCone->Vertex;
dot = E.Dot(mCone->Axis);
if (dot > (Real)0)
{
mIntersectionType = IT_RAY;
mQuantity = 2;
mPoint[1] = mLine->Direction;
}
else
{
mIntersectionType = IT_EMPTY;
mQuantity = 0;
}
}
else if (Math<Real>::FAbs(c0) >= Math<Real>::ZERO_TOLERANCE)
{
// c2 = c1 = 0, c0 != 0
mIntersectionType = IT_EMPTY;
mQuantity = 0;
}
else
{
// c2 = c1 = c0 = 0, cone contains ray V+t*D where V is cone vertex
// and D is the line direction.
mIntersectionType = IT_RAY;
mQuantity = 2;
mPoint[0] = mCone->Vertex;
mPoint[1] = mLine->Direction;
}
return mIntersectionType != IT_EMPTY;
}
//----------------------------------------------------------------------------
template <typename Real>
int IntrLine3Cone3<Real>::GetQuantity () const
{
return mQuantity;
}
//----------------------------------------------------------------------------
template <typename Real>
const Vector3<Real>& IntrLine3Cone3<Real>::GetPoint (int i) const
{
return mPoint[i];
}
//----------------------------------------------------------------------------
//----------------------------------------------------------------------------
// Explicit instantiation.
//----------------------------------------------------------------------------
template WM5_MATHEMATICS_ITEM
class IntrLine3Cone3<float>;
template WM5_MATHEMATICS_ITEM
class IntrLine3Cone3<double>;
//----------------------------------------------------------------------------
}
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