void CubicPolynomialCurve3<Real>::Tessellate ( int level )
{
// Vertices V = (2^L+1).
int twoPowL = ( 1 << level );
mNumVertices = twoPowL + 1;
delete1( mVertices );
mVertices = new1<Vector3<Real> >( mNumVertices );
// Indices of endpoints, I[t].
IntervalParameters IP;
IP.I0 = 0;
IP.I1 = twoPowL;
// Vertices for subdivision.
Vector3<Real>* X = mVertices;
X[IP.I0] = GetPosition( mTMin );
X[IP.I1] = GetPosition( mTMax );
// Recursive subdivision.
if ( level > 0 )
{
IP.Xuu[0] = GetSecondDerivative( mTMin );
IP.Xuu[1] = GetSecondDerivative( mTMax );
Subdivide( --level, ( Real )0.25, X, IP );
}
}
Vector3<Real> Curve3<Real>::GetNormal ( Real t ) const
{
Vector3<Real> velocity = GetFirstDerivative( t );
Vector3<Real> acceleration = GetSecondDerivative( t );
Real VDotV = velocity.Dot( velocity );
Real VDotA = velocity.Dot( acceleration );
Vector3<Real> normal = VDotV * acceleration - VDotA * velocity;
normal.Normalize();
return normal;
}
Vector3 Curve<Real>::GetNormal (Real t)
{
Vector3 velocity = GetFirstDerivative(t);
Vector3 acceleration = GetSecondDerivative(t);
Real VDotV = dot(velocity,velocity);
Real VDotA = dot(velocity,acceleration);
Vector3 normal = VDotV*acceleration - VDotA*velocity;
normal.normalize();
return normal;
}
void Curve3<Real>::GetFrame ( Real t, Vector3<Real>& position,
Vector3<Real>& tangent, Vector3<Real>& normal, Vector3<Real>& binormal )
const
{
position = GetPosition( t );
Vector3<Real> velocity = GetFirstDerivative( t );
Vector3<Real> acceleration = GetSecondDerivative( t );
Real VDotV = velocity.Dot( velocity );
Real VDotA = velocity.Dot( acceleration );
normal = VDotV * acceleration - VDotA * velocity;
normal.Normalize();
tangent = velocity;
tangent.Normalize();
binormal = tangent.Cross( normal );
}
void Curve<Real>::GetFrame (Real t, Vector3& position,
Vector3& tangent, Vector3& normal, Vector3& binormal)
{
position = GetPosition(t);
Vector3 velocity = GetFirstDerivative(t);
Vector3 acceleration = GetSecondDerivative(t);
Real VDotV = dot(velocity,velocity);
Real VDotA = dot(velocity,acceleration);
normal = VDotV*acceleration - VDotA*velocity;
normal.normalize();
tangent = velocity;
tangent.normalize();
binormal = tangent.cross(normal);
}
Real Curve3<Real>::GetTorsion ( Real t ) const
{
Vector3<Real> velocity = GetFirstDerivative( t );
Vector3<Real> acceleration = GetSecondDerivative( t );
Vector3<Real> cross = velocity.Cross( acceleration );
Real denom = cross.SquaredLength();
if ( denom >= Math<Real>::ZERO_TOLERANCE )
{
Vector3<Real> jerk = GetThirdDerivative( t );
Real numer = cross.Dot( jerk );
return numer / denom;
}
else
{
// Torsion is indeterminate, just return 0.
return ( Real )0;
}
}
Real Curve3<Real>::GetCurvature ( Real t ) const
{
Vector3<Real> velocity = GetFirstDerivative( t );
Real speedSqr = velocity.SquaredLength();
if ( speedSqr >= Math<Real>::ZERO_TOLERANCE )
{
Vector3<Real> acceleration = GetSecondDerivative( t );
Vector3<Real> cross = velocity.Cross( acceleration );
Real numer = cross.Length();
Real denom = Math<Real>::Pow( speedSqr, ( Real )1.5 );
return numer / denom;
}
else
{
// Curvature is indeterminate, just return 0.
return ( Real )0;
}
}
Real Curve<Real>::GetTorsion (Real t)
{
Vector3 velocity = GetFirstDerivative(t);
Vector3 acceleration = GetSecondDerivative(t);
Vector3 _cross = velocity.cross(acceleration);
Real denom = _cross.squaredNorm();
if (denom >= REAL_ZERO_TOLERANCE)
{
Vector3 jerk = GetThirdDerivative(t);
Real numer = dot(_cross,jerk);
return numer/denom;
}
else
{
// Torsion is indeterminate, just return 0.
return (Real)0;
}
}
Real Curve<Real>::GetCurvature (Real t)
{
Vector3 velocity = GetFirstDerivative(t);
Real speedSqr = velocity.squaredNorm();
if (speedSqr >= REAL_ZERO_TOLERANCE)
{
Vector3 acceleration = GetSecondDerivative(t);
Vector3 _cross = velocity.cross(acceleration);
Real numer = _cross.norm();
Real denom = pow(speedSqr, (Real)1.5);
return numer/denom;
}
else
{
// Curvature is indeterminate, just return 0.
return (Real)0;
}
}
