double VerdictVector::interior_angle(const VerdictVector &otherVector)
{
double cosAngle=0., angleRad=0., len1, len2=0.;
if (((len1 = this->length()) > 0) && ((len2 = otherVector.length()) > 0))
cosAngle = (*this % otherVector)/(len1 * len2);
else
{
assert(len1 > 0);
assert(len2 > 0);
}
if ((cosAngle > 1.0) && (cosAngle < 1.0001))
{
cosAngle = 1.0;
angleRad = acos(cosAngle);
}
else if (cosAngle < -1.0 && cosAngle > -1.0001)
{
cosAngle = -1.0;
angleRad = acos(cosAngle);
}
else if (cosAngle >= -1.0 && cosAngle <= 1.0)
angleRad = acos(cosAngle);
else
{
assert(cosAngle < 1.0001 && cosAngle > -1.0001);
}
return( (angleRad * 180.) / VERDICT_PI );
}
/*!
The scaled jacobian of a tri
minimum of the jacobian divided by the lengths of 2 edge vectors
*/
C_FUNC_DEF double v_tri_scaled_jacobian( int /*num_nodes*/, double coordinates[][3])
{
static const double detw = 2./sqrt(3.0);
VerdictVector first, second;
double jacobian;
VerdictVector edge[3];
edge[0].set(coordinates[1][0] - coordinates[0][0],
coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2]);
edge[1].set(coordinates[2][0] - coordinates[0][0],
coordinates[2][1] - coordinates[0][1],
coordinates[2][2] - coordinates[0][2]);
edge[2].set(coordinates[2][0] - coordinates[1][0],
coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2]);
first = edge[1]-edge[0];
second = edge[2]-edge[0];
VerdictVector cross = first * second;
jacobian = cross.length();
double max_edge_length_product;
max_edge_length_product = VERDICT_MAX( edge[0].length()*edge[1].length(),
VERDICT_MAX( edge[1].length()*edge[2].length(),
edge[0].length()*edge[2].length() ) );
if( max_edge_length_product < VERDICT_DBL_MIN )
return (double)0.0;
jacobian *= detw;
jacobian /= max_edge_length_product;
if( compute_normal )
{
//center of tri
double point[3], surf_normal[3];
point[0] = (coordinates[0][0] + coordinates[1][0] + coordinates[2][0]) / 3;
point[1] = (coordinates[0][1] + coordinates[1][1] + coordinates[2][1]) / 3;
point[2] = (coordinates[0][2] + coordinates[1][2] + coordinates[2][2]) / 3;
//dot product
compute_normal( point, surf_normal );
if( (cross.x()*surf_normal[0] +
cross.y()*surf_normal[1] +
cross.z()*surf_normal[2] ) < 0 )
jacobian *= -1;
}
if( jacobian > 0 )
return (double) VERDICT_MIN( jacobian, VERDICT_DBL_MAX );
return (double) VERDICT_MAX( jacobian, -VERDICT_DBL_MAX );
}
/*!
the aspect of a tet
CR / (3.0*IR) where CR is the circumsphere radius and IR is the inscribed sphere radius
*/
C_FUNC_DEF VERDICT_REAL v_tet_aspect_beta( int /*num_nodes*/, VERDICT_REAL coordinates[][3] )
{
//Determine side vectors
VerdictVector side[6];
side[0].set( coordinates[1][0] - coordinates[0][0],
coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2] );
side[1].set( coordinates[2][0] - coordinates[1][0],
coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2] );
side[2].set( coordinates[0][0] - coordinates[2][0],
coordinates[0][1] - coordinates[2][1],
coordinates[0][2] - coordinates[2][2] );
side[3].set( coordinates[3][0] - coordinates[0][0],
coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2] );
side[4].set( coordinates[3][0] - coordinates[1][0],
coordinates[3][1] - coordinates[1][1],
coordinates[3][2] - coordinates[1][2] );
side[5].set( coordinates[3][0] - coordinates[2][0],
coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2] );
VerdictVector numerator = side[3].length_squared() * ( side[2] * side[0]) +
side[2].length_squared() * ( side[3] * side[0]) +
side[0].length_squared() * ( side[3] * side[2]);
double area_sum = 0.0;
area_sum = ((side[2] * side[0]).length() +
(side[3] * side[0]).length() +
(side[4] * side[1]).length() +
(side[3] * side[2]).length() ) * 0.5;
double volume = v_tet_volume(4, coordinates);
if( volume < VERDICT_DBL_MIN )
return (VERDICT_REAL)VERDICT_DBL_MAX;
else
{
double aspect_ratio;
aspect_ratio = numerator.length() * area_sum / (108*volume*volume);
if( aspect_ratio > 0 )
return (VERDICT_REAL) VERDICT_MIN( aspect_ratio, VERDICT_DBL_MAX );
return (VERDICT_REAL) VERDICT_MAX( aspect_ratio, -VERDICT_DBL_MAX );
}
}
/*!
the aspect ratio of a triangle
NB (P. Pebay 01/14/07):
Hmax / ( 2.0 * sqrt(3.0) * IR) where Hmax is the maximum edge length
and IR is the inradius
note that previous incarnations of verdict used "v_tri_aspect_ratio" to denote
what is now called "v_tri_aspect_frobenius"
*/
C_FUNC_DEF double v_tri_aspect_ratio( int /*num_nodes*/, double coordinates[][3] )
{
static const double normal_coeff = sqrt( 3. ) / 6.;
// three vectors for each side
VerdictVector a( coordinates[1][0] - coordinates[0][0],
coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2] );
VerdictVector b( coordinates[2][0] - coordinates[1][0],
coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2] );
VerdictVector c( coordinates[0][0] - coordinates[2][0],
coordinates[0][1] - coordinates[2][1],
coordinates[0][2] - coordinates[2][2] );
double a1 = a.length();
double b1 = b.length();
double c1 = c.length();
double hm = a1 > b1 ? a1 : b1;
hm = hm > c1 ? hm : c1;
VerdictVector ab = a * b;
double denominator = ab.length();
if( denominator < VERDICT_DBL_MIN )
return (double)VERDICT_DBL_MAX;
else
{
double aspect_ratio;
aspect_ratio = normal_coeff * hm * (a1 + b1 + c1) / denominator;
if( aspect_ratio > 0 )
return (double) VERDICT_MIN( aspect_ratio, VERDICT_DBL_MAX );
return (double) VERDICT_MAX( aspect_ratio, -VERDICT_DBL_MAX );
}
}
/*!
The area of a tri
0.5 * jacobian at a node
*/
C_FUNC_DEF double v_tri_area( int /*num_nodes*/, double coordinates[][3] )
{
// two vectors for two sides
VerdictVector side1( coordinates[1][0] - coordinates[0][0],
coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2] );
VerdictVector side3( coordinates[2][0] - coordinates[0][0],
coordinates[2][1] - coordinates[0][1],
coordinates[2][2] - coordinates[0][2] );
// the cross product of the two vectors representing two sides of the
// triangle
VerdictVector tmp = side1 * side3;
// return the magnitude of the vector divided by two
double area = 0.5 * tmp.length();
if( area > 0 )
return (double) VERDICT_MIN( area, VERDICT_DBL_MAX );
return (double) VERDICT_MAX( area, -VERDICT_DBL_MAX );
}
/*!
the quality metrics of a tet
*/
C_FUNC_DEF void v_tet_quality( int num_nodes, VERDICT_REAL coordinates[][3],
unsigned int metrics_request_flag, TetMetricVals *metric_vals )
{
memset( metric_vals, 0, sizeof(TetMetricVals) );
/*
node numbers and edge numbers below
3
+ edge 0 is node 0 to 1
+|+ edge 1 is node 1 to 2
3/ | \5 edge 2 is node 0 to 2
/ 4| \ edge 3 is node 0 to 3
0 - -|- + 2 edge 4 is node 1 to 3
\ | + edge 5 is node 2 to 3
0\ | /1
+|/ edge 2 is behind edge 4
1
*/
// lets start with making the vectors
VerdictVector edges[6];
edges[0].set( coordinates[1][0] - coordinates[0][0],
coordinates[1][1] - coordinates[0][1],
coordinates[1][2] - coordinates[0][2] );
edges[1].set( coordinates[2][0] - coordinates[1][0],
coordinates[2][1] - coordinates[1][1],
coordinates[2][2] - coordinates[1][2] );
edges[2].set( coordinates[0][0] - coordinates[2][0],
coordinates[0][1] - coordinates[2][1],
coordinates[0][2] - coordinates[2][2] );
edges[3].set( coordinates[3][0] - coordinates[0][0],
coordinates[3][1] - coordinates[0][1],
coordinates[3][2] - coordinates[0][2] );
edges[4].set( coordinates[3][0] - coordinates[1][0],
coordinates[3][1] - coordinates[1][1],
coordinates[3][2] - coordinates[1][2] );
edges[5].set( coordinates[3][0] - coordinates[2][0],
coordinates[3][1] - coordinates[2][1],
coordinates[3][2] - coordinates[2][2] );
// common numbers
static const double root_of_2 = sqrt(2.0);
// calculate the jacobian
static const int do_jacobian = V_TET_JACOBIAN | V_TET_VOLUME |
V_TET_ASPECT_BETA | V_TET_ASPECT_GAMMA | V_TET_SHAPE |
V_TET_RELATIVE_SIZE_SQUARED | V_TET_SHAPE_AND_SIZE |
V_TET_SCALED_JACOBIAN | V_TET_CONDITION;
if(metrics_request_flag & do_jacobian )
{
metric_vals->jacobian = (VERDICT_REAL)(edges[3] % (edges[2] * edges[0]));
}
// calculate the volume
if(metrics_request_flag & V_TET_VOLUME)
{
metric_vals->volume = (VERDICT_REAL)(metric_vals->jacobian / 6.0);
}
// calculate aspect ratio
if(metrics_request_flag & V_TET_ASPECT_BETA)
{
double surface_area = ((edges[2] * edges[0]).length() +
(edges[3] * edges[0]).length() +
(edges[4] * edges[1]).length() +
(edges[3] * edges[2]).length() ) * 0.5;
VerdictVector numerator = edges[3].length_squared() * ( edges[2] * edges[0] ) +
edges[2].length_squared() * ( edges[3] * edges[0] ) +
edges[0].length_squared() * ( edges[3] * edges[2] );
double volume = metric_vals->jacobian / 6.0;
if(volume < VERDICT_DBL_MIN )
metric_vals->aspect_beta = (VERDICT_REAL)(VERDICT_DBL_MAX);
else
metric_vals->aspect_beta =
(VERDICT_REAL)( numerator.length() * surface_area/ (108*volume*volume) );
}
// calculate the aspect gamma
if(metrics_request_flag & V_TET_ASPECT_GAMMA)
{
double volume = fabs( metric_vals->jacobian / 6.0 );
if( fabs( volume ) < VERDICT_DBL_MIN )
metric_vals->aspect_gamma = VERDICT_DBL_MAX;
else
{
double srms = sqrt((
edges[0].length_squared() + edges[1].length_squared() +
edges[2].length_squared() + edges[3].length_squared() +
edges[4].length_squared() + edges[5].length_squared()
) / 6.0 );
// cube the srms
srms *= (srms * srms);
metric_vals->aspect_gamma = (VERDICT_REAL)( srms / (8.47967 * volume ));
}
}
// calculate the shape of the tet
if(metrics_request_flag & ( V_TET_SHAPE | V_TET_SHAPE_AND_SIZE ) )
{
static const double two_thirds = 2.0/3.0;
double num = 3.0 * pow(root_of_2 * metric_vals->jacobian, two_thirds);
double den = 1.5 *
(edges[0] % edges[0] + edges[2] % edges[2] + edges[3] % edges[3]) -
(edges[0] % -edges[2] + -edges[2] % edges[3] + edges[3] % edges[0]);
if( den < VERDICT_DBL_MIN )
metric_vals->shape = (VERDICT_REAL)0.0;
else
metric_vals->shape = (VERDICT_REAL)VERDICT_MAX( num/den, 0 );
}
// calculate the relative size of the tet
if(metrics_request_flag & (V_TET_RELATIVE_SIZE_SQUARED | V_TET_SHAPE_AND_SIZE ))
{
VerdictVector w1, w2, w3;
get_weight(w1,w2,w3);
double avg_vol = (w1 % (w2 *w3))/6;
if( avg_vol < VERDICT_DBL_MIN )
metric_vals->relative_size_squared = 0.0;
else
{
double tmp = metric_vals->jacobian / (6*avg_vol);
if( tmp < VERDICT_DBL_MIN )
metric_vals->relative_size_squared = 0.0;
else
{
tmp *= tmp;
metric_vals->relative_size_squared = (VERDICT_REAL)VERDICT_MIN(tmp, 1/tmp);
}
}
}
// calculate the shape and size
if(metrics_request_flag & V_TET_SHAPE_AND_SIZE)
{
metric_vals->shape_and_size = (VERDICT_REAL)(metric_vals->shape * metric_vals->relative_size_squared);
}
// calculate the scaled jacobian
if(metrics_request_flag & V_TET_SCALED_JACOBIAN)
{
//find out which node the normalized jacobian can be calculated at
//and it will be the smaller than at other nodes
double length_squared[4] = {
edges[0].length_squared() * edges[2].length_squared() * edges[3].length_squared(),
edges[0].length_squared() * edges[1].length_squared() * edges[4].length_squared(),
edges[1].length_squared() * edges[2].length_squared() * edges[5].length_squared(),
edges[3].length_squared() * edges[4].length_squared() * edges[5].length_squared()
};
int which_node = 0;
if(length_squared[1] > length_squared[which_node])
which_node = 1;
if(length_squared[2] > length_squared[which_node])
which_node = 2;
if(length_squared[3] > length_squared[which_node])
which_node = 3;
// find the scaled jacobian at this node
double length_product = sqrt( length_squared[which_node] );
if(length_product < fabs(metric_vals->jacobian))
length_product = fabs(metric_vals->jacobian);
if( length_product < VERDICT_DBL_MIN )
metric_vals->scaled_jacobian = (VERDICT_REAL) VERDICT_DBL_MAX;
else
metric_vals->scaled_jacobian =
(VERDICT_REAL)(root_of_2 * metric_vals->jacobian / length_product);
}
// calculate the condition number
if(metrics_request_flag & V_TET_CONDITION)
{
static const double root_of_3 = sqrt(3.0);
static const double root_of_6 = sqrt(6.0);
VerdictVector c_1, c_2, c_3;
c_1 = edges[0];
c_2 = (-2*edges[2] - edges[0])/root_of_3;
c_3 = (3*edges[3] + edges[2] - edges[0])/root_of_6;
double term1 = c_1 % c_1 + c_2 % c_2 + c_3 % c_3;
double term2 = ( c_1 * c_2 ) % ( c_1 * c_2 ) +
( c_2 * c_3 ) % ( c_2 * c_3 ) +
( c_3 * c_1 ) % ( c_3 * c_1 );
double det = c_1 % ( c_2 * c_3 );
if(det <= VERDICT_DBL_MIN)
metric_vals->condition = (VERDICT_REAL)VERDICT_DBL_MAX;
else
metric_vals->condition = (VERDICT_REAL)(sqrt(term1 * term2) / (3.0*det));
}
// calculate the distortion
if(metrics_request_flag & V_TET_DISTORTION)
{
metric_vals->distortion = v_tet_distortion(num_nodes, coordinates);
}
//check for overflow
if(metrics_request_flag & V_TET_ASPECT_BETA )
{
if( metric_vals->aspect_beta > 0 )
metric_vals->aspect_beta = (VERDICT_REAL) VERDICT_MIN( metric_vals->aspect_beta, VERDICT_DBL_MAX );
metric_vals->aspect_beta = (VERDICT_REAL) VERDICT_MAX( metric_vals->aspect_beta, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_ASPECT_GAMMA)
{
if( metric_vals->aspect_gamma > 0 )
metric_vals->aspect_gamma = (VERDICT_REAL) VERDICT_MIN( metric_vals->aspect_gamma, VERDICT_DBL_MAX );
metric_vals->aspect_gamma = (VERDICT_REAL) VERDICT_MAX( metric_vals->aspect_gamma, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_VOLUME)
{
if( metric_vals->volume > 0 )
metric_vals->volume = (VERDICT_REAL) VERDICT_MIN( metric_vals->volume, VERDICT_DBL_MAX );
metric_vals->volume = (VERDICT_REAL) VERDICT_MAX( metric_vals->volume, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_CONDITION)
{
if( metric_vals->condition > 0 )
metric_vals->condition = (VERDICT_REAL) VERDICT_MIN( metric_vals->condition, VERDICT_DBL_MAX );
metric_vals->condition = (VERDICT_REAL) VERDICT_MAX( metric_vals->condition, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_JACOBIAN)
{
if( metric_vals->jacobian > 0 )
metric_vals->jacobian = (VERDICT_REAL) VERDICT_MIN( metric_vals->jacobian, VERDICT_DBL_MAX );
metric_vals->jacobian = (VERDICT_REAL) VERDICT_MAX( metric_vals->jacobian, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_SCALED_JACOBIAN)
{
if( metric_vals->scaled_jacobian > 0 )
metric_vals->scaled_jacobian = (VERDICT_REAL) VERDICT_MIN( metric_vals->scaled_jacobian, VERDICT_DBL_MAX );
metric_vals->scaled_jacobian = (VERDICT_REAL) VERDICT_MAX( metric_vals->scaled_jacobian, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_SHAPE)
{
if( metric_vals->shape > 0 )
metric_vals->shape = (VERDICT_REAL) VERDICT_MIN( metric_vals->shape, VERDICT_DBL_MAX );
metric_vals->shape = (VERDICT_REAL) VERDICT_MAX( metric_vals->shape, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_RELATIVE_SIZE_SQUARED)
{
if( metric_vals->relative_size_squared > 0 )
metric_vals->relative_size_squared = (VERDICT_REAL) VERDICT_MIN( metric_vals->relative_size_squared, VERDICT_DBL_MAX );
metric_vals->relative_size_squared = (VERDICT_REAL) VERDICT_MAX( metric_vals->relative_size_squared, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_SHAPE_AND_SIZE)
{
if( metric_vals->shape_and_size > 0 )
metric_vals->shape_and_size = (VERDICT_REAL) VERDICT_MIN( metric_vals->shape_and_size, VERDICT_DBL_MAX );
metric_vals->shape_and_size = (VERDICT_REAL) VERDICT_MAX( metric_vals->shape_and_size, -VERDICT_DBL_MAX );
}
if(metrics_request_flag & V_TET_DISTORTION)
{
if( metric_vals->distortion > 0 )
metric_vals->distortion = (VERDICT_REAL) VERDICT_MIN( metric_vals->distortion, VERDICT_DBL_MAX );
metric_vals->distortion = (VERDICT_REAL) VERDICT_MAX( metric_vals->distortion, -VERDICT_DBL_MAX );
}
}