# Rectangular Prism - rectangular_prism.c

```    ```static double
form_volume(double length_a, double b2a_ratio, double c2a_ratio)
{
return length_a * (length_a*b2a_ratio) * (length_a*c2a_ratio);
}

static double
radius_from_excluded_volume(double length_a, double b2a_ratio, double c2a_ratio)
{
double const r_equiv   = sqrt(length_a*length_a*b2a_ratio/M_PI);
double const length_c  = c2a_ratio*length_a;
return 0.5*cbrt(0.75*r_equiv*(2.0*r_equiv*length_c + (r_equiv + length_c)*(M_PI*r_equiv + length_c)));
}

static double
radius_effective(int mode, double length_a, double b2a_ratio, double c2a_ratio)
{
switch (mode) {
default:
case 1: // equivalent cylinder excluded volume
case 2: // equivalent volume sphere
return cbrt(cube(length_a)*b2a_ratio*c2a_ratio/M_4PI_3);
case 3: // half length_a
return 0.5 * length_a;
case 4: // half length_b
return 0.5 * length_a*b2a_ratio;
case 5: // half length_c
return 0.5 * length_a*c2a_ratio;
case 6: // equivalent circular cross-section
return length_a*sqrt(b2a_ratio/M_PI);
case 7: // half ab diagonal
return 0.5*sqrt(square(length_a) * (1.0 + square(b2a_ratio)));
case 8: // half diagonal
return 0.5*sqrt(square(length_a) * (1.0 + square(b2a_ratio) + square(c2a_ratio)));
}
}

static double
Iq(double q,
double sld,
double solvent_sld,
double length_a,
double b2a_ratio,
double c2a_ratio)
{
const double length_b = length_a * b2a_ratio;
const double length_c = length_a * c2a_ratio;
const double a_half = 0.5 * length_a;
const double b_half = 0.5 * length_b;
const double c_half = 0.5 * length_c;

//Integration limits to use in Gaussian quadrature
const double v1a = 0.0;
const double v1b = M_PI_2;  //theta integration limits
const double v2a = 0.0;
const double v2b = M_PI_2;  //phi integration limits

double outer_sum = 0.0;
for(int i=0; i<GAUSS_N; i++) {
const double theta = 0.5 * ( GAUSS_Z[i]*(v1b-v1a) + v1a + v1b );
double sin_theta, cos_theta;
SINCOS(theta, sin_theta, cos_theta);

const double termC = sas_sinx_x(q * c_half * cos_theta);

double inner_sum = 0.0;
for(int j=0; j<GAUSS_N; j++) {
double phi = 0.5 * ( GAUSS_Z[j]*(v2b-v2a) + v2a + v2b );
double sin_phi, cos_phi;
SINCOS(phi, sin_phi, cos_phi);

// Amplitude AP from eqn. (12), rewritten to avoid round-off effects when arg=0
const double termA = sas_sinx_x(q * a_half * sin_theta * sin_phi);
const double termB = sas_sinx_x(q * b_half * sin_theta * cos_phi);
const double AP = termA * termB * termC;
inner_sum += GAUSS_W[j] * AP * AP;
}
inner_sum = 0.5 * (v2b-v2a) * inner_sum;
outer_sum += GAUSS_W[i] * inner_sum * sin_theta;
}

// Normalize by Pi (Eqn. 16).
// The term (ABC)^2 does not appear because it was introduced before on
// the definitions of termA, termB, termC.
// The factor 2 appears because the theta integral has been defined between
// 0 and pi/2, instead of 0 to pi.
answer /= M_PI_2; //Form factor P(q)

// Multiply by contrast^2 and volume^2
const double volume = length_a * length_b * length_c;

// Convert from [1e-12 A-1] to [cm-1]

}

static void
Fq(double q,
double *F1,
double *F2,
double sld,
double solvent_sld,
double length_a,
double b2a_ratio,
double c2a_ratio)
{
const double length_b = length_a * b2a_ratio;
const double length_c = length_a * c2a_ratio;
const double a_half = 0.5 * length_a;
const double b_half = 0.5 * length_b;
const double c_half = 0.5 * length_c;

//Integration limits to use in Gaussian quadrature
const double v1a = 0.0;
const double v1b = M_PI_2;  //theta integration limits
const double v2a = 0.0;
const double v2b = M_PI_2;  //phi integration limits

double outer_sum_F1 = 0.0;
double outer_sum_F2 = 0.0;
for(int i=0; i<GAUSS_N; i++) {
const double theta = 0.5 * ( GAUSS_Z[i]*(v1b-v1a) + v1a + v1b );
double sin_theta, cos_theta;
SINCOS(theta, sin_theta, cos_theta);

const double termC = sas_sinx_x(q * c_half * cos_theta);

double inner_sum_F1 = 0.0;
double inner_sum_F2 = 0.0;
for(int j=0; j<GAUSS_N; j++) {
double phi = 0.5 * ( GAUSS_Z[j]*(v2b-v2a) + v2a + v2b );
double sin_phi, cos_phi;
SINCOS(phi, sin_phi, cos_phi);

// Amplitude AP from eqn. (12), rewritten to avoid round-off effects when arg=0
const double termA = sas_sinx_x(q * a_half * sin_theta * sin_phi);
const double termB = sas_sinx_x(q * b_half * sin_theta * cos_phi);
const double AP = termA * termB * termC;
inner_sum_F1 += GAUSS_W[j] * AP;
inner_sum_F2 += GAUSS_W[j] * AP * AP;
}
inner_sum_F1 = 0.5 * (v2b-v2a) * inner_sum_F1;
inner_sum_F2 = 0.5 * (v2b-v2a) * inner_sum_F2;
outer_sum_F1 += GAUSS_W[i] * inner_sum_F1 * sin_theta;
outer_sum_F2 += GAUSS_W[i] * inner_sum_F2 * sin_theta;
}

outer_sum_F1 *= 0.5*(v1b-v1a);
outer_sum_F2 *= 0.5*(v1b-v1a);

// Normalize by Pi (Eqn. 16).
// The term (ABC)^2 does not appear because it was introduced before on
// the definitions of termA, termB, termC.
// The factor 2 appears because the theta integral has been defined between
// 0 and pi/2, instead of 0 to pi.
outer_sum_F1 /= M_PI_2;
outer_sum_F2 /= M_PI_2;

// Multiply by contrast and volume
const double s = (sld-solvent_sld) * (length_a * length_b * length_c);

// Convert from [1e-12 A-1] to [cm-1]
*F1 = 1e-2 * s * outer_sum_F1;
*F2 = 1e-4 * s * s * outer_sum_F2;
}

static double
Iqabc(double qa, double qb, double qc,
double sld,
double solvent_sld,
double length_a,
double b2a_ratio,
double c2a_ratio)
{
const double length_b = length_a * b2a_ratio;
const double length_c = length_a * c2a_ratio;
const double a_half = 0.5 * length_a;
const double b_half = 0.5 * length_b;
const double c_half = 0.5 * length_c;

// Amplitude AP from eqn. (13)
const double termA = sas_sinx_x(qa * a_half);
const double termB = sas_sinx_x(qb * b_half);
const double termC = sas_sinx_x(qc * c_half);
const double AP = termA * termB * termC;

// Multiply by contrast and volume
const double s = (sld-solvent_sld) * (length_a * length_b * length_c);

// Convert from [1e-12 A-1] to [cm-1]
return 1.0e-4 * square(s * AP);
}
```
```