// exact solution
scalar Fn(double u)
{
double s, c;
fresnl(sqrt(2/M_PI) * u, &s , &c);
scalar fres = cplx(c,-s);
scalar a = cplx(0.0, M_PI/4);
scalar b = cplx(0.0, u*u);
return 0.5*sqrt(M_PI) * exp(b) * (exp(-a) - sqrt(2.0)*(fres));
}
scalar der_Hr(double x, double y)
{
double r = sqrt(x*x + y*y);
double t = atan2(y,x);
scalar a = cplx(0.0, M_PI/4 - k*r);
scalar i = cplx(0.0,1.0);
return 1/sqrt(M_PI) * exp(a) *
( (-i*k)*(Fn(sqrt(2*k*r)*sin(t/2 - M_PI/8)) + Fn(sqrt(2*k*r)*sin(t/2 + M_PI/8))) +
(Fder(sqrt(2*k*r)*sin(t/2 - M_PI/8))*(sqrt(k)/sqrt(2*r)*sin(t/2 - M_PI/8)) +
Fder(sqrt(2*k*r)*sin(t/2 + M_PI/8))*(sqrt(k)/sqrt(2*r)*sin(t/2 + M_PI/8))));
}
scalar Fder(double u)
{
scalar a = cplx(0.0, M_PI/4);
scalar b = cplx(0.0, u*u);
scalar d = cplx(0.0, 2.0*u);
double s, c;
fresnl(sqrt(2/M_PI) * u, &s , &c);
scalar fres = cplx(c,-s);
scalar fresder = exp(-b);
return 0.5*sqrt(M_PI) * exp(b) * ( d * (exp(-a) - sqrt(2.0)*(fres)) - sqrt(2.0)*fresder*sqrt(2.0/M_PI) );
}
scalar Fder2(double u)
{
scalar a = cplx(0.0, M_PI/4);
scalar i = cplx(0.0,1.0);
scalar b = cplx(0.0, u*u);
scalar d = cplx(0.0, 2.0*u);
double s, c;
fresnl(sqrt(2/M_PI) * u, &s , &c);
scalar fres = cplx(c,-s);
scalar fresder = exp(-b);
scalar fresder2 = exp(-b)*(-2.0 * i * u);
return 2.0 * u * i * Fder(u) +
0.5 * sqrt(M_PI) * exp(b) *
( 2.0 * i * (exp(-a) - sqrt(2.0)*(fres)) + d * (-sqrt(2.0)*fresder*sqrt(2.0/M_PI)) - sqrt(2.0) * fresder2 * sqrt(2.0/M_PI) );
}
Scalar bilinear_form(int n, double *wt, Func<Scalar> *u_ext[], Func<Real> *u, Func<Real> *v, Geom<Real> *e, ExtData<Scalar> *ext)
{
cplx ikappa = cplx(0.0, kappa);
return 1.0/mu_r * int_curl_e_curl_f<Real, Scalar>(n, wt, u, v) -
ikappa * sqrt(mu_0 / e_0) * int_F_e_f<Real, Scalar>(n, wt, gam, u, v, e) -
sqr(kappa) * int_F_e_f<Real, Scalar>(n, wt, er, u, v, e);
}
void process_periods(cplx tau1, cplx tau2) {
if (tau1 == cplx(0.0)) {
throw std::string("period cannot be zero");
}
cplx tau(tau2/tau1);
cplx f(1.0/tau1);
if (tau.imag() == 0) {
throw std::string("degenerate periods");
}
if (tau.imag() < 0) {
tau = 1.0/tau;
f = 1.0/tau2;
}
tau.real() -= floor(tau.real() + 0.5);
while (norm(tau) < 1.0) {
tau = -1.0/tau;
tau.real() -= floor(tau.real() + 0.5);
f *= tau;
}
f_ = f;
f2_ = f*f;
tau_ = tau;
}
scalar liform_surf(int n, double *wt, Func<scalar> *u_ext[], Func<double> *v, Geom<double> *e, ExtData<scalar> *ext)
{
cplx ii = cplx(0.0, 1.0);
scalar result = 0;
for (int i = 0; i < n; i++) {
scalar dx[3], dy[3], dz[3];
scalar3 ev;
ev[0] = exact(e->x[i], e->y[i], e->z[i], dx, dy, dz)[0];
ev[1] = exact(e->x[i], e->y[i], e->z[i], dx, dy, dz)[0];
ev[2] = exact(e->x[i], e->y[i], e->z[i], dx, dy, dz)[0];
scalar curl_e[3];
calc_curl(dx, dy, dz, curl_e);
scalar tpe[3];
calc_tan_proj(e->nx[i], e->ny[i], e->nz[i], ev, tpe);
scalar g[3] = {
(e->nz[i] * curl_e[1] - e->ny[i] * curl_e[2]) - ii * kappa * tpe[0],
(e->nx[i] * curl_e[2] - e->nz[i] * curl_e[0]) - ii * kappa * tpe[1],
(e->ny[i] * curl_e[0] - e->nx[i] * curl_e[1]) - ii * kappa * tpe[2],
};
// tpv is tangencial projection of v (test function)
scalar vv[3] = { v->val0[i], v->val1[i], v->val2[i] };
scalar tpv[3];
calc_tan_proj(e->nx[i], e->ny[i], e->nz[i], vv, tpv);
result += wt[i] * (g[0] * tpv[0] + g[1] * tpv[1] + g[2] * tpv[2]);
}
return result;
}
void CooMatrix::copy_into(Matrix *m)
{
m->free_data();
int index = 0;
if (this->complex)
{
for(std::map<size_t, std::map<size_t, cplx> >::const_iterator it_row = A_cplx.begin(); it_row != A_cplx.end(); ++it_row)
{
for(std::map<size_t, cplx>::const_iterator it_col = it_row->second.begin(); it_col != it_row->second.end(); ++it_col)
{
m->add(it_row->first,
it_col->first,
cplx(it_col->second.real(), it_col->second.imag()));
index++;
}
}
}
else
{
for(std::map<size_t, std::map<size_t, double> >::const_iterator it_row = A.begin(); it_row != A.end(); ++it_row)
{
for(std::map<size_t, double>::const_iterator it_col = it_row->second.begin(); it_col != it_row->second.end(); ++it_col)
{
m->add(it_row->first,
it_col->first,
it_col->second);
index++;
}
}
}
}
scalar der_Hrr(double x, double y)
{
double r = sqrt(x*x + y*y);
double t = atan2(y,x);
scalar a = cplx(0.0, M_PI/4 - k*r);
scalar i = cplx(0.0,1.0);
scalar f1_d = Fder(sqrt(2*k*r)*sin(t/2 - M_PI/8));
scalar f2_d = Fder(sqrt(2*k*r)*sin(t/2 + M_PI/8));
scalar f1_d2 = Fder2(sqrt(2*k*r)*sin(t/2 - M_PI/8));
scalar f2_d2 = Fder2(sqrt(2*k*r)*sin(t/2 + M_PI/8));
scalar b1 = (sqrt(k/(2*r))*sin(t/2 - M_PI/8));
scalar b2 = (sqrt(k/(2*r))*sin(t/2 + M_PI/8));
return -i*k*der_Hr(x,y) + 1/sqrt(M_PI) * exp(a) *
( (-i*k)*(f1_d*b1 + f2_d*b2) +
( f1_d2*b1*b1 + f2_d2*b2*b2) +
f1_d*(-0.5*sqrt(k/(2*r*r*r))*sin(t/2 - M_PI/8)) + f2_d*(-0.5*sqrt(k/(2*r*r*r))*sin(t/2 + M_PI/8)));
}
scalar exact1(double x, double y, scalar& dx, scalar& dy)
{
double r = sqrt(x*x + y*y);
double theta = atan2(y,x);
scalar Hr = der_Hr(x,y);
scalar Ht = der_Ht(x,y);
scalar i = cplx(0.0,1.0);
return i * ( Hr * x/r - Ht * y/(r*r));
}
scalar der_Ht(double x, double y)
{
double r = sqrt(x*x + y*y);
double t = atan2(y,x);
scalar a = cplx(0.0, M_PI/4 - k*r);
return 1/sqrt(M_PI) * exp(a) *
(Fder(sqrt(2*k*r)*sin(t/2 - M_PI/8))*(sqrt(k*r/2)*cos(t/2 - M_PI/8)) +
Fder(sqrt(2*k*r)*sin(t/2 + M_PI/8))*(sqrt(k*r/2)*cos(t/2 + M_PI/8)));
}
scalar der_Htr(double x, double y)
{
double r = sqrt(x*x + y*y);
double t = atan2(y,x);
scalar i = cplx(0.0,1.0);
scalar a = cplx(0.0, M_PI/4 - k*r);
scalar f1_d = Fder(sqrt(2*k*r)*sin(t/2 - M_PI/8));
scalar f2_d = Fder(sqrt(2*k*r)*sin(t/2 + M_PI/8));
scalar f1_d2 = Fder2(sqrt(2*k*r)*sin(t/2 - M_PI/8));
scalar f2_d2 = Fder2(sqrt(2*k*r)*sin(t/2 + M_PI/8));
scalar b1 = (sqrt(k)/sqrt(2*r)*sin(t/2 - M_PI/8));
scalar b2 = (sqrt(k)/sqrt(2*r)*sin(t/2 + M_PI/8));
scalar c1 = (sqrt(k*r)/sqrt(2.0)*cos(t/2 - M_PI/8));
scalar c2 = (sqrt(k*r)/sqrt(2.0)*cos(t/2 + M_PI/8));
return -i*k*der_Ht(x,y) + 1/sqrt(M_PI) * exp(a) *
((f1_d2*b1*c1 + f2_d2*b2*c2) +
f1_d*(0.5*sqrt(k/(2*r))*cos(t/2 - M_PI/8)) + f2_d*(0.5*sqrt(k/(2*r))*cos(t/2 + M_PI/8)));
}
t_cplx cplx_cos(t_cplx z)
{
double a;
double b;
a = exp(z.i);
b = 1.0 / a;
a = a / 2.0;
b = b/2.0;
return (cplx(cos(z.r) * (a + b), - sin(z.r ) * (a - b)));
}
void burningship(t_env *env, int x, int y)
{
size_t i;
t_cplx z;
t_cplx c;
t_cplx a;
c.r = env->ptx + ((x - (SCREEN_W / 2)) / env->zoom);
c.i = env->pty + ((y - (SCREEN_H / 2)) / env->zoom);
z = cplx(0.0, 0.0);
a = cplx(c.r, c.i);
i = -1;
while (++i < env->max_i && mod(z) < 2)
{
z = cplx(fabs(z.r), fabs(z.i));
z = cplx_add(cplx_mult(z, z), c);
}
plotpixel(env, x, y, get_color(env, z, a, i));
}
Scalar matrix_form(int n, double *wt, Func<Scalar> *u_ext[], Func<Real> *u,
Func<Real> *v, Geom<Real> *e, ExtData<Scalar> *ext) const {
cplx ikappa = cplx(0.0, kappa);
Scalar result1 = 0;
Scalar result2 = 0;
for (int i = 0; i < n; i++)
result1 += wt[i] * gamma(e->elem_marker, e->x[i], e->y[i]) * (u->val0[i] * conj(v->val0[i]) + u->val1[i] * conj(v->val1[i]));
for (int i = 0; i < n; i++)
result2 += wt[i] * er(e->elem_marker, e->x[i], e->y[i]) * (u->val0[i] * conj(v->val0[i]) + u->val1[i] * conj(v->val1[i]));
return 1.0/mu_r * int_curl_e_curl_f<Real, Scalar>(n, wt, u, v) -
ikappa * sqrt(mu_0 / e_0) * result1 - sqr(kappa) * result2;
}
void init_from_periods(int n, cplx tau1, cplx tau2) {
process_periods(tau1, tau2);
cplx q(std::exp(tau_*cplx(0, M_PI)));
cplx q2(q*q);
/* e1 */
cplx e1(0.0);
for (cplx q2k(q2); ; q2k *= q2) {
cplx t1(1.0 + q2k);
cplx t2(1.0 - q2k);
cplx term(q2k*(1.0/(t1*t1) + 1.0/(t2*t2)));
if (std::abs(term) <= 1e-10) break;
e1 += term;
}
e1 = (e1*8.0 + 2.0/3)*M_PI*M_PI;
/* e2 */
cplx e2(0.0);
cplx qk;
double s = -1;
for (qk = q, s = -1; ; qk *= q, s = -s) {
cplx t1(1.0 - qk);
cplx term(s*qk/(t1*t1));
if (std::abs(term) <= 1e-10) break;
e2 += term;
}
e2 = (e2*8.0 - 1.0/3)*M_PI*M_PI;
/* e3 */
cplx e3(0.0);
for (qk = q, s = -1; ; qk *= q, s = -s) {
cplx t1(1.0 - s*qk);
cplx term(qk/(t1*t1));
if (std::abs(term) <= 1e-10) break;
e3 += term;
}
e3 = (e3*8.0 - 1.0/3)*M_PI*M_PI;
init(n, e1, e2, e3);
}
t_cplx cplx_pow(t_cplx z, double n)
{
double h;
double lr;
double li;
double a;
h = mod(z);
lr = pow(h, n);
a = atan2(z.i, z.r);
li = n * a;
return (cplx(lr * cos(li), lr * sin(li)));
}
Scalar J_cranic(int n, double *wt, Func<Scalar> *u_ext[], Func<Real> *u, Func<Real> *v, Geom<Real> *e, ExtData<Scalar> *ext)
{
scalar ii = cplx(0.0, 1.0); // imaginary unit, ii^2 = -1
Scalar result = 0;
Func<Scalar>* psi_prev_newton = u_ext[0];
for (int i = 0; i < n; i++)
result += wt[i] * (ii * H * u->val[i] * v->val[i] / time_step
- 0.5*H*H/(2*M) * (u->dx[i] * v->dx[i] + u->dy[i] * v->dy[i])
- 0.5*2* G * u->val[i] * psi_prev_newton->val[i] * conj(psi_prev_newton->val[i]) * v->val[i]
- 0.5*G * psi_prev_newton->val[i] * psi_prev_newton->val[i] * u->val[i] * v->val[i]
- 0.5*.5*M*OMEGA*OMEGA * (e->x[i] * e->x[i] + e->y[i] * e->y[i]) * u->val[i] * v->val[i]);
return result;
}
CustomWeakFormAcoustics(std::string bdy_newton, double rho, double sound_speed, double omega) : WeakForm(1)
{
scalar ii = cplx(0.0, 1.0);
// Jacobian.
add_matrix_form(new WeakFormsH1::DefaultJacobianDiffusion(0, 0, HERMES_ANY, new HermesFunction(1.0/rho), HERMES_SYM));
add_matrix_form(new WeakFormsH1::DefaultMatrixFormVol(0, 0, HERMES_ANY, new HermesFunction(-sqr(omega) / rho / sqr(sound_speed)), HERMES_SYM));
add_matrix_form_surf(new WeakFormsH1::DefaultMatrixFormSurf(0, 0, bdy_newton, new HermesFunction(-ii * omega / rho / sound_speed)));
// Residual.
add_vector_form(new WeakFormsH1::DefaultResidualDiffusion(0, HERMES_ANY, new HermesFunction(1.0/rho)));
add_vector_form(new WeakFormsH1::DefaultResidualVol(0, HERMES_ANY, new HermesFunction(-sqr(omega) / rho / sqr(sound_speed))));
add_vector_form_surf(new WeakFormsH1::DefaultResidualSurf(0, bdy_newton, new HermesFunction(-ii * omega / rho / sound_speed)));
};
Scalar CustomWeakFormGPRK::CustomFormMatrixFormVol::matrix_form_rk(int n, double *wt, Func<Scalar> *u_ext[], Func<Real> *u,
Func<Real> *v, Geom<Real> *e, ExtData<Scalar> *ext) const
{
scalar ii = cplx(0.0, 1.0); // imaginary unit, ii^2 = -1
Scalar result = 0;
Func<Scalar>* psi_prev_newton = u_ext[0];
for (int i = 0; i < n; i++)
result += wt[i] * (h*h/(2*m*ii*h) * (u->dx[i] * v->dx[i] + u->dy[i] * v->dy[i])
+ 2*g/(ii*h)* u->val[i] * psi_prev_newton->val[i] * conj(psi_prev_newton->val[i]) * v->val[i]
+ (g/ii*h) * psi_prev_newton->val[i] * psi_prev_newton->val[i] * u->val[i] * v->val[i]
+ .5*m*omega*omega/(ii*h) * (e->x[i] * e->x[i] + e->y[i] * e->y[i]) * u->val[i] * v->val[i]);
return result;
}
scalar MumpsMatrix::get(int m, int n)
{
_F_
// Find m-th row in the n-th column.
int mid = find_position(Ai + Ap[n], Ap[n + 1] - Ap[n], m);
// Return 0 if the entry has not been found.
if (mid < 0) return 0.0;
// Otherwise, add offset to the n-th column and return the value.
if (mid >= 0) mid += Ap[n];
#if !defined(H2D_COMPLEX) && !defined(H3D_COMPLEX)
return Ax[mid];
#else
return cplx(Ax[mid].r, Ax[mid].i);
#endif
}
CustomWeakForm(double mu_r, double kappa) : WeakForm(1)
{
cplx ii = cplx(0.0, 1.0);
// Jacobian.
add_matrix_form(new WeakFormsHcurl::DefaultJacobianCurlCurl(0, 0, HERMES_ANY, 1.0/mu_r));
add_matrix_form(new WeakFormsHcurl::DefaultMatrixFormVol(0, 0, HERMES_ANY, -sqr(kappa)));
add_matrix_form_surf(new WeakFormsHcurl::DefaultMatrixFormSurf(0, 0, HERMES_ANY, -kappa*ii));
// Residual.
add_vector_form(new WeakFormsHcurl::DefaultResidualCurlCurl(0, HERMES_ANY, 1.0/mu_r));
add_vector_form(new WeakFormsHcurl::DefaultResidualVol(0, HERMES_ANY, -sqr(kappa)));
add_vector_form_surf(new WeakFormsHcurl::DefaultResidualSurf(0, HERMES_ANY, -kappa*ii));
add_vector_form_surf(new CustomVectorFormSurf());
};
scalar der_Htt(double x, double y)
{
double r = sqrt(x*x + y*y);
double t = atan2(y,x);
scalar a = cplx(0.0, M_PI/4 - k*r);
scalar f1_d = Fder(sqrt(2*k*r)*sin(t/2 - M_PI/8));
scalar f2_d = Fder(sqrt(2*k*r)*sin(t/2 + M_PI/8));
scalar f1_d2 = Fder2(sqrt(2*k*r)*sin(t/2 - M_PI/8));
scalar f2_d2 = Fder2(sqrt(2*k*r)*sin(t/2 + M_PI/8));
scalar c1 = (sqrt(k*r/(2))*cos(t/2 - M_PI/8));
scalar c2 = (sqrt(k*r/(2))*cos(t/2 + M_PI/8));
return 1/sqrt(M_PI) * exp(a) *
((f1_d2*c1*c1 + f2_d2*c2*c2) +
f1_d*(-0.5*sqrt(k*r/2)*sin(t/2 - M_PI/8)) + f2_d*(-0.5*sqrt(k*r/2)*sin(t/2 + M_PI/8)));
}
Scalar F_euler(int n, double *wt, Func<Scalar> *u_ext[], Func<Real> *v, Geom<Real> *e, ExtData<Scalar> *ext)
{
scalar ii = cplx(0.0, 1.0); // imaginary unit, ii^2 = -1
Scalar result = 0;
Func<Scalar>* psi_prev_newton = u_ext[0];
Func<Scalar>* psi_prev_time = ext->fn[0];
for (int i = 0; i < n; i++)
result += wt[i] * (ii * H * (psi_prev_newton->val[i] - psi_prev_time->val[i]) * v->val[i] / time_step
- H*H/(2*M) * (psi_prev_newton->dx[i] * v->dx[i] + psi_prev_newton->dy[i] * v->dy[i])
- G * psi_prev_newton->val[i] * psi_prev_newton->val[i] * conj(psi_prev_newton->val[i]) * v->val[i]
- .5*M*OMEGA*OMEGA * (e->x[i] * e->x[i] + e->y[i] * e->y[i]) * psi_prev_newton->val[i] * v->val[i]);
return result;
}
t_cplx rings(t_coeff col, double x, double y)
{
t_cplx z;
double r;
double theta;
double prefix;
r = sqrt(x * x + y * y);
theta = atan2(y, x);
prefix = mod(cplx((r + col.pa2 * col.pa2)
, (2.0 * col.pa2 * col.pa2))) - (col.pa2 * col.pa2)
+ (r * (1.0 - col.pa2 * col.pa2));
z.r = prefix * cos(theta);
z.i = prefix * sin(theta);
return (z);
}
void fcmul(cplxfloat_t *c, const cplxfloat_t *a, const cplxfloat_t *b, unsigned int d1, unsigned int d2, unsigned int d3)
{
cplxfloat_t *r = c, s;
unsigned int i, j, k;
if (c == a || c == b)
r = alloca(d1 * d3 * sizeof(r[0]));
for (i = 0; i < d1; i++)
for (k = 0; k < d3; k++) {
cplx(s, 0, 0);
for (j = 0; j < d2; j++)
cmac(s, a[i*d2+j], b[j*d3+k]);
r[i*d3+k] = s;
}
if (r != c)
memcpy(c, r, d1 * d3 * sizeof(c[0]));
}
scalar linear_form_surf(int n, double *wt, Func<double> *v, Geom<double> *e, ExtData<scalar> *ext)
{
scalar result = 0;
for (int i = 0; i < n; i++)
{
double r = sqrt(e->x[i] * e->x[i] + e->y[i] * e->y[i]);
double theta = atan2(e->y[i], e->x[i]);
if (theta < 0) theta += 2.0*M_PI;
double j13 = jv(-1.0/3.0, r), j23 = jv(+2.0/3.0, r);
double cost = cos(theta), sint = sin(theta);
double cos23t = cos(2.0/3.0*theta), sin23t = sin(2.0/3.0*theta);
double Etau = e->tx[i] * (cos23t*sint*j13 - 2.0/(3.0*r)*j23*(cos23t*sint + sin23t*cost)) +
e->ty[i] * (-cos23t*cost*j13 + 2.0/(3.0*r)*j23*(cos23t*cost - sin23t*sint));
result += wt[i] * cplx(cos23t*j23, -Etau) * ((v->val0[i] * e->tx[i] + v->val1[i] * e->ty[i]));
}
return result;
}
scalar biform_surf(int n, double *wt, Func<scalar> *u_ext[], Func<double> *u, Func<double> *v, Geom<double> *e, ExtData<scalar> *ext)
{
// j * kappa * E_T * F_T
// E_T = nu x E x nu (nu is outer normal)
cplx ii = cplx(0.0, 1.0);
scalar result = 0;
for (int i = 0; i < n; i++) {
scalar uu[3] = { u->val0[i], u->val1[i], u->val2[i] };
scalar tpu[3];
calc_tan_proj(e->nx[i], e->ny[i], e->nz[i], uu, tpu);
scalar vv[3] = { v->val0[i], v->val1[i], v->val2[i] };
scalar tpv[3];
calc_tan_proj(e->nx[i], e->ny[i], e->nz[i], vv, tpv);
result += wt[i] * (uu[0] * vv[0] + uu[1] * vv[1] + uu[2] * vv[2]);
}
return ii * (-kappa) * result;
}
static void exact_sol(double x, double y, scalar& u0, scalar& u1, scalar& u1dx, scalar& u0dy)
{
scalar dx,dy;
u0 = exact0(x,y,dx,dy);
u1 = exact1(x,y,dx,dy);
scalar Hr = der_Hr(x,y);
scalar Ht = der_Ht(x,y);
scalar Hrr = der_Hrr(x,y);
scalar Hrt = der_Hrt(x,y);
scalar Htr = der_Htr(x,y);
scalar Htt = der_Htt(x,y);
double r = sqrt(x*x + y*y);
double theta = atan2(y,x);
scalar i = cplx(0.0,1.0);
u1dx = i * (( Hrr * x/r + Hrt * (-y/(r*r))) * x/r + Hr * (y*y)/(r*r*r) -
((Htr * x/r + Htt * (-y/(r*r))) * y/(r*r) + Ht * (-2.0*x*y/(r*r*r*r))));
u0dy = -i * (( Hrr * y/r + Hrt * x/(r*r)) * y/r + Hr * (x*x)/(r*r*r) +
(Htr * y/r + Htt * x/(r*r)) * x/(r*r) + Ht * (-2.0*x*y/(r*r*r*r)));
}
t_cplx fan(t_coeff col, double x, double y)
{
double r;
double theta;
double t;
t_cplx z;
r = sqrt(x * x + y * y);
theta = atan2(y, x);
t = M_PI * col.cc * col.cc;
if (mod(cplx(theta, t)) > (t * 0.5))
{
z.r = r * cos(theta - (t * .5));
z.i = r * sin(theta - (t * .5));
}
else
{
z.r = r * cos(theta + (t * 0.5));
z.i = r * sin(theta + (t * 0.5));
}
return (z);
}
