+/* Draw a squeezed torus Klein bottle projected into 3D. */
+static int squeezed_torus(ModeInfo *mi, double umin, double umax, double vmin,
+ double vmax)
+{
+ int polys = 0;
+ static const GLfloat mat_diff_red[] = { 1.0, 0.0, 0.0, 1.0 };
+ static const GLfloat mat_diff_green[] = { 0.0, 1.0, 0.0, 1.0 };
+ static const GLfloat mat_diff_trans_red[] = { 1.0, 0.0, 0.0, 0.7 };
+ static const GLfloat mat_diff_trans_green[] = { 0.0, 1.0, 0.0, 0.7 };
+ float p[3], pu[3], pv[3], pm[3], n[3], b[3], mat[4][4];
+ int i, j, k, l, m, o;
+ double u, v;
+ double xx[4], xxu[4], xxv[4], y[4], yu[4], yv[4];
+ double q, r, s, t;
+ double cu, su, cv, sv, cv2, sv2;
+ float q1[4], q2[4], r1[4][4], r2[4][4];
+ kleinstruct *kb = &klein[MI_SCREEN(mi)];
+
+ if (view == VIEW_WALK || view == VIEW_WALKTURN)
+ {
+ /* Compute the rotation that rotates the Klein bottle in 4D without the
+ trackball rotations. */
+ rotateall4d(kb->zeta,kb->eta,kb->theta,mat);
+
+ u = kb->umove;
+ v = kb->vmove;
+ cu = cos(u);
+ su = sin(u);
+ cv = cos(v);
+ sv = sin(v);
+ cv2 = cos(0.5*v);
+ sv2 = sin(0.5*v);
+ xx[0] = (SQUEEZED_TORUS_RADIUS+cu)*cv;
+ xx[1] = (SQUEEZED_TORUS_RADIUS+cu)*sv;
+ xx[2] = su*cv2;
+ xx[3] = su*sv2;
+ xxu[0] = -su*cv;
+ xxu[1] = -su*sv;
+ xxu[2] = cu*cv2;
+ xxu[3] = cu*sv2;
+ xxv[0] = -(SQUEEZED_TORUS_RADIUS+cu)*sv;
+ xxv[1] = (SQUEEZED_TORUS_RADIUS+cu)*cv;
+ xxv[2] = -0.5*su*sv2;
+ xxv[3] = 0.5*su*cv2;
+ for (l=0; l<4; l++)
+ {
+ xx[l] /= SQUEEZED_TORUS_RADIUS+1.25;
+ xxu[l] /= SQUEEZED_TORUS_RADIUS+1.25;
+ xxv[l] /= SQUEEZED_TORUS_RADIUS+1.25;
+ }
+ for (l=0; l<4; l++)
+ {
+ y[l] = (mat[l][0]*xx[0]+mat[l][1]*xx[1]+
+ mat[l][2]*xx[2]+mat[l][3]*xx[3]);
+ yu[l] = (mat[l][0]*xxu[0]+mat[l][1]*xxu[1]+
+ mat[l][2]*xxu[2]+mat[l][3]*xxu[3]);
+ yv[l] = (mat[l][0]*xxv[0]+mat[l][1]*xxv[1]+
+ mat[l][2]*xxv[2]+mat[l][3]*xxv[3]);
+ }
+ if (projection_4d == DISP_4D_ORTHOGRAPHIC)
+ {
+ for (l=0; l<3; l++)
+ {
+ p[l] = y[l]+kb->offset4d[l];
+ pu[l] = yu[l];
+ pv[l] = yv[l];
+ }
+ }
+ else
+ {
+ s = y[3]+kb->offset4d[3];
+ q = 1.0/s;
+ t = q*q;
+ for (l=0; l<3; l++)
+ {
+ r = y[l]+kb->offset4d[l];
+ p[l] = r*q;
+ pu[l] = (yu[l]*s-r*yu[3])*t;
+ pv[l] = (yv[l]*s-r*yv[3])*t;
+ }
+ }
+ n[0] = pu[1]*pv[2]-pu[2]*pv[1];
+ n[1] = pu[2]*pv[0]-pu[0]*pv[2];
+ n[2] = pu[0]*pv[1]-pu[1]*pv[0];
+ t = 1.0/(kb->side*4.0*sqrt(n[0]*n[0]+n[1]*n[1]+n[2]*n[2]));
+ n[0] *= t;
+ n[1] *= t;
+ n[2] *= t;
+ pm[0] = pu[0]*kb->dumove+pv[0]*kb->dvmove;
+ pm[1] = pu[1]*kb->dumove+pv[1]*kb->dvmove;
+ pm[2] = pu[2]*kb->dumove+pv[2]*kb->dvmove;
+ t = 1.0/(4.0*sqrt(pm[0]*pm[0]+pm[1]*pm[1]+pm[2]*pm[2]));
+ pm[0] *= t;
+ pm[1] *= t;
+ pm[2] *= t;
+ b[0] = n[1]*pm[2]-n[2]*pm[1];
+ b[1] = n[2]*pm[0]-n[0]*pm[2];
+ b[2] = n[0]*pm[1]-n[1]*pm[0];
+ t = 1.0/(4.0*sqrt(b[0]*b[0]+b[1]*b[1]+b[2]*b[2]));
+ b[0] *= t;
+ b[1] *= t;
+ b[2] *= t;
+
+ /* Compute alpha, beta, delta from the three basis vectors.
+ | -b[0] -b[1] -b[2] |
+ m = | n[0] n[1] n[2] |
+ | -pm[0] -pm[1] -pm[2] |
+ */
+ kb->alpha = atan2(-n[2],-pm[2])*180/M_PI;
+ kb->beta = atan2(-b[2],sqrt(b[0]*b[0]+b[1]*b[1]))*180/M_PI;
+ kb->delta = atan2(b[1],-b[0])*180/M_PI;
+
+ /* Compute the rotation that rotates the Klein bottle in 4D. */
+ rotateall(kb->alpha,kb->beta,kb->delta,kb->zeta,kb->eta,kb->theta,mat);
+
+ u = kb->umove;
+ v = kb->vmove;
+ cu = cos(u);
+ su = sin(u);
+ cv = cos(v);
+ sv = sin(v);
+ cv2 = cos(0.5*v);
+ sv2 = sin(0.5*v);
+ xx[0] = (SQUEEZED_TORUS_RADIUS+cu)*cv;
+ xx[1] = (SQUEEZED_TORUS_RADIUS+cu)*sv;
+ xx[2] = su*cv2;
+ xx[3] = su*sv2;
+ for (l=0; l<4; l++)
+ xx[l] /= SQUEEZED_TORUS_RADIUS+1.25;
+ for (l=0; l<4; l++)
+ {
+ r = 0.0;
+ for (m=0; m<4; m++)
+ r += mat[l][m]*xx[m];
+ y[l] = r;
+ }
+ if (projection_4d == DISP_4D_ORTHOGRAPHIC)
+ {
+ for (l=0; l<3; l++)
+ p[l] = y[l]+kb->offset4d[l];
+ }
+ else
+ {
+ s = y[3]+kb->offset4d[3];
+ for (l=0; l<3; l++)
+ p[l] = (y[l]+kb->offset4d[l])/s;
+ }
+
+ kb->offset3d[0] = -p[0];
+ kb->offset3d[1] = -p[1]-DELTAY;
+ kb->offset3d[2] = -p[2];
+ }
+ else
+ {
+ /* Compute the rotation that rotates the Klein bottle in 4D, including
+ the trackball rotations. */
+ rotateall(kb->alpha,kb->beta,kb->delta,kb->zeta,kb->eta,kb->theta,r1);
+
+ gltrackball_get_quaternion(kb->trackballs[0],q1);
+ gltrackball_get_quaternion(kb->trackballs[1],q2);
+ quats_to_rotmat(q1,q2,r2);
+
+ mult_rotmat(r2,r1,mat);
+ }
+
+ /* Project the points from 4D to 3D. */
+ for (i=0; i<=NUMU; i++)
+ {
+ for (j=0; j<=NUMV; j++)
+ {
+ o = i*(NUMV+1)+j;
+ for (l=0; l<4; l++)
+ {
+ y[l] = (mat[l][0]*kb->x[o][0]+mat[l][1]*kb->x[o][1]+
+ mat[l][2]*kb->x[o][2]+mat[l][3]*kb->x[o][3]);
+ yu[l] = (mat[l][0]*kb->xu[o][0]+mat[l][1]*kb->xu[o][1]+
+ mat[l][2]*kb->xu[o][2]+mat[l][3]*kb->xu[o][3]);
+ yv[l] = (mat[l][0]*kb->xv[o][0]+mat[l][1]*kb->xv[o][1]+
+ mat[l][2]*kb->xv[o][2]+mat[l][3]*kb->xv[o][3]);
+ }
+ if (projection_4d == DISP_4D_ORTHOGRAPHIC)
+ {
+ for (l=0; l<3; l++)
+ {
+ kb->pp[o][l] = (y[l]+kb->offset4d[l])+kb->offset3d[l];
+ pu[l] = yu[l];
+ pv[l] = yv[l];
+ }
+ }
+ else
+ {
+ s = y[3]+kb->offset4d[3];
+ q = 1.0/s;
+ t = q*q;
+ for (l=0; l<3; l++)
+ {
+ r = y[l]+kb->offset4d[l];
+ kb->pp[o][l] = r*q+kb->offset3d[l];
+ pu[l] = (yu[l]*s-r*yu[3])*t;
+ pv[l] = (yv[l]*s-r*yv[3])*t;
+ }
+ }
+ kb->pn[o][0] = pu[1]*pv[2]-pu[2]*pv[1];
+ kb->pn[o][1] = pu[2]*pv[0]-pu[0]*pv[2];
+ kb->pn[o][2] = pu[0]*pv[1]-pu[1]*pv[0];
+ t = 1.0/sqrt(kb->pn[o][0]*kb->pn[o][0]+kb->pn[o][1]*kb->pn[o][1]+
+ kb->pn[o][2]*kb->pn[o][2]);
+ kb->pn[o][0] *= t;
+ kb->pn[o][1] *= t;
+ kb->pn[o][2] *= t;
+ }
+ }
+
+ if (colors == COLORS_TWOSIDED)
+ {
+ glColor3fv(mat_diff_red);
+ if (display_mode == DISP_TRANSPARENT)
+ {
+ glMaterialfv(GL_FRONT,GL_AMBIENT_AND_DIFFUSE,mat_diff_trans_red);
+ glMaterialfv(GL_BACK,GL_AMBIENT_AND_DIFFUSE,mat_diff_trans_green);
+ }
+ else
+ {
+ glMaterialfv(GL_FRONT,GL_AMBIENT_AND_DIFFUSE,mat_diff_red);
+ glMaterialfv(GL_BACK,GL_AMBIENT_AND_DIFFUSE,mat_diff_green);
+ }
+ }
+ glBindTexture(GL_TEXTURE_2D,kb->tex_name);
+
+ for (i=0; i<NUMU; i++)
+ {
+ if (appearance == APPEARANCE_BANDS && ((i & (NUMB-1)) >= NUMB/2))
+ continue;
+ if (display_mode == DISP_WIREFRAME)
+ glBegin(GL_QUAD_STRIP);
+ else
+ glBegin(GL_TRIANGLE_STRIP);
+ for (j=0; j<=NUMV; j++)
+ {
+ for (k=0; k<=1; k++)
+ {
+ l = (i+k);
+ m = j;
+ o = l*(NUMV+1)+m;
+ glNormal3fv(kb->pn[o]);
+ glTexCoord2fv(kb->tex[o]);
+ if (colors != COLORS_TWOSIDED)
+ {
+ glColor3fv(kb->col[o]);
+ glMaterialfv(GL_FRONT_AND_BACK,GL_AMBIENT_AND_DIFFUSE,kb->col[o]);
+ }
+ glVertex3fv(kb->pp[o]);
+ polys++;
+ }
+ }
+ glEnd();
+ }
+ polys /= 2;
+ return polys;
+}
+
+