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    /*
       HYPERB.I

       Yorick functions to manipulate 3-D rays on a cylindrical mesh.

       $Id: rays.i,v 1.1 1993/09/02 22:48:38 munro Exp $
     */
    /*    Copyright (c) 1994.  The Regents of the University of California.
                        All rights reserved.  */

    /*

       form_rays      -- convert from nrays lists to (5, 3, or 6)-by-nrays

       best_rays      -- convert to 5-by-nrays representation

       dirt_rays      -- convert to 3-by-nrays representation

       internal_rays  -- convert to 6-by-nrays representation

       picture_rays   -- return rays at centers of arbitrary "pixel mesh"

       get_s0         -- compute origins for slimits (internally, slimits
                         is based on an s coordinate which is zero at the
    		     point of closest approach to the origin, rather

    		     than at the x,y,z point selected in the best_rays

    		     or internal_rays coordinate systems)

       plray          -- plot a ray

       plray_lims     -- set limits for plray
     */

    func form_rays (rays)

    /* DOCUMENT best= form_rays( [x, y, z, theta, phi] )

             or dirt= form_rays( [x, y, theta] )

    	 or internal= form_rays( [cos, sin, y, z, x, r] )
         forms 5-by-nrays, 3-by-nrays, or 6-by-nrays ray representation

         given individual lists of array coordinates.  The [...]
         operator builds an nrays-by-5, nrays-by-3, or nrays-by-6

         array, which form_rays transposes.  The "nrays" may represent
         zero or more actual dimensions.

       SEE ALSO: best_rays, dirt_rays, internal_rays, picture_rays
     */

    { return transpose(rays, 2); }

    func best_rays (rays)

    /* DOCUMENT best_rays(rays)
         returns 5-element (x,y,z,theta,phi) representation of RAYS.
         The first dimension of RAYS may be length 3, 5, or 6 to represent
         the ray(s) in TDG/DIRT coordinates (x,y,theta), "best" coordinates

         (x,y,z,theta,phi), or internal coordinates (cos,sin,y,z,x,r),
         respectively.  The first dimension of the result always has length 5.

         The "best" coordinate system is the easiest to visualize:
         (x,y,z) represents any point on the ray, while (theta,phi)
         represents the ray direction in standard spherical coordinates
         relative to the +z-axis.  Namely, theta is the angle from the

         +z-direction to the ray direction (between 0 and pi), and phi is
         the counterclockwise angle from the +x-axis to the projection of
         the ray direction into the xy-plane, assuming xyz is a right-handed
         coordinate system.

         As a specification of a ray, this system is doubly redundant because
         the point (x,y,z) could be any point on the ray, and the underlying
         mesh through which the ray propagates is cylindrically symmetric about
         the z-axis.

         However, the slimits parameter -- used to specify the points along

         a ray where the transport integration starts and stops -- is
         measured from the point (x,y,z) specified as a part of the
         (x,y,z,theta,phi) ray coordinate.  Thus, any change in the point
         (x,y,z) on a ray must be accompanied by a corresponding change in
         the slimits for that ray.

       SEE ALSO: form_rays, dirt_rays, internal_rays, get_s0, picture_rays
     */
    {
      dummy= use_origins(0);

      dim= dimsof(rays)(2);
      if (dim==5) return rays;

      if (dim==3) {
        /* convert from DIRT/TDG (x,y,theta) to (x,y,z,theta,phi) */
        x= rays(1,..);
        y= rays(2,..);
        theta= rays(3,..);

        return form_rays([x*cos(theta), y, x*sin(theta),

    		      abs(theta), (theta>=0.0)*pi]);

      } else if (dim==6) {

        /* convert from internal (cos,sin,y,z,x,r) to (x,y,z,theta,phi) */

        theta= atan(rays(2,..), rays(1,..));
        y= rays(3,..);
        z= rays(4,..);
        x= rays(5,..);

        return form_rays([x, y, z, abs(theta), (theta<0)*pi]);
      }

      return [];
    }

    func dirt_rays (rays)

    /* DOCUMENT dirt_rays(rays)
         returns 3-element (x,y,theta) representation of RAYS.
         The first dimension of RAYS may be length 3, 5, or 6 to represent
         the ray(s) in TDG/DIRT coordinates (x,y,theta), "best" coordinates

         (x,y,z,theta,phi), or internal coordinates (cos,sin,y,z,x,r),
         respectively.  The first dimension of the result always has length 3.

         The TDG/DIRT coordinate system is based on the coordinates (x,y)
         in a plane normal to the ray.  Unfortunately, the old TDG and DIRT
         codes used an angle theta which has the opposite sense from the
         "best" and internal coordinates.  Therefore, conversion from
         TDG/DIRT coordinates to internal coordinates will reverse the

         sign of theta.  Conversion from TDG/DIRT coordinates to "best"
         coordinates always results in positive theta, but the angle phi
         will be pi for positive input theta.

         The slimits parameter -- used to specify the points along

         a ray where the transport integration starts and stops -- is
         measured from the point of closest approach of the ray described
         by (x,y,theta) to the origin x=y=z=0.  Therefore, slimits is
         independent of the TDG/DIRT ray coordinate representation.

       SEE ALSO: form_rays, best_rays, internal_rays, get_s0, picture_rays
     */
    {
      dummy= use_origins(0);

      dim= dimsof(rays)(2);
      if (dim==3) return rays;

      if (dim==5) {
        /* convert from best (x,y,z,theta,phi) to DIRT/TDG (x,y,theta) */
        x= rays(1,..);
        y= rays(2,..);
        z= rays(3,..);
        theta= rays(4,..);
        phi= rays(5,..);
        cosp= cos(phi);
        sinp= sin(phi);
        sgn= double((cosp>=0.0)*2-1);

        return form_rays(sgn*[(x*cosp+y*sinp)*cos(theta) - z*sin(theta),
    			  (y*cosp-x*sinp), -theta]);

      } else if (dim==6) {

        /* convert from internal (cos,sin,y,z,x,r) to DIRT/TDG (x,y,theta) */
        cost= rays(1,..);
        sint= rays(2,..)
        y= rays(3,..);
        z= rays(4,..);
        x= rays(5,..);
        /* NOTE SIGN CHANGE IN THETA */

        return form_rays([x*cost-z*sint, y, -atan(sint, cost)]);
      }

      return [];
    }

    func internal_rays (rays)

    /* DOCUMENT internal_rays(rays)

         returns 6-element (cos,sin,y,z,x,r) representation of RAYS.
         The first dimension of RAYS may be length 3, 5, or 6 to represent
         the ray(s) in TDG/DIRT coordinates (x,y,theta), "best" coordinates

         (x,y,z,theta,phi), or internal coordinates (cos,sin,y,z,x,r),
         respectively.  The first dimension of the result always has length 6.

         The internal coordinates are what Drat uses internally to

         describe the ray.  The coordinate system is rotated about the
         z-axis until the ray lies in a plane of constant y (there are at
         least two ways to do this).  The point (x,y,z) can be any point on

         the ray, and r=sqrt(x^2+y^2) is the corresponding cylindrical radius.
         The clockwise angle theta from the +z-axis to the ray direction

         (which always lies in the zx-plane) determines cos=cos(theta) and

         sin=sin(theta).

         As a specification of a ray, this system is triply redundant because
         the point (x,y,z) could be any point on the ray, both the sine and

         cosine of theta appear, and r=sqrt(x^2+y^2).

         However, the slimits parameter -- used to specify the points along

         a ray where the transport integration starts and stops -- is
         measured from the point (x,y,z) specified as a part of the

         (cos,sin,y,z,x,r) ray coordinate.  Thus, any change in the point
         (x,y,z) on a ray must be accompanied by a corresponding change in
         the slimits for that ray.

       SEE ALSO: form_rays, best_rays, dirt_rays, get_s0, picture_rays
     */
    {
      dummy= use_origins(0);

      dim= dimsof(rays)(2);

      if (dim==6) {
        /* assure that r is consistent with x and y */

        rays(6,..)= abs(rays(3,..), rays(5,..));
        return rays;

      } else if (dim==5) {

        /* convert from best (x,y,z,theta,phi) to internal (cos,sin,y,z,x,r) */
        x= rays(1,..);
        y= rays(2,..);
        z= rays(3,..);
        theta= rays(4,..);
        phi= rays(5,..);
        cosp= cos(phi);
        sinp= sin(phi);
        sgn= double((cosp>=0.0)*2-1);

        return form_rays([cos(theta), sgn*sin(theta), sgn*(y*cosp-x*sinp), z,

    		      sgn*(x*cosp+y*sinp), abs(x, y)]);

      } else if (dim==3) {

        /* convert from DIRT/TDG (x,y,theta) to internal (cos,sin,y,z,x,r) */
        x= rays(1,..);
        y= rays(2,..);
        theta= rays(3,..);
        cost= cos(theta);
        sint= sin(theta);
        xcost= x*cost;
        /* NOTE SIGN CHANGE IN THETA */

        return form_rays([cost, -sint, y, x*sint, xcost, abs(xcost, y)]);
      }

      return [];
    }

    func get_s0 (rays)

    /* DOCUMENT get_s0(rays)
         returns the s-coordinate of the point of closest approach of
         the RAYS to the origin x=y=z=0.  The length of the first dimension
         of RAYS may be either 3, 5, or 6; this first dimension will not
         be present in the result.

         The s-coordinate represents distance along the ray, increasing in
         the direction the ray moves.  The 5 and 6 component ray coordinates

         include a reference point (x,y,z) on the ray; s=0 at that point.

         For the 3 component ray coordinate, get_s0 always returns 0.

       SEE ALSO: best_rays, dirt_rays, internal_rays
     */
    {
      dummy= use_origins(0);

      dim= dimsof(rays)(2);

      if (dim==3) {

        return array(0.0, dimsof(rays(1,..)));

      } else if (dim==5) {
        x= rays(1,..);
        y= rays(2,..);
        z= rays(3,..);
        theta= rays(4,..);
        phi= rays(5,..);
        cost= cos(theta);
        sint= sin(theta);

        x= x*cos(phi)+y*sin(phi);

      } else if (dim==6) {
        cost= rays(1,..);
        sint= rays(2,..)
        z= rays(4,..);
        x= rays(5,..);
      }

      return -z*cost-x*sint;
    }

    func picture_rays (xpict, ypict, ray, theta_up, phi_up)

    /* DOCUMENT picture_rays(xpict, ypict, ray)

             or picture_rays(xpict, ypict, ray, theta_up, phi_up)

         returns 2-D array of rays, one at the center of each zone (which

         represents a pixel here) of the mesh (XPICT, YPICT).  The rays are
         all parallel to the given RAY (a 3, 5, or 6 element vector).  The

         (XPICT, YPICT) coordinates are in the plane perpendicular to the rays,
         with the origin XPICT=YPICT=0 at the given RAY.

         If (THETA_UP, PHI_UP) are given, the +YPICT-axis will lie along the
         projection of the (THETA_UP, PHI_UP) direction into the (XPICT, YPICT)

         plane.  The default (THETA_UP, PHI_UP) is (pi/2, pi/2) -- the +y-axis

         -- unless (THETA, PHI) is the y-axis, in which case it is (pi/2, 0)
         -- the +x-axis.  This matches the DIRT/TDG ray coordinate convention
         in the sense that if RAY is [0,0,theta], then

         (zncen(XPICT),zncen(YPICT)) are the DIRT/TDG (x,y) coordinates for
         the rays.

         If XPICT and YPICT are imax-by-jmax, the returned array will have
         dimensions 5-by-(imax-1)-by-(jmax-1).  That is, "best" coordinates

         are returned.  The (x,y,z) of all of the returned rays will lie in
         the plane perpendicular to the ray passing through the given central
         RAY.

       SEE ALSO: form_rays, best_rays, dirt_rays, internal_rays, area
     */
    {
      dummy= use_origins(0);
      ray= best_rays(ray);
      x= ray(1);
      y= ray(2);
      z= ray(3);
      theta= ray(4);
      phi= ray(5);

      /* form (rx,ry,rz) ray direction cosines */
      sint= sin(theta);
      rx= sint*cos(phi);
      ry= sint*sin(phi);
      rz= cos(theta);

      /* form (ux,uy,uz) picture up direction cosines */
      if (!is_void(theta_up)) {
        sint= sin(theta_up);
        ux= sint*cos(phi_up);
        uy= sint*sin(phi_up);
        uz= cos(theta_up);
      } else {
        if (rz || rx) { ux= 0.0; uy= 1.0; }
        else { ux= 1.0; uy= 0.0; }
        uz= 0.0;
      }

      /* form (px,py,pz) picture right direction cosines */
      px= uy*rz - uz*ry;
      py= uz*rx - ux*rz;
      pz= ux*ry - uy*rx;
      len= abs(px, py, pz);
      px/= len;
      py/= len;
      pz/= len;

      /* project (ux,uy,uz) perpendicular to (rx,ry,rz) */
      ux= ry*pz - rz*py;
      uy= rz*px - rx*pz;
      uz= rx*py - ry*px;

      /* get coordinates of zone centers in picture plane */
      xp= xpict(zcen, zcen);
      yp= ypict(zcen, zcen);

      return form_rays([x+xp*px+yp*ux, y+xp*py+yp*uy, z+xp*pz+yp*uz,
    		    theta, phi]);
    }

    func plray (ray)
    /* DOCUMENT plray, ray

         where RAY is a vector of length 5 representing [x, y, z, theta, phi]
         -- a point (x,y,z) on the ray, and the ray direction (theta,phi)
         relative to the z-axis.  The ray hyperbola is plotted with the
         z-axis horizontal.  The portion of the ray plotted is determined

         by the plray_lims command, which must be issued prior to the first
         plray.
         The 3 and 6 component ray formats are also accepted.

       SEE ALSO: plray_lims, dirt_rays, internal_rays
     */
    {
      dims= dimsof(ray);

      if (dims(1)!=1) error, "ray must be a vector of length 5 (or 3 or 6)";

      if (dims(2)!=5) ry= best_rays(ray);
      else ry= ray;

      dummy= use_origins(0);

      x0= ry(1);
      y0= ry(2);
      z0= ry(3);
      theta= ry(4);
      phi= ry(5);

      sint= sin(theta);
      cost= cos(theta);
      sinp= sin(phi);
      cosp= cos(phi);

      rtp= abs(x0*sinp - y0*cosp);  /* turning point radius */

      /**/ extern plray_zx, plray_zn, plray_rx;
      /**/ extern plray_n;   /* always odd! */

      if (sint < 1.e-4) {

        /* rays which are nearly parallel to the axis need special treatment */
        z= span(plray_zn, plray_zx, 5);
        if (cost<0.0) z= z(::-1);
        r= (z-z0)*sint/cost;
        r= abs(x0+r*cosp, y0+r*sinp);

      } else {

        /* form th == tan(angle)/abs(tan(theta)) where angle is evenly
           spaced as th varies from -1 to 1 */

        abstan= abs(sint)/(abs(cost)+1.e-20);
        absth= atan(abstan);
        fuzz= 1.e-6;
        th= span(-absth, absth, plray_n);
        th(1)= -(1.0-0.5*fuzz*fuzz);
        th(2:-1)= tan(th(2:-1))/abstan;
        th(0)= -th(1);

        sqth= sqrt((1.0-th)*(1.0+th));
        sqth(1)= sqth(0)= fuzz;

        /* rays which nearly hit the axis need special treatment */
        if (rtp < fuzz*plray_rx) rt= fuzz*plray_rx/sqth;
        else rt= rtp/sqth;

        zt= z0 + (rt*th - (x0*cosp-y0*sinp)*sign(sint))*sign(cost)/abstan;

        /* clip (zt,rt) to (z,r) with z between plray_zn and plray_zx */

        list= where(zt>plray_zn & zt<plray_zx);

        if (min(abs(zt(dif))) > 0.0) {
          if (cost>=0.0) {  /* z increasing */
            if (min(zt)<=plray_zn) {
    	  z= [plray_zn];
    	  r= interp(rt, zt, z);
    	} else {
    	  z= [];
    	  r= [];
    	}
    	grow, z, zt(list);
    	grow, r, rt(list);
            if (max(zt)>=plray_zx) {
    	  grow, z, [plray_zx];

    	  grow, r, interp(rt, zt, [plray_zx]);
    	}
          } else {          /* z decreasing */
            if (max(zt)>=plray_zx) {
    	  z= [plray_zx];
    	  r= interp(rt, zt, z);
    	} else {
    	  z= [];
    	  r= [];
    	}
    	grow, z, zt(list);
    	grow, r, rt(list);
            if (min(zt)<=plray_zn) {
    	  grow, z, [plray_zn];

    	  grow, r, interp(rt, zt, [plray_zn]);
    	}
          }
        } else {
          z= zt(list);
          r= rt(list);
        }
      }

      /* clip (z,r) to r < rx */

      if (numberof(r) && numberof((list= where(r<plray_rx)))) {

        if (numberof(list) < numberof(r)) {
          rt= r;
          zt= z;

          i= rt(mnx);  /* the interp operation seems to be safe even if many
    		      of the rt are clustered at this minimum value */

          if (min(list)<=i && max(rt(:i))>=plray_rx) {
    	r= [plray_rx];
    	z= interp(zt(:i), rt(:i), r);
          } else {
    	r= [];
    	z= [];
          }
          grow, r, rt(list);
          grow, z, zt(list);

          if (max(list)>=i && max(rt(i:))>=plray_rx) {
    	grow, r, [plray_rx];

    	grow, z, interp(zt(i:), rt(i:), [plray_rx]);
          }
        }

        /* finally, plot the result */

        /* Note: Gist substitutes the marker for \001 if that is the
                 first character of the legend.  */
        legend=

          swrite(format="\001: theta= %.4g, phi= %.4g, thru (%.4g, %.4g, %.4g)",
    	     theta, phi, x0, y0, z0)(1);

        plg, r, z, rays=1, legend=legend;

      } else {

        write, "WARNING (plray) ray not within limits (use plray_lims)";
      }
    }

    func plray_lims (zmin, zmax, rmax, rx)

    /* DOCUMENT plray_lims, zmin, zmax, rmax

         sets the (z,r) coordinate limits for the plray command.  Subsequent

         rays will be clipped to the region from z=ZMIN to z=ZMAX, and from
         r=0 to r=RMAX.
       SEE ALSO: plray
     */
    {
      /**/ extern plray_zx, plray_zn, plray_rx;

      if (!is_void(rx)) rmax= rx;  /* allow for limits syntax as well */
      if (zmin!=zmax && rmax>0.0) {
        plray_zx= double(max(zmin, zmax));
        plray_zn= double(min(zmin, zmax));
        plray_rx= double(rmax);

        limits, plray_zn, plray_zx, 0.0, plray_rx;
      } else {

        error, "zmin==zmax or rmax<=0.0 are not legal ray limits";
      }
    }

    plray_zx= 1.0;
    plray_zn= -1.0;
    plray_rx= 1.0;
    plray_n= 31;    /* This should always be an odd number to ensure that
    		   the turning point of the ray is always one of the
    		   points plotted.  */