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+/* projectiveplane --- Shows a 4d embedding of the real projective plane
+   that rotates in 4d or on which you can walk */
+
+#if 0
+static const char sccsid[] = "@(#)projectiveplane.c  1.1 14/01/01 xlockmore";
+#endif
+
+/* Copyright (c) 2005-2014 Carsten Steger <carsten@mirsanmir.org>. */
+
+/*
+ * Permission to use, copy, modify, and distribute this software and its
+ * documentation for any purpose and without fee is hereby granted,
+ * provided that the above copyright notice appear in all copies and that
+ * both that copyright notice and this permission notice appear in
+ * supporting documentation.
+ *
+ * This file is provided AS IS with no warranties of any kind.  The author
+ * shall have no liability with respect to the infringement of copyrights,
+ * trade secrets or any patents by this file or any part thereof.  In no
+ * event will the author be liable for any lost revenue or profits or
+ * other special, indirect and consequential damages.
+ *
+ * REVISION HISTORY:
+ * C. Steger - 14/01/03: Initial version
+ * C. Steger - 14/10/03: Moved the curlicue texture to curlicue.h
+ */
+
+/*
+ * This program shows a 4d embedding of the real projective plane.
+ * You can walk on the projective plane, see it turn in 4d, or walk on
+ * it while it turns in 4d.  The fact that the surface is an embedding
+ * of the real projective plane in 4d can be seen in the depth colors
+ * mode: set all rotation speeds to 0 and the projection mode to 4d
+ * orthographic projection.  In its default orientation, the embedding
+ * of the real projective plane will then project to the Roman
+ * surface, which has three lines of self-intersection.  However, at
+ * the three lines of self-intersection the parts of the surface that
+ * intersect have different colors, i.e., different 4d depths.
+ *
+ * The real projective plane is a non-orientable surface.  To make
+ * this apparent, the two-sided color mode can be used.
+ * Alternatively, orientation markers (curling arrows) can be drawn as
+ * a texture map on the surface of the projective plane.  While
+ * walking on the projective plane, you will notice that the
+ * orientation of the curling arrows changes (which it must because
+ * the projective plane is non-orientable).
+ *
+ * The real projective plane is a model for the projective geometry in
+ * 2d space.  One point can be singled out as the origin.  A line can
+ * be singled out as the line at infinity, i.e., a line that lies at
+ * an infinite distance to the origin.  The line at infinity is
+ * topologically a circle.  Points on the line at infinity are also
+ * used to model directions in projective geometry.  The origin can be
+ * visualized in different manners.  When using distance colors, the
+ * origin is the point that is displayed as fully saturated red, which
+ * is easier to see as the center of the reddish area on the
+ * projective plane.  Alternatively, when using distance bands, the
+ * origin is the center of the only band that projects to a disc.
+ * When using direction bands, the origin is the point where all
+ * direction bands collapse to a point.  Finally, when orientation
+ * markers are being displayed, the origin the the point where all
+ * orientation markers are compressed to a point.  The line at
+ * infinity can also be visualized in different ways.  When using
+ * distance colors, the line at infinity is the line that is displayed
+ * as fully saturated magenta.  When two-sided colors are used, the
+ * line at infinity lies at the points where the red and green "sides"
+ * of the projective plane meet (of course, the real projective plane
+ * only has one side, so this is a design choice of the
+ * visualization).  Alternatively, when orientation markers are being
+ * displayed, the line at infinity is the place where the orientation
+ * markers change their orientation.
+ *
+ * Note that when the projective plane is displayed with bands, the
+ * orientation markers are placed in the middle of the bands.  For
+ * distance bands, the bands are chosen in such a way that the band at
+ * the origin is only half as wide as the remaining bands, which
+ * results in a disc being displayed at the origin that has the same
+ * diameter as the remaining bands.  This choice, however, also
+ * implies that the band at infinity is half as wide as the other
+ * bands.  Since the projective plane is attached to itself (in a
+ * complicated fashion) at the line at infinity, effectively the band
+ * at infinity is again as wide as the remaining bands.  However,
+ * since the orientation markers are displayed in the middle of the
+ * bands, this means that only one half of the orientation markers
+ * will be displayed twice at the line at infinity if distance bands
+ * are used.  If direction bands are used or if the projective plane
+ * is displayed as a solid surface, the orientation markers are
+ * displayed fully at the respective sides of the line at infinity.
+ *
+ * The program projects the 4d projective plane to 3d using either a
+ * perspective or an orthographic projection.  Which of the two
+ * alternatives looks more appealing is up to you.  However, two
+ * famous surfaces are obtained if orthographic 4d projection is used:
+ * The Roman surface and the cross cap.  If the projective plane is
+ * rotated in 4d, the result of the projection for certain rotations
+ * is a Roman surface and for certain rotations it is a cross cap.
+ * The easiest way to see this is to set all rotation speeds to 0 and
+ * the rotation speed around the yz plane to a value different from 0.
+ * However, for any 4d rotation speeds, the projections will generally
+ * cycle between the Roman surface and the cross cap.  The difference
+ * is where the origin and the line at infinity will lie with respect
+ * to the self-intersections in the projections to 3d.
+ *
+ * The projected projective plane can then be projected to the screen
+ * either perspectively or orthographically.  When using the walking
+ * modes, perspective projection to the screen will be used.
+ *
+ * There are three display modes for the projective plane: mesh
+ * (wireframe), solid, or transparent.  Furthermore, the appearance of
+ * the projective plane can be as a solid object or as a set of
+ * see-through bands.  The bands can be distance bands, i.e., bands
+ * that lie at increasing distances from the origin, or direction
+ * bands, i.e., bands that lie at increasing angles with respect to
+ * the origin.
+ *
+ * When the projective plane is displayed with direction bands, you
+ * will be able to see that each direction band (modulo the "pinching"
+ * at the origin) is a Moebius strip, which also shows that the
+ * projective plane is non-orientable.
+ *
+ * Finally, the colors with with the projective plane is drawn can be
+ * set to two-sided, distance, direction, or depth.  In two-sided
+ * mode, the projective plane is drawn with red on one "side" and
+ * green on the "other side".  As described above, the projective
+ * plane only has one side, so the color jumps from red to green along
+ * the line at infinity.  This mode enables you to see that the
+ * projective plane is non-orientable.  In distance mode, the
+ * projective plane is displayed with fully saturated colors that
+ * depend on the distance of the points on the projective plane to the
+ * origin.  The origin is displayed in red, the line at infinity is
+ * displayed in magenta.  If the projective plane is displayed as
+ * distance bands, each band will be displayed with a different color.
+ * In direction mode, the projective plane is displayed with fully
+ * saturated colors that depend on the angle of the points on the
+ * projective plane with respect to the origin.  Angles in opposite
+ * directions to the origin (e.g., 15 and 205 degrees) are displayed
+ * in the same color since they are projectively equivalent.  If the
+ * projective plane is displayed as direction bands, each band will be
+ * displayed with a different color.  Finally, in depth mode the
+ * projective plane with colors chosen depending on the 4d "depth"
+ * (i.e., the w coordinate) of the points on the projective plane at
+ * its default orientation in 4d.  As discussed above, this mode
+ * enables you to see that the projective plane does not intersect
+ * itself in 4d.
+ *
+ * The rotation speed for each of the six planes around which the
+ * projective plane rotates can be chosen.  For the walk-and-turn
+ * more, only the rotation speeds around the true 4d planes are used
+ * (the xy, xz, and yz planes).
+ *
+ * Furthermore, in the walking modes the walking direction in the 2d
+ * base square of the projective plane and the walking speed can be
+ * chosen.  The walking direction is measured as an angle in degrees
+ * in the 2d square that forms the coordinate system of the surface of
+ * the projective plane.  A value of 0 or 180 means that the walk is
+ * along a circle at a randomly chosen distance from the origin
+ * (parallel to a distance band).  A value of 90 or 270 means that the
+ * walk is directly from the origin to the line at infinity and back
+ * (analogous to a direction band).  Any other value results in a
+ * curved path from the origin to the line at infinity and back.
+ *
+ * This program is somewhat inspired by Thomas Banchoff's book "Beyond
+ * the Third Dimension: Geometry, Computer Graphics, and Higher
+ * Dimensions", Scientific American Library, 1990.
+ */
+
+#include "curlicue.h"
+
+#ifndef M_PI
+#define M_PI 3.14159265358979323846
+#endif
+
+#define DISP_WIREFRAME             0
+#define DISP_SURFACE               1
+#define DISP_TRANSPARENT           2
+#define NUM_DISPLAY_MODES          3
+
+#define APPEARANCE_SOLID           0
+#define APPEARANCE_DISTANCE_BANDS  1
+#define APPEARANCE_DIRECTION_BANDS 2
+#define NUM_APPEARANCES            3
+
+#define COLORS_TWOSIDED            0
+#define COLORS_DISTANCE            1
+#define COLORS_DIRECTION           2
+#define COLORS_DEPTH               3
+#define NUM_COLORS                 4
+
+#define VIEW_WALK                  0
+#define VIEW_TURN                  1
+#define VIEW_WALKTURN              2
+#define NUM_VIEW_MODES             3
+
+#define DISP_3D_PERSPECTIVE        0
+#define DISP_3D_ORTHOGRAPHIC       1
+#define NUM_DISP_3D_MODES          2
+
+#define DISP_4D_PERSPECTIVE        0
+#define DISP_4D_ORTHOGRAPHIC       1
+#define NUM_DISP_4D_MODES          2
+
+#define DEF_DISPLAY_MODE           "random"
+#define DEF_APPEARANCE             "random"
+#define DEF_COLORS                 "random"
+#define DEF_VIEW_MODE              "random"
+#define DEF_MARKS                  "False"
+#define DEF_PROJECTION_3D          "random"
+#define DEF_PROJECTION_4D          "random"
+#define DEF_SPEEDWX                "1.1"
+#define DEF_SPEEDWY                "1.3"
+#define DEF_SPEEDWZ                "1.5"
+#define DEF_SPEEDXY                "1.7"
+#define DEF_SPEEDXZ                "1.9"
+#define DEF_SPEEDYZ                "2.1"
+#define DEF_WALK_DIRECTION         "83.0"
+#define DEF_WALK_SPEED             "20.0"
+
+#ifdef STANDALONE
+# define DEFAULTS           "*delay:      10000 \n" \
+                            "*showFPS:    False \n" \
+
+# define free_projectiveplane 0
+# define release_projectiveplane 0
+# include "xlockmore.h"         /* from the xscreensaver distribution */
+#else  /* !STANDALONE */
+# include "xlock.h"             /* from the xlockmore distribution */
+#endif /* !STANDALONE */
+
+#ifdef USE_GL
+
+#ifndef HAVE_JWXYZ
+# include <X11/keysym.h>
+#endif
+
+#include "gltrackball.h"
+
+#include <float.h>
+
+
+#ifdef USE_MODULES
+ModStruct projectiveplane_description =
+{"projectiveplane", "init_projectiveplane", "draw_projectiveplane",
+ NULL, "draw_projectiveplane", "change_projectiveplane",
+ NULL, &projectiveplane_opts, 25000, 1, 1, 1, 1.0, 4, "",
+ "Rotate a 4d embedding of the real projective plane in 4d or walk on it",
+ 0, NULL};
+
+#endif
+
+
+static char *mode;
+static char *appear;
+static char *color_mode;
+static char *view_mode;
+static Bool marks;
+static char *proj_3d;
+static char *proj_4d;
+static float speed_wx;
+static float speed_wy;
+static float speed_wz;
+static float speed_xy;
+static float speed_xz;
+static float speed_yz;
+static float walk_direction;
+static float walk_speed;
+
+
+static XrmOptionDescRec opts[] =
+{
+  {"-mode",              ".displayMode",   XrmoptionSepArg, 0 },
+  {"-wireframe",         ".displayMode",   XrmoptionNoArg,  "wireframe" },
+  {"-surface",           ".displayMode",   XrmoptionNoArg,  "surface" },
+  {"-transparent",       ".displayMode",   XrmoptionNoArg,  "transparent" },
+  {"-appearance",        ".appearance",    XrmoptionSepArg, 0 },
+  {"-solid",             ".appearance",    XrmoptionNoArg,  "solid" },
+  {"-distance-bands",    ".appearance",    XrmoptionNoArg,  "distance-bands" },
+  {"-direction-bands",   ".appearance",    XrmoptionNoArg,  "direction-bands" },
+  {"-colors",            ".colors",        XrmoptionSepArg, 0 },
+  {"-twosided-colors",   ".colors",        XrmoptionNoArg,  "two-sided" },
+  {"-distance-colors",   ".colors",        XrmoptionNoArg,  "distance" },
+  {"-direction-colors",  ".colors",        XrmoptionNoArg,  "direction" },
+  {"-depth-colors",      ".colors",        XrmoptionNoArg,  "depth" },
+  {"-view-mode",         ".viewMode",      XrmoptionSepArg, 0 },
+  {"-walk",              ".viewMode",      XrmoptionNoArg,  "walk" },
+  {"-turn",              ".viewMode",      XrmoptionNoArg,  "turn" },
+  {"-walk-turn",         ".viewMode",      XrmoptionNoArg,  "walk-turn" },
+  {"-orientation-marks", ".marks",         XrmoptionNoArg, "on"},
+  {"+orientation-marks", ".marks",         XrmoptionNoArg, "off"},
+  {"-projection-3d",     ".projection3d",  XrmoptionSepArg, 0 },
+  {"-perspective-3d",    ".projection3d",  XrmoptionNoArg,  "perspective" },
+  {"-orthographic-3d",   ".projection3d",  XrmoptionNoArg,  "orthographic" },
+  {"-projection-4d",     ".projection4d",  XrmoptionSepArg, 0 },
+  {"-perspective-4d",    ".projection4d",  XrmoptionNoArg,  "perspective" },
+  {"-orthographic-4d",   ".projection4d",  XrmoptionNoArg,  "orthographic" },
+  {"-speed-wx",          ".speedwx",       XrmoptionSepArg, 0 },
+  {"-speed-wy",          ".speedwy",       XrmoptionSepArg, 0 },
+  {"-speed-wz",          ".speedwz",       XrmoptionSepArg, 0 },
+  {"-speed-xy",          ".speedxy",       XrmoptionSepArg, 0 },
+  {"-speed-xz",          ".speedxz",       XrmoptionSepArg, 0 },
+  {"-speed-yz",          ".speedyz",       XrmoptionSepArg, 0 },
+  {"-walk-direction",    ".walkDirection", XrmoptionSepArg, 0 },
+  {"-walk-speed",        ".walkSpeed",     XrmoptionSepArg, 0 }
+};
+
+static argtype vars[] =
+{
+  { &mode,           "displayMode",   "DisplayMode",   DEF_DISPLAY_MODE,   t_String },
+  { &appear,         "appearance",    "Appearance",    DEF_APPEARANCE,     t_String },
+  { &color_mode,     "colors",        "Colors",        DEF_COLORS,         t_String },
+  { &view_mode,      "viewMode",      "ViewMode",      DEF_VIEW_MODE,      t_String },
+  { &marks,          "marks",         "Marks",         DEF_MARKS,          t_Bool },
+  { &proj_3d,        "projection3d",  "Projection3d",  DEF_PROJECTION_3D,  t_String },
+  { &proj_4d,        "projection4d",  "Projection4d",  DEF_PROJECTION_4D,  t_String },
+  { &speed_wx,       "speedwx",       "Speedwx",       DEF_SPEEDWX,        t_Float},
+  { &speed_wy,       "speedwy",       "Speedwy",       DEF_SPEEDWY,        t_Float},
+  { &speed_wz,       "speedwz",       "Speedwz",       DEF_SPEEDWZ,        t_Float},
+  { &speed_xy,       "speedxy",       "Speedxy",       DEF_SPEEDXY,        t_Float},
+  { &speed_xz,       "speedxz",       "Speedxz",       DEF_SPEEDXZ,        t_Float},
+  { &speed_yz,       "speedyz",       "Speedyz",       DEF_SPEEDYZ,        t_Float},
+  { &walk_direction, "walkDirection", "WalkDirection", DEF_WALK_DIRECTION, t_Float},
+  { &walk_speed,     "walkSpeed",     "WalkSpeed",     DEF_WALK_SPEED,     t_Float}
+};
+
+ENTRYPOINT ModeSpecOpt projectiveplane_opts =
+{sizeof opts / sizeof opts[0], opts, sizeof vars / sizeof vars[0], vars, NULL};
+
+
+/* Offset by which we walk above the projective plane */
+#define DELTAY  0.01
+
+/* Number of subdivisions of the projective plane */
+#define NUMU 128
+#define NUMV 128
+
+/* Number of subdivisions per band */
+#define NUMB 8
+
+
+typedef struct {
+  GLint      WindH, WindW;
+  GLXContext *glx_context;
+  /* Options */
+  int display_mode;
+  int appearance;
+  int colors;
+  int view;
+  Bool marks;
+  int projection_3d;
+  int projection_4d;
+  /* 4D rotation angles */
+  float alpha, beta, delta, zeta, eta, theta;
+  /* Movement parameters */
+  float umove, vmove, dumove, dvmove;
+  int side, dir;
+  /* The viewing offset in 4d */
+  float offset4d[4];
+  /* The viewing offset in 3d */
+  float offset3d[4];
+  /* The 4d coordinates of the projective plane and their derivatives */
+  float x[(NUMU+1)*(NUMV+1)][4];
+  float xu[(NUMU+1)*(NUMV+1)][4];
+  float xv[(NUMU+1)*(NUMV+1)][4];
+  float pp[(NUMU+1)*(NUMV+1)][3];
+  float pn[(NUMU+1)*(NUMV+1)][3];
+  /* The precomputed colors of the projective plane */
+  float col[(NUMU+1)*(NUMV+1)][4];
+  /* The precomputed texture coordinates of the projective plane */
+  float tex[(NUMU+1)*(NUMV+1)][2];
+  /* The "curlicue" texture */
+  GLuint tex_name;
+  /* Aspect ratio of the current window */
+  float aspect;
+  /* Trackball states */
+  trackball_state *trackballs[2];
+  int current_trackball;
+  Bool button_pressed;
+  /* A random factor to modify the rotation speeds */
+  float speed_scale;
+} projectiveplanestruct;
+
+static projectiveplanestruct *projectiveplane = (projectiveplanestruct *) NULL;
+
+
+/* Add a rotation around the wx-plane to the matrix m. */
+static void rotatewx(float m[4][4], float phi)
+{
+  float c, s, u, v;
+  int i;
+
+  phi *= M_PI/180.0;
+  c = cos(phi);
+  s = sin(phi);
+  for (i=0; i<4; i++)
+  {
+    u = m[i][1];
+    v = m[i][2];
+    m[i][1] = c*u+s*v;
+    m[i][2] = -s*u+c*v;
+  }
+}
+
+
+/* Add a rotation around the wy-plane to the matrix m. */
+static void rotatewy(float m[4][4], float phi)
+{
+  float c, s, u, v;
+  int i;
+
+  phi *= M_PI/180.0;
+  c = cos(phi);
+  s = sin(phi);
+  for (i=0; i<4; i++)
+  {
+    u = m[i][0];
+    v = m[i][2];
+    m[i][0] = c*u-s*v;
+    m[i][2] = s*u+c*v;
+  }
+}
+
+
+/* Add a rotation around the wz-plane to the matrix m. */
+static void rotatewz(float m[4][4], float phi)
+{
+  float c, s, u, v;
+  int i;
+
+  phi *= M_PI/180.0;
+  c = cos(phi);
+  s = sin(phi);
+  for (i=0; i<4; i++)
+  {
+    u = m[i][0];
+    v = m[i][1];
+    m[i][0] = c*u+s*v;
+    m[i][1] = -s*u+c*v;
+  }
+}
+
+
+/* Add a rotation around the xy-plane to the matrix m. */
+static void rotatexy(float m[4][4], float phi)
+{
+  float c, s, u, v;
+  int i;
+
+  phi *= M_PI/180.0;
+  c = cos(phi);
+  s = sin(phi);
+  for (i=0; i<4; i++)
+  {
+    u = m[i][2];
+    v = m[i][3];
+    m[i][2] = c*u+s*v;
+    m[i][3] = -s*u+c*v;
+  }
+}
+
+
+/* Add a rotation around the xz-plane to the matrix m. */
+static void rotatexz(float m[4][4], float phi)
+{
+  float c, s, u, v;
+  int i;
+
+  phi *= M_PI/180.0;
+  c = cos(phi);
+  s = sin(phi);
+  for (i=0; i<4; i++)
+  {
+    u = m[i][1];
+    v = m[i][3];
+    m[i][1] = c*u-s*v;
+    m[i][3] = s*u+c*v;
+  }
+}
+
+
+/* Add a rotation around the yz-plane to the matrix m. */
+static void rotateyz(float m[4][4], float phi)
+{
+  float c, s, u, v;
+  int i;
+
+  phi *= M_PI/180.0;
+  c = cos(phi);
+  s = sin(phi);
+  for (i=0; i<4; i++)
+  {
+    u = m[i][0];
+    v = m[i][3];
+    m[i][0] = c*u-s*v;
+    m[i][3] = s*u+c*v;
+  }
+}
+
+
+/* Compute the rotation matrix m from the rotation angles. */
+static void rotateall(float al, float be, float de, float ze, float et,
+                      float th, float m[4][4])
+{
+  int i, j;
+
+  for (i=0; i<4; i++)
+    for (j=0; j<4; j++)
+      m[i][j] = (i==j);
+  rotatewx(m,al);
+  rotatewy(m,be);
+  rotatewz(m,de);
+  rotatexy(m,ze);
+  rotatexz(m,et);
+  rotateyz(m,th);
+}
+
+
+/* Compute the rotation matrix m from the 4d rotation angles. */
+static void rotateall4d(float ze, float et, float th, float m[4][4])
+{
+  int i, j;
+
+  for (i=0; i<4; i++)
+    for (j=0; j<4; j++)
+      m[i][j] = (i==j);
+  rotatexy(m,ze);
+  rotatexz(m,et);
+  rotateyz(m,th);
+}
+
+
+/* Multiply two rotation matrices: o=m*n. */
+static void mult_rotmat(float m[4][4], float n[4][4], float o[4][4])
+{
+  int i, j, k;
+
+  for (i=0; i<4; i++)
+  {
+    for (j=0; j<4; j++)
+    {
+      o[i][j] = 0.0;
+      for (k=0; k<4; k++)
+        o[i][j] += m[i][k]*n[k][j];
+    }
+  }
+}
+
+
+/* Compute a 4D rotation matrix from two unit quaternions. */
+static void quats_to_rotmat(float p[4], float q[4], float m[4][4])
+{
+  double al, be, de, ze, et, th;
+  double r00, r01, r02, r12, r22;
+
+  r00 = 1.0-2.0*(p[1]*p[1]+p[2]*p[2]);
+  r01 = 2.0*(p[0]*p[1]+p[2]*p[3]);
+  r02 = 2.0*(p[2]*p[0]-p[1]*p[3]);
+  r12 = 2.0*(p[1]*p[2]+p[0]*p[3]);
+  r22 = 1.0-2.0*(p[1]*p[1]+p[0]*p[0]);
+
+  al = atan2(-r12,r22)*180.0/M_PI;
+  be = atan2(r02,sqrt(r00*r00+r01*r01))*180.0/M_PI;
+  de = atan2(-r01,r00)*180.0/M_PI;
+
+  r00 = 1.0-2.0*(q[1]*q[1]+q[2]*q[2]);
+  r01 = 2.0*(q[0]*q[1]+q[2]*q[3]);
+  r02 = 2.0*(q[2]*q[0]-q[1]*q[3]);
+  r12 = 2.0*(q[1]*q[2]+q[0]*q[3]);
+  r22 = 1.0-2.0*(q[1]*q[1]+q[0]*q[0]);
+
+  et = atan2(-r12,r22)*180.0/M_PI;
+  th = atan2(r02,sqrt(r00*r00+r01*r01))*180.0/M_PI;
+  ze = atan2(-r01,r00)*180.0/M_PI;
+
+  rotateall(al,be,de,ze,et,-th,m);
+}
+
+
+/* Compute a fully saturated and bright color based on an angle. */
+static void color(projectiveplanestruct *pp, double angle, float col[4])
+{
+  int s;
+  double t;
+
+  if (pp->colors == COLORS_TWOSIDED)
+    return;
+
+  if (angle >= 0.0)
+    angle = fmod(angle,2.0*M_PI);
+  else
+    angle = fmod(angle,-2.0*M_PI);
+  s = floor(angle/(M_PI/3));
+  t = angle/(M_PI/3)-s;
+  if (s >= 6)
+    s = 0;
+  switch (s)
+  {
+    case 0:
+      col[0] = 1.0;
+      col[1] = t;
+      col[2] = 0.0;
+      break;
+    case 1:
+      col[0] = 1.0-t;
+      col[1] = 1.0;
+      col[2] = 0.0;
+      break;
+    case 2:
+      col[0] = 0.0;
+      col[1] = 1.0;
+      col[2] = t;
+      break;
+    case 3:
+      col[0] = 0.0;
+      col[1] = 1.0-t;
+      col[2] = 1.0;
+      break;
+    case 4:
+      col[0] = t;
+      col[1] = 0.0;
+      col[2] = 1.0;
+      break;
+    case 5:
+      col[0] = 1.0;
+      col[1] = 0.0;
+      col[2] = 1.0-t;
+      break;
+  }
+  if (pp->display_mode == DISP_TRANSPARENT)
+    col[3] = 0.7;
+  else
+    col[3] = 1.0;
+}
+
+
+/* Set up the projective plane coordinates, colors, and texture. */
+static void setup_projective_plane(ModeInfo *mi, double umin, double umax,
+                                   double vmin, double vmax)
+{
+  int i, j, k;
+  double u, v, ur, vr;
+  double cu, su, cv2, sv2, cv4, sv4, c2u, s2u;
+  projectiveplanestruct *pp = &projectiveplane[MI_SCREEN(mi)];
+
+  ur = umax-umin;
+  vr = vmax-vmin;
+  for (i=0; i<=NUMV; i++)
+  {
+    for (j=0; j<=NUMU; j++)
+    {
+      k = i*(NUMU+1)+j;
+      if (pp->appearance != APPEARANCE_DIRECTION_BANDS)
+        u = -ur*j/NUMU+umin;
+      else
+        u = ur*j/NUMU+umin;
+      v = vr*i/NUMV+vmin;
+      cu = cos(u);
+      su = sin(u);
+      c2u = cos(2.0*u);
+      s2u = sin(2.0*u);
+      sv2 = sin(0.5*v);
+      cv4 = cos(0.25*v);
+      sv4 = sin(0.25*v);
+      if (pp->colors == COLORS_DEPTH)
+        color(pp,((su*su*sv4*sv4-cv4*cv4)+1.0)*M_PI*2.0/3.0,pp->col[k]);
+      else if (pp->colors == COLORS_DIRECTION)
+        color(pp,2.0*M_PI+fmod(2.0*u,2.0*M_PI),pp->col[k]);
+      else /* pp->colors == COLORS_DISTANCE */
+        color(pp,v*(5.0/6.0),pp->col[k]);
+      pp->tex[k][0] = -32*u/(2.0*M_PI);
+      if (pp->appearance != APPEARANCE_DISTANCE_BANDS)
+        pp->tex[k][1] = 32*v/(2.0*M_PI);
+      else
+        pp->tex[k][1] = 32*v/(2.0*M_PI)-0.5;
+      pp->x[k][0] = 0.5*s2u*sv4*sv4;
+      pp->x[k][1] = 0.5*su*sv2;
+      pp->x[k][2] = 0.5*cu*sv2;
+      pp->x[k][3] = 0.5*(su*su*sv4*sv4-cv4*cv4);
+      /* Avoid degenerate tangential plane basis vectors. */
+      if (v < FLT_EPSILON)
+        v = FLT_EPSILON;
+      cv2 = cos(0.5*v);
+      sv2 = sin(0.5*v);
+      sv4 = sin(0.25*v);
+      pp->xu[k][0] = c2u*sv4*sv4;
+      pp->xu[k][1] = 0.5*cu*sv2;
+      pp->xu[k][2] = -0.5*su*sv2;
+      pp->xu[k][3] = 0.5*s2u*sv4*sv4;
+      pp->xv[k][0] = 0.125*s2u*sv2;
+      pp->xv[k][1] = 0.25*su*cv2;
+      pp->xv[k][2] = 0.25*cu*cv2;
+      pp->xv[k][3] = 0.125*(su*su+1.0)*sv2;
+    }
+  }
+}
+
+
+/* Draw a 4d embedding of the projective plane projected into 3D. */
+static int projective_plane(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, cv2, sv2, cv4, sv4, c2u, s2u;
+  float q1[4], q2[4], r1[4][4], r2[4][4];
+  projectiveplanestruct *pp = &projectiveplane[MI_SCREEN(mi)];
+
+  if (pp->view == VIEW_WALK || pp->view == VIEW_WALKTURN)
+  {
+    /* Compute the rotation that rotates the projective plane in 4D without
+       the trackball rotations. */
+    rotateall4d(pp->zeta,pp->eta,pp->theta,mat);
+
+    u = pp->umove;
+    v = pp->vmove;
+    cu = cos(u);
+    su = sin(u);
+    c2u = cos(2.0*u);
+    s2u = sin(2.0*u);
+    sv2 = sin(0.5*v);
+    cv4 = cos(0.25*v);
+    sv4 = sin(0.25*v);
+    xx[0] = 0.5*s2u*sv4*sv4;
+    xx[1] = 0.5*su*sv2;
+    xx[2] = 0.5*cu*sv2;
+    xx[3] = 0.5*(su*su*sv4*sv4-cv4*cv4);
+    /* Avoid degenerate tangential plane basis vectors. */
+    if (v < FLT_EPSILON)
+      v = FLT_EPSILON;
+    cv2 = cos(0.5*v);
+    sv2 = sin(0.5*v);
+    sv4 = sin(0.25*v);
+    xxu[0] = c2u*sv4*sv4;
+    xxu[1] = 0.5*cu*sv2;
+    xxu[2] = -0.5*su*sv2;
+    xxu[3] = 0.5*s2u*sv4*sv4;
+    xxv[0] = 0.125*s2u*sv2;
+    xxv[1] = 0.25*su*cv2;
+    xxv[2] = 0.25*cu*cv2;
+    xxv[3] = 0.125*(su*su+1.0)*sv2;
+    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 (pp->projection_4d == DISP_4D_ORTHOGRAPHIC)
+    {
+      for (l=0; l<3; l++)
+      {
+        p[l] = y[l]+pp->offset4d[l];
+        pu[l] = yu[l];
+        pv[l] = yv[l];
+      }
+    }
+    else
+    {
+      s = y[3]+pp->offset4d[3];
+      q = 1.0/s;
+      t = q*q;
+      for (l=0; l<3; l++)
+      {
+        r = y[l]+pp->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/(pp->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]*pp->dumove+pv[0]*pp->dvmove;
+    pm[1] = pu[1]*pp->dumove+pv[1]*pp->dvmove;
+    pm[2] = pu[2]*pp->dumove+pv[2]*pp->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] |
+    */
+    pp->alpha = atan2(-n[2],-pm[2])*180/M_PI;
+    pp->beta = atan2(-b[2],sqrt(b[0]*b[0]+b[1]*b[1]))*180/M_PI;
+    pp->delta = atan2(b[1],-b[0])*180/M_PI;
+
+    /* Compute the rotation that rotates the projective plane in 4D. */
+    rotateall(pp->alpha,pp->beta,pp->delta,pp->zeta,pp->eta,pp->theta,mat);
+
+    u = pp->umove;
+    v = pp->vmove;
+    cu = cos(u);
+    su = sin(u);
+    s2u = sin(2.0*u);
+    sv2 = sin(0.5*v);
+    cv4 = cos(0.25*v);
+    sv4 = sin(0.25*v);
+    xx[0] = 0.5*s2u*sv4*sv4;
+    xx[1] = 0.5*su*sv2;
+    xx[2] = 0.5*cu*sv2;
+    xx[3] = 0.5*(su*su*sv4*sv4-cv4*cv4);
+    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 (pp->projection_4d == DISP_4D_ORTHOGRAPHIC)
+    {
+      for (l=0; l<3; l++)
+        p[l] = y[l]+pp->offset4d[l];
+    }
+    else
+    {
+      s = y[3]+pp->offset4d[3];
+      for (l=0; l<3; l++)
+        p[l] = (y[l]+pp->offset4d[l])/s;
+    }
+
+    pp->offset3d[0] = -p[0];
+    pp->offset3d[1] = -p[1]-DELTAY;
+    pp->offset3d[2] = -p[2];
+  }
+  else
+  {
+    /* Compute the rotation that rotates the projective plane in 4D,
+       including the trackball rotations. */
+    rotateall(pp->alpha,pp->beta,pp->delta,pp->zeta,pp->eta,pp->theta,r1);
+
+    gltrackball_get_quaternion(pp->trackballs[0],q1);
+    gltrackball_get_quaternion(pp->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<=NUMV; i++)
+  {
+    for (j=0; j<=NUMU; j++)
+    {
+      o = i*(NUMU+1)+j;
+      for (l=0; l<4; l++)
+      {
+        y[l] = (mat[l][0]*pp->x[o][0]+mat[l][1]*pp->x[o][1]+
+                mat[l][2]*pp->x[o][2]+mat[l][3]*pp->x[o][3]);
+        yu[l] = (mat[l][0]*pp->xu[o][0]+mat[l][1]*pp->xu[o][1]+
+                 mat[l][2]*pp->xu[o][2]+mat[l][3]*pp->xu[o][3]);
+        yv[l] = (mat[l][0]*pp->xv[o][0]+mat[l][1]*pp->xv[o][1]+
+                 mat[l][2]*pp->xv[o][2]+mat[l][3]*pp->xv[o][3]);
+      }
+      if (pp->projection_4d == DISP_4D_ORTHOGRAPHIC)
+      {
+        for (l=0; l<3; l++)
+        {
+          pp->pp[o][l] = (y[l]+pp->offset4d[l])+pp->offset3d[l];
+          pu[l] = yu[l];
+          pv[l] = yv[l];
+        }
+      }
+      else
+      {
+        s = y[3]+pp->offset4d[3];
+        q = 1.0/s;
+        t = q*q;
+        for (l=0; l<3; l++)
+        {
+          r = y[l]+pp->offset4d[l];
+          pp->pp[o][l] = r*q+pp->offset3d[l];
+          pu[l] = (yu[l]*s-r*yu[3])*t;
+          pv[l] = (yv[l]*s-r*yv[3])*t;
+        }
+      }
+      pp->pn[o][0] = pu[1]*pv[2]-pu[2]*pv[1];
+      pp->pn[o][1] = pu[2]*pv[0]-pu[0]*pv[2];
+      pp->pn[o][2] = pu[0]*pv[1]-pu[1]*pv[0];
+      t = 1.0/sqrt(pp->pn[o][0]*pp->pn[o][0]+pp->pn[o][1]*pp->pn[o][1]+
+                   pp->pn[o][2]*pp->pn[o][2]);
+      pp->pn[o][0] *= t;
+      pp->pn[o][1] *= t;
+      pp->pn[o][2] *= t;
+    }
+  }
+
+  if (pp->colors == COLORS_TWOSIDED)
+  {
+    glColor3fv(mat_diff_red);
+    if (pp->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,pp->tex_name);
+
+  if (pp->appearance != APPEARANCE_DIRECTION_BANDS)
+  {
+    for (i=0; i<NUMV; i++)
+    {
+      if (pp->appearance == APPEARANCE_DISTANCE_BANDS &&
+          ((i & (NUMB-1)) >= NUMB/4) && ((i & (NUMB-1)) < 3*NUMB/4))
+        continue;
+      if (pp->display_mode == DISP_WIREFRAME)
+        glBegin(GL_QUAD_STRIP);
+      else
+        glBegin(GL_TRIANGLE_STRIP);
+      for (j=0; j<=NUMU; j++)
+      {
+        for (k=0; k<=1; k++)
+        {
+          l = (i+k);
+          m = j;
+          o = l*(NUMU+1)+m;
+          glNormal3fv(pp->pn[o]);
+          glTexCoord2fv(pp->tex[o]);
+          if (pp->colors != COLORS_TWOSIDED)
+          {
+            glColor3fv(pp->col[o]);
+            glMaterialfv(GL_FRONT_AND_BACK,GL_AMBIENT_AND_DIFFUSE,pp->col[o]);
+          }
+          glVertex3fv(pp->pp[o]);
+          polys++;
+        }
+      }
+      glEnd();
+    }
+  }
+  else /* pp->appearance == APPEARANCE_DIRECTION_BANDS */
+  {
+    for (j=0; j<NUMU; j++)
+    {
+      if ((j & (NUMB-1)) >= NUMB/2)
+        continue;
+      if (pp->display_mode == DISP_WIREFRAME)
+        glBegin(GL_QUAD_STRIP);
+      else
+        glBegin(GL_TRIANGLE_STRIP);
+      for (i=0; i<=NUMV; i++)
+      {
+        for (k=0; k<=1; k++)
+        {
+          l = i;
+          m = (j+k);
+          o = l*(NUMU+1)+m;
+          glNormal3fv(pp->pn[o]);
+          glTexCoord2fv(pp->tex[o]);
+          if (pp->colors != COLORS_TWOSIDED)
+          {
+            glColor3fv(pp->col[o]);
+            glMaterialfv(GL_FRONT_AND_BACK,GL_AMBIENT_AND_DIFFUSE,pp->col[o]);
+          }
+          glVertex3fv(pp->pp[o]);
+          polys++;
+        }
+      }
+      glEnd();
+    }
+  }
+
+  polys /= 2;
+  return polys;
+}
+
+
+/* Generate a texture image that shows the orientation reversal. */
+static void gen_texture(ModeInfo *mi)
+{
+  projectiveplanestruct *pp = &projectiveplane[MI_SCREEN(mi)];
+
+  glGenTextures(1,&pp->tex_name);
+  glBindTexture(GL_TEXTURE_2D,pp->tex_name);
+  glPixelStorei(GL_UNPACK_ALIGNMENT,1);
+  glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_WRAP_S,GL_REPEAT);
+  glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_WRAP_T,GL_REPEAT);
+  glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_LINEAR);
+  glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTER,GL_LINEAR);
+  glTexEnvf(GL_TEXTURE_ENV,GL_TEXTURE_ENV_MODE,GL_MODULATE);
+  glTexImage2D(GL_TEXTURE_2D,0,GL_RGB,TEX_DIMENSION,TEX_DIMENSION,0,
+               GL_LUMINANCE,GL_UNSIGNED_BYTE,texture);
+}
+
+
+static void init(ModeInfo *mi)
+{
+  static const GLfloat light_ambient[]  = { 0.0, 0.0, 0.0, 1.0 };
+  static const GLfloat light_diffuse[]  = { 1.0, 1.0, 1.0, 1.0 };
+  static const GLfloat light_specular[] = { 1.0, 1.0, 1.0, 1.0 };
+  static const GLfloat light_position[] = { 1.0, 1.0, 1.0, 0.0 };
+  static const GLfloat mat_specular[]   = { 1.0, 1.0, 1.0, 1.0 };
+  projectiveplanestruct *pp = &projectiveplane[MI_SCREEN(mi)];
+
+  if (walk_speed == 0.0)
+    walk_speed = 20.0;
+
+  if (pp->view == VIEW_TURN)
+  {
+    pp->alpha = frand(360.0);
+    pp->beta = frand(360.0);
+    pp->delta = frand(360.0);
+    pp->zeta = 0.0;
+    pp->eta = 0.0;
+    pp->theta = 0.0;
+  }
+  else
+  {
+    pp->alpha = 0.0;
+    pp->beta = 0.0;
+    pp->delta = 0.0;
+    pp->zeta = 120.0;
+    pp->eta = 180.0;
+    pp->theta = 90.0;
+  }
+  pp->umove = frand(2.0*M_PI);
+  pp->vmove = frand(2.0*M_PI);
+  pp->dumove = 0.0;
+  pp->dvmove = 0.0;
+  pp->side = 1;
+  if (sin(walk_direction*M_PI/180.0) >= 0.0)
+    pp->dir = 1;
+  else
+    pp->dir = -1;
+
+  pp->offset4d[0] = 0.0;
+  pp->offset4d[1] = 0.0;
+  pp->offset4d[2] = 0.0;
+  pp->offset4d[3] = 1.2;
+  pp->offset3d[0] = 0.0;
+  pp->offset3d[1] = 0.0;
+  pp->offset3d[2] = -1.2;
+  pp->offset3d[3] = 0.0;
+
+  gen_texture(mi);
+  setup_projective_plane(mi,0.0,2.0*M_PI,0.0,2.0*M_PI);
+
+  if (pp->marks)
+    glEnable(GL_TEXTURE_2D);
+  else
+    glDisable(GL_TEXTURE_2D);
+
+  glMatrixMode(GL_PROJECTION);
+  glLoadIdentity();
+  if (pp->projection_3d == DISP_3D_PERSPECTIVE ||
+      pp->view == VIEW_WALK || pp->view == VIEW_WALKTURN)
+  {
+    if (pp->view == VIEW_WALK || pp->view == VIEW_WALKTURN)
+      gluPerspective(60.0,1.0,0.01,10.0);
+    else
+      gluPerspective(60.0,1.0,0.1,10.0);
+  }
+  else
+  {
+    glOrtho(-0.6,0.6,-0.6,0.6,0.1,10.0);
+  }
+  glMatrixMode(GL_MODELVIEW);
+  glLoadIdentity();
+
+# ifdef HAVE_JWZGLES /* #### glPolygonMode other than GL_FILL unimplemented */
+  if (pp->display_mode == DISP_WIREFRAME)
+    pp->display_mode = DISP_SURFACE;
+# endif
+
+  if (pp->display_mode == DISP_SURFACE)
+  {
+    glEnable(GL_DEPTH_TEST);
+    glDepthFunc(GL_LESS);
+    glShadeModel(GL_SMOOTH);
+    glPolygonMode(GL_FRONT_AND_BACK,GL_FILL);
+    glLightModeli(GL_LIGHT_MODEL_TWO_SIDE,GL_TRUE);
+    glEnable(GL_LIGHTING);
+    glEnable(GL_LIGHT0);
+    glLightfv(GL_LIGHT0,GL_AMBIENT,light_ambient);
+    glLightfv(GL_LIGHT0,GL_DIFFUSE,light_diffuse);
+    glLightfv(GL_LIGHT0,GL_SPECULAR,light_specular);
+    glLightfv(GL_LIGHT0,GL_POSITION,light_position);
+    glMaterialfv(GL_FRONT_AND_BACK,GL_SPECULAR,mat_specular);
+    glMaterialf(GL_FRONT_AND_BACK,GL_SHININESS,50.0);
+    glDepthMask(GL_TRUE);
+    glDisable(GL_BLEND);
+  }
+  else if (pp->display_mode == DISP_TRANSPARENT)
+  {
+    glDisable(GL_DEPTH_TEST);
+    glShadeModel(GL_SMOOTH);
+    glPolygonMode(GL_FRONT_AND_BACK,GL_FILL);
+    glLightModeli(GL_LIGHT_MODEL_TWO_SIDE,GL_TRUE);
+    glEnable(GL_LIGHTING);
+    glEnable(GL_LIGHT0);
+    glLightfv(GL_LIGHT0,GL_AMBIENT,light_ambient);
+    glLightfv(GL_LIGHT0,GL_DIFFUSE,light_diffuse);
+    glLightfv(GL_LIGHT0,GL_SPECULAR,light_specular);
+    glLightfv(GL_LIGHT0,GL_POSITION,light_position);
+    glMaterialfv(GL_FRONT_AND_BACK,GL_SPECULAR,mat_specular);
+    glMaterialf(GL_FRONT_AND_BACK,GL_SHININESS,50.0);
+    glDepthMask(GL_FALSE);
+    glEnable(GL_BLEND);
+    glBlendFunc(GL_SRC_ALPHA,GL_ONE);
+  }
+  else  /* pp->display_mode == DISP_WIREFRAME */
+  {
+    glDisable(GL_DEPTH_TEST);
+    glShadeModel(GL_FLAT);
+    glPolygonMode(GL_FRONT_AND_BACK,GL_LINE);
+    glDisable(GL_LIGHTING);
+    glDisable(GL_LIGHT0);
+    glDisable(GL_BLEND);
+  }
+}
+
+
+/* Redisplay the Klein bottle. */
+static void display_projectiveplane(ModeInfo *mi)
+{
+  projectiveplanestruct *pp = &projectiveplane[MI_SCREEN(mi)];
+
+  if (!pp->button_pressed)
+  {
+    if (pp->view == VIEW_TURN)
+    {
+      pp->alpha += speed_wx * pp->speed_scale;
+      if (pp->alpha >= 360.0)
+        pp->alpha -= 360.0;
+      pp->beta += speed_wy * pp->speed_scale;
+      if (pp->beta >= 360.0)
+        pp->beta -= 360.0;
+      pp->delta += speed_wz * pp->speed_scale;
+      if (pp->delta >= 360.0)
+        pp->delta -= 360.0;
+      pp->zeta += speed_xy * pp->speed_scale;
+      if (pp->zeta >= 360.0)
+        pp->zeta -= 360.0;
+      pp->eta += speed_xz * pp->speed_scale;
+      if (pp->eta >= 360.0)
+        pp->eta -= 360.0;
+      pp->theta += speed_yz * pp->speed_scale;
+      if (pp->theta >= 360.0)
+        pp->theta -= 360.0;
+    }
+    if (pp->view == VIEW_WALKTURN)
+    {
+      pp->zeta += speed_xy * pp->speed_scale;
+      if (pp->zeta >= 360.0)
+        pp->zeta -= 360.0;
+      pp->eta += speed_xz * pp->speed_scale;
+      if (pp->eta >= 360.0)
+        pp->eta -= 360.0;
+      pp->theta += speed_yz * pp->speed_scale;
+      if (pp->theta >= 360.0)
+        pp->theta -= 360.0;
+    }
+    if (pp->view == VIEW_WALK || pp->view == VIEW_WALKTURN)
+    {
+      pp->dvmove = (pp->dir*sin(walk_direction*M_PI/180.0)*
+                    walk_speed*M_PI/4096.0);
+      pp->vmove += pp->dvmove;
+      if (pp->vmove > 2.0*M_PI)
+      {
+        pp->vmove = 4.0*M_PI-pp->vmove;
+        pp->umove = pp->umove-M_PI;
+        if (pp->umove < 0.0)
+          pp->umove += 2.0*M_PI;
+        pp->side = -pp->side;
+        pp->dir = -pp->dir;
+        pp->dvmove = -pp->dvmove;
+      }
+      if (pp->vmove < 0.0)
+      {
+        pp->vmove = -pp->vmove;
+        pp->umove = pp->umove-M_PI;
+        if (pp->umove < 0.0)
+          pp->umove += 2.0*M_PI;
+        pp->dir = -pp->dir;
+        pp->dvmove = -pp->dvmove;
+      }
+      pp->dumove = cos(walk_direction*M_PI/180.0)*walk_speed*M_PI/4096.0;
+      pp->umove += pp->dumove;
+      if (pp->umove >= 2.0*M_PI)
+        pp->umove -= 2.0*M_PI;
+      if (pp->umove < 0.0)
+        pp->umove += 2.0*M_PI;
+    }
+  }
+
+  glMatrixMode(GL_PROJECTION);
+  glLoadIdentity();
+  if (pp->projection_3d == DISP_3D_PERSPECTIVE ||
+      pp->view == VIEW_WALK || pp->view == VIEW_WALKTURN)
+  {
+    if (pp->view == VIEW_WALK || pp->view == VIEW_WALKTURN)
+      gluPerspective(60.0,pp->aspect,0.01,10.0);
+    else
+      gluPerspective(60.0,pp->aspect,0.1,10.0);
+  }
+  else
+  {
+    if (pp->aspect >= 1.0)
+      glOrtho(-0.6*pp->aspect,0.6*pp->aspect,-0.6,0.6,0.1,10.0);
+    else
+      glOrtho(-0.6,0.6,-0.6/pp->aspect,0.6/pp->aspect,0.1,10.0);
+  }
+  glMatrixMode(GL_MODELVIEW);
+  glLoadIdentity();
+
+  mi->polygon_count = projective_plane(mi,0.0,2.0*M_PI,0.0,2.0*M_PI);
+}
+
+
+ENTRYPOINT void reshape_projectiveplane(ModeInfo *mi, int width, int height)
+{
+  projectiveplanestruct *pp = &projectiveplane[MI_SCREEN(mi)];
+
+  pp->WindW = (GLint)width;
+  pp->WindH = (GLint)height;
+  glViewport(0,0,width,height);
+  pp->aspect = (GLfloat)width/(GLfloat)height;
+}
+
+
+ENTRYPOINT Bool projectiveplane_handle_event(ModeInfo *mi, XEvent *event)
+{
+  projectiveplanestruct *pp = &projectiveplane[MI_SCREEN(mi)];
+  KeySym  sym = 0;
+  char c = 0;
+
+  if (event->xany.type == KeyPress || event->xany.type == KeyRelease)
+    XLookupString (&event->xkey, &c, 1, &sym, 0);
+
+  if (event->xany.type == ButtonPress &&
+      event->xbutton.button == Button1)
+  {
+    pp->button_pressed = True;
+    gltrackball_start(pp->trackballs[pp->current_trackball],
+                      event->xbutton.x, event->xbutton.y,
+                      MI_WIDTH(mi), MI_HEIGHT(mi));
+    return True;
+  }
+  else if (event->xany.type == ButtonRelease &&
+           event->xbutton.button == Button1)
+  {
+    pp->button_pressed = False;
+    return True;
+  }
+  else if (event->xany.type == KeyPress)
+  {
+    if (sym == XK_Shift_L || sym == XK_Shift_R)
+    {
+      pp->current_trackball = 1;
+      if (pp->button_pressed)
+        gltrackball_start(pp->trackballs[pp->current_trackball],
+                          event->xbutton.x, event->xbutton.y,
+                          MI_WIDTH(mi), MI_HEIGHT(mi));
+      return True;
+    }
+  }
+  else if (event->xany.type == KeyRelease)
+  {
+    if (sym == XK_Shift_L || sym == XK_Shift_R)
+    {
+      pp->current_trackball = 0;
+      if (pp->button_pressed)
+        gltrackball_start(pp->trackballs[pp->current_trackball],
+                          event->xbutton.x, event->xbutton.y,
+                          MI_WIDTH(mi), MI_HEIGHT(mi));
+      return True;
+    }
+  }
+  else if (event->xany.type == MotionNotify && pp->button_pressed)
+  {
+    gltrackball_track(pp->trackballs[pp->current_trackball],
+                      event->xmotion.x, event->xmotion.y,
+                      MI_WIDTH(mi), MI_HEIGHT(mi));
+    return True;
+  }
+
+  return False;
+}
+
+
+/*
+ *-----------------------------------------------------------------------------
+ *-----------------------------------------------------------------------------
+ *    Xlock hooks.
+ *-----------------------------------------------------------------------------
+ *-----------------------------------------------------------------------------
+ */
+
+/*
+ *-----------------------------------------------------------------------------
+ *    Initialize projectiveplane.  Called each time the window changes.
+ *-----------------------------------------------------------------------------
+ */
+
+ENTRYPOINT void init_projectiveplane(ModeInfo *mi)
+{
+  projectiveplanestruct *pp;
+
+  MI_INIT(mi, projectiveplane);
+  pp = &projectiveplane[MI_SCREEN(mi)];
+
+  
+  pp->trackballs[0] = gltrackball_init(True);
+  pp->trackballs[1] = gltrackball_init(True);
+  pp->current_trackball = 0;
+  pp->button_pressed = False;
+
+  /* Set the display mode. */
+  if (!strcasecmp(mode,"random"))
+  {
+    pp->display_mode = random() % NUM_DISPLAY_MODES;
+  }
+  else if (!strcasecmp(mode,"wireframe"))
+  {
+    pp->display_mode = DISP_WIREFRAME;
+  }
+  else if (!strcasecmp(mode,"surface"))
+  {
+    pp->display_mode = DISP_SURFACE;
+  }
+  else if (!strcasecmp(mode,"transparent"))
+  {
+    pp->display_mode = DISP_TRANSPARENT;
+  }
+  else
+  {
+    pp->display_mode = random() % NUM_DISPLAY_MODES;
+  }
+
+  /* Orientation marks don't make sense in wireframe mode. */
+  pp->marks = marks;
+  if (pp->display_mode == DISP_WIREFRAME)
+    pp->marks = False;
+
+  /* Set the appearance. */
+  if (!strcasecmp(appear,"random"))
+  {
+    pp->appearance = random() % NUM_APPEARANCES;
+  }
+  else if (!strcasecmp(appear,"solid"))
+  {
+    pp->appearance = APPEARANCE_SOLID;
+  }
+  else if (!strcasecmp(appear,"distance-bands"))
+  {
+    pp->appearance = APPEARANCE_DISTANCE_BANDS;
+  }
+  else if (!strcasecmp(appear,"direction-bands"))
+  {
+    pp->appearance = APPEARANCE_DIRECTION_BANDS;
+  }
+  else
+  {
+    pp->appearance = random() % NUM_APPEARANCES;
+  }
+
+  /* Set the color mode. */
+  if (!strcasecmp(color_mode,"random"))
+  {
+    pp->colors = random() % NUM_COLORS;
+  }
+  else if (!strcasecmp(color_mode,"two-sided"))
+  {
+    pp->colors = COLORS_TWOSIDED;
+  }
+  else if (!strcasecmp(color_mode,"distance"))
+  {
+    pp->colors = COLORS_DISTANCE;
+  }
+  else if (!strcasecmp(color_mode,"direction"))
+  {
+    pp->colors = COLORS_DIRECTION;
+  }
+  else if (!strcasecmp(color_mode,"depth"))
+  {
+    pp->colors = COLORS_DEPTH;
+  }
+  else
+  {
+    pp->colors = random() % NUM_COLORS;
+  }
+
+  /* Set the view mode. */
+  if (!strcasecmp(view_mode,"random"))
+  {
+    pp->view = random() % NUM_VIEW_MODES;
+  }
+  else if (!strcasecmp(view_mode,"walk"))
+  {
+    pp->view = VIEW_WALK;
+  }
+  else if (!strcasecmp(view_mode,"turn"))
+  {
+    pp->view = VIEW_TURN;
+  }
+  else if (!strcasecmp(view_mode,"walk-turn"))
+  {
+    pp->view = VIEW_WALKTURN;
+  }
+  else
+  {
+    pp->view = random() % NUM_VIEW_MODES;
+  }
+
+  /* Set the 3d projection mode. */
+  if (!strcasecmp(proj_3d,"random"))
+  {
+    /* Orthographic projection only makes sense in turn mode. */
+    if (pp->view == VIEW_TURN)
+      pp->projection_3d = random() % NUM_DISP_3D_MODES;
+    else
+      pp->projection_3d = DISP_3D_PERSPECTIVE;
+  }
+  else if (!strcasecmp(proj_3d,"perspective"))
+  {
+    pp->projection_3d = DISP_3D_PERSPECTIVE;
+  }
+  else if (!strcasecmp(proj_3d,"orthographic"))
+  {
+    pp->projection_3d = DISP_3D_ORTHOGRAPHIC;
+  }
+  else
+  {
+    /* Orthographic projection only makes sense in turn mode. */
+    if (pp->view == VIEW_TURN)
+      pp->projection_3d = random() % NUM_DISP_3D_MODES;
+    else
+      pp->projection_3d = DISP_3D_PERSPECTIVE;
+  }
+
+  /* Set the 4d projection mode. */
+  if (!strcasecmp(proj_4d,"random"))
+  {
+    pp->projection_4d = random() % NUM_DISP_4D_MODES;
+  }
+  else if (!strcasecmp(proj_4d,"perspective"))
+  {
+    pp->projection_4d = DISP_4D_PERSPECTIVE;
+  }
+  else if (!strcasecmp(proj_4d,"orthographic"))
+  {
+    pp->projection_4d = DISP_4D_ORTHOGRAPHIC;
+  }
+  else
+  {
+    pp->projection_4d = random() % NUM_DISP_4D_MODES;
+  }
+
+  /* Modify the speeds to a useful range in walk-and-turn mode. */
+  if (pp->view == VIEW_WALKTURN)
+  {
+    speed_wx *= 0.2;
+    speed_wy *= 0.2;
+    speed_wz *= 0.2;
+    speed_xy *= 0.2;
+    speed_xz *= 0.2;
+    speed_yz *= 0.2;
+  }
+
+  /* make multiple screens rotate at slightly different rates. */
+  pp->speed_scale = 0.9 + frand(0.3);
+
+  if ((pp->glx_context = init_GL(mi)) != NULL)
+  {
+    reshape_projectiveplane(mi,MI_WIDTH(mi),MI_HEIGHT(mi));
+    glDrawBuffer(GL_BACK);
+    init(mi);
+  }
+  else
+  {
+    MI_CLEARWINDOW(mi);
+  }
+}
+
+/*
+ *-----------------------------------------------------------------------------
+ *    Called by the mainline code periodically to update the display.
+ *-----------------------------------------------------------------------------
+ */
+ENTRYPOINT void draw_projectiveplane(ModeInfo *mi)
+{
+  Display          *display = MI_DISPLAY(mi);
+  Window           window = MI_WINDOW(mi);
+  projectiveplanestruct *pp;
+
+  if (projectiveplane == NULL)
+    return;
+  pp = &projectiveplane[MI_SCREEN(mi)];
+
+  MI_IS_DRAWN(mi) = True;
+  if (!pp->glx_context)
+    return;
+
+  glXMakeCurrent(display,window,*(pp->glx_context));
+
+  glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT);
+  glLoadIdentity();
+
+  display_projectiveplane(mi);
+
+  if (MI_IS_FPS(mi))
+    do_fps (mi);
+
+  glFlush();
+
+  glXSwapBuffers(display,window);
+}
+
+
+#ifndef STANDALONE
+ENTRYPOINT void change_projectiveplane(ModeInfo *mi)
+{
+  projectiveplanestruct *pp = &projectiveplane[MI_SCREEN(mi)];
+
+  if (!pp->glx_context)
+    return;
+
+  glXMakeCurrent(MI_DISPLAY(mi),MI_WINDOW(mi),*(pp->glx_context));
+  init(mi);
+}
+#endif /* !STANDALONE */
+
+XSCREENSAVER_MODULE ("ProjectivePlane", projectiveplane)
+
+#endif /* USE_GL */