diff --git a/basegraph.cpp b/basegraph.cpp
index 5e9f7465..91192d63 100644
--- a/basegraph.cpp
+++ b/basegraph.cpp
@@ -263,6 +263,10 @@ void stereo::set_viewport(int ed) {
     glViewport(vid.xres/2, 0, vid.xres/2, vid.yres);
   }    
 
+bool model_needs_depth() {
+  return pmodel == mdBall || pmodel == mdHyperboloid || pmodel == mdHemisphere;
+  }
+
 void setGLProjection(int col) {
   DEBB(DF_GRAPH, (debugfile,"setGLProjection\n"));
   GLERR("pre_setGLProjection");
@@ -299,17 +303,17 @@ void setGLProjection(int col) {
   //glLineWidth(1.0f);
   glBlendFunc(GL_SRC_ALPHA, GL_ONE_MINUS_SRC_ALPHA);
   
-  if(pmodel == mdBall || pmodel == mdHyperboloid || pmodel == mdHemisphere) {
+  if(model_needs_depth()) {
 #ifdef GL_ES
     glClearDepthf(1.0f);
 #else
     glClearDepth(1.0f);
 #endif
-    glEnable(GL_DEPTH_TEST);
     glDepthFunc(GL_LEQUAL);
+    glhr::set_depthtest(true);
     }
   else
-    glDisable(GL_DEPTH_TEST);
+    glhr::set_depthtest(false);
   
   GLERR("setGLProjection");
   
@@ -523,6 +527,8 @@ bool gl_print(int x, int y, int shift, int size, const char *s, int color, int a
   int ysiz = f.heights[32] * size / gsiz;
 
   bool clicked = (mousex >= x && mousey <= y && mousex <= x+tsize && mousey >= y-ysiz);
+  
+  glhr::set_depthtest(false);
 
   for(int i=0; s[i];) {
   
@@ -917,6 +923,7 @@ void drawCircle(int x, int y, int size, int color) {
       glcoords.push_back(make_array<GLfloat>(x + size * sin(rr), y + size * cos(rr), stereo::scrdist));
       }
     glhr::vertices(glcoords);
+    glhr::set_depthtest(false);
     glDrawArrays(GL_LINE_LOOP, 0, pts);
     return;
     }
diff --git a/config.cpp b/config.cpp
index 508ec2a0..397e5e77 100644
--- a/config.cpp
+++ b/config.cpp
@@ -337,6 +337,7 @@ void initConfig() {
   addsaver(stereo::anaglyph_eyewidth, "eyewidth-anaglyph", 0.1);
   addsaver(stereo::fov, "field-of-vision", 90);
   addsaverenum(stereo::mode, "stereo-mode");
+  addsaver(vid.euclid_to_sphere, "euclid to sphere projection", 1.5);
   
 #if CAP_SHMUP  
   shmup::initConfig();
@@ -1130,6 +1131,17 @@ void showStereo() {
     };
   }
 
+void config_camera_rotation() {
+  dialog::editNumber(vid.ballangle, 0, 90, 5, 0, XLAT("camera rotation in 3D models"), 
+    "Rotate the camera in 3D models (ball model, hyperboloid, and hemisphere). "
+    "Note that hyperboloid and hemisphere models are also available in the "
+    "Hypersian Rug surfaces menu, but they are rendered differently there -- "
+    "they are rendered by making a flat picture first, then mapping it to a surface. "
+    "This makes the output better in some ways, but 3D effects are lost. "
+    "Hypersian Rug model also allows more camera freedom."
+    );
+  }
+
 void show3D() {
   cmode = sm::SIDE | sm::A3 | sm::MAYDARK;
   gamescreen(0);
@@ -1158,7 +1170,7 @@ void show3D() {
   dialog::addBreak(50);
   dialog::addBoolItem(XLAT("ball model"), pmodel == mdBall, 'B');
   dialog::addBoolItem(XLAT("hyperboloid model"), pmodel == mdHyperboloid, 'M');
-  dialog::addSelItem(XLAT("camera rotation in ball model"), fts3(vid.ballangle), 'b');
+  dialog::addSelItem(XLAT("camera rotation in 3D models"), fts3(vid.ballangle), 'b');
   dialog::addSelItem(XLAT("projection in ball model"), fts3(vid.ballproj), 'x');
   
   if(sphere)
@@ -1181,10 +1193,8 @@ void show3D() {
     dialog::handleNavigation(sym, uni);
     
     if(uni == 'n') 
-      cmode &= sm::A3,
       dialog::editNumber(geom3::highdetail, 0, 5, .5, 7, XLAT("High detail range"), "");
     else if(uni == 'm') 
-      cmode &= sm::A3,
       dialog::editNumber(geom3::middetail, 0, 5, .5, 7, XLAT("Mid detail range"), "");
     else if(uni == 'c') 
       tc_camera = ticks,
@@ -1209,22 +1219,17 @@ void show3D() {
       pushScreen(showStereo);
     
     else if(uni == 'y') 
-      cmode &= sm::A3,
       dialog::editNumber(vid.yshift, 0, 1, .1, 0, XLAT("Y shift"), 
         "Don't center on the player character."
         );
     else if(uni == 's') 
-      cmode &= sm::A3,
       dialog::editNumber(vid.camera_angle, -180, 180, 5, 0, XLAT("camera rotation"), 
         "Rotate the camera. Can be used to obtain a first person perspective, "
         "or third person perspective when combined with Y shift."
         );
     else if(uni == 'b') 
-      cmode &= sm::A3,
-      dialog::editNumber(vid.ballangle, 0, 90, 5, 0, XLAT("camera rotation in ball model"), 
-        "Rotate the camera in ball/hyperboloid model.");
+      config_camera_rotation();
     else if(uni == 'x') 
-      cmode &= sm::A3,
       dialog::editNumber(vid.ballproj, 0, 100, .1, 0, XLAT("projection in ball model"), 
         "This parameter affects the ball model the same way as the projection parameter affects the disk model.");
     else if(uni == 'i') 
diff --git a/conformal.cpp b/conformal.cpp
index 4064aa02..3872e616 100644
--- a/conformal.cpp
+++ b/conformal.cpp
@@ -537,6 +537,7 @@ namespace conformal {
      { "right", "up", "left", "down" },
      { "right", "up", "left", "down" },
      { "right", "up", "left", "down" },
+     { "right", "up", "left", "down" },
      { "right", "up", "left", "down" }
      };
      
@@ -577,7 +578,7 @@ namespace conformal {
       dialog::addBoolItem("Switch", false, '6');
       }      
     
-    if(pmodel == 4) {
+    if(pmodel == mdPolynomial) {
       dialog::addSelItem(XLAT("coefficient"), 
         fts4(polygonal::coefr[polygonal::coefid]), 'x');
       dialog::addSelItem(XLAT("coefficient (imaginary)"), 
@@ -585,12 +586,20 @@ namespace conformal {
       dialog::addSelItem(XLAT("which coefficient"), its(polygonal::coefid), 'n');
       }
 
-    if(pmodel == 3) {
+    if(pmodel == mdPolygonal) {
       dialog::addSelItem(XLAT("polygon sides"), its(polygonal::SI), 'x');
       dialog::addSelItem(XLAT("star factor"), fts(polygonal::STAR), 'y');
       dialog::addSelItem(XLAT("degree of the approximation"), its(polygonal::deg), 'n');
       }
     
+    if(pmodel == mdBall || pmodel == mdHyperboloid || pmodel == mdHemisphere) {
+      dialog::addSelItem(XLAT("camera rotation in 3D models"), fts3(vid.ballangle), 'b');
+      }    
+    
+    if(pmodel == mdHemisphere) {
+      dialog::addSelItem(XLAT("euclid to sphere projection"), fts3(vid.euclid_to_sphere), 'l');
+      }
+      
     if(!bounded && !euclid) dialog::addBoolItem(XLAT("prepare the line animation"), (on), 'e');
     if(on) dialog::addSelItem(XLAT("animation speed"), fts(lvspeed), 'a');
     
@@ -633,6 +642,14 @@ namespace conformal {
         create();
         }
       }
+    else if(uni == 'b') 
+      config_camera_rotation();
+    else if(uni == 'l')  {
+      dialog::editNumber(vid.euclid_to_sphere, 0, 10, .1, 1, XLAT("euclid to sphere projection"), 
+        "Stereographic projection to a sphere. Choose the radius of the sphere."
+        );
+      dialog::scaleLog();
+      }
     else if(uni == 'o') 
       autoband = !autoband;
     else if(uni == 'm' || uni == 'M') {
diff --git a/dialogs.cpp b/dialogs.cpp
index edc164c9..2fc8f605 100644
--- a/dialogs.cpp
+++ b/dialogs.cpp
@@ -829,7 +829,7 @@ namespace dialog {
     ne.intval = NULL;
     ne.positive = false;
     dialogflags = (cmode & sm::A3);
-    if(cmode & sm::SIDE) dialogflags |= sm::MAYDARK;
+    if(cmode & sm::SIDE) dialogflags |= sm::MAYDARK | sm::SIDE;
     cmode |= sm::NUMBER;
     pushScreen(drawNumberDialog);
     reaction = reaction_t();
diff --git a/graph.cpp b/graph.cpp
index b9c20d30..45f0deac 100644
--- a/graph.cpp
+++ b/graph.cpp
@@ -2309,7 +2309,8 @@ void drawaura() {
   glhr::switch_mode(glhr::gmVarColored);
   glhr::id_modelview();
   glhr::prepare(auravertices);
-  glDrawArrays(GL_TRIANGLES, 0, size(auravertices));    
+  glhr::set_depthtest(false);
+  glDrawArrays(GL_TRIANGLES, 0, size(auravertices));
 
 
   setcameraangle(false);
diff --git a/hyper.h b/hyper.h
index 0c337ee2..69121689 100644
--- a/hyper.h
+++ b/hyper.h
@@ -647,7 +647,7 @@ extern reaction_t help_delegate;
 
 struct videopar {
   ld scale, alpha, sspeed, mspeed, yshift, camera_angle;
-  ld ballangle, ballproj;
+  ld ballangle, ballproj, euclid_to_sphere;
   int mobilecompasssize;
   int aurastr, aurasmoothen;
 
diff --git a/hypgraph.cpp b/hypgraph.cpp
index 8b91ad62..dd0d2f50 100644
--- a/hypgraph.cpp
+++ b/hypgraph.cpp
@@ -129,20 +129,23 @@ void applymodel(hyperpoint H, hyperpoint& ret) {
     
     switch(cgclass) {
       case gcHyperbolic: {
+        ld zl = zlevel(H);
         ret = H / H[2];
         ret[2] = sqrt(1 - sqhypot2(ret));
+        ret = ret * (1 + (zl - 1) * ret[2]);
         break;
         }
         
       case gcEuclid: {
         // stereographic projection to a sphere
-        auto hd = hdist0(H) / H[2];
-        if(hd == 0) H[2] = -1;
+        auto hd = hdist0(H) / vid.euclid_to_sphere;
+        if(hd == 0) ret = hpxyz(0, 0, -1);
         else {
           ld x = 2 * hd / (1 + hd * hd);
           ld y = x / hd;
-          H = H * x / (hd * H[2]);
-          H[2] = 1 - y;
+          ret = H * x / hd / vid.euclid_to_sphere;
+          ret[2] = (1 - y);
+          ret = ret * (1 + (H[2]-1) * y / vid.euclid_to_sphere);
           }
         break;
         }
@@ -153,7 +156,7 @@ void applymodel(hyperpoint H, hyperpoint& ret) {
         }
       }
     
-    ret = rotmatrix(0, 2, ball) * ret;
+    ret = rotmatrix(M_PI/2 + ball, 1, 2) * ret;
     
     ghcheck(ret, H);
     return;
diff --git a/polygons.cpp b/polygons.cpp
index f2252128..55aecd5a 100644
--- a/polygons.cpp
+++ b/polygons.cpp
@@ -250,7 +250,7 @@ void setmatrix(int useV, const transmatrix& V) {
     GLfloat mat[16] = {
       1, 0, 0, 0,
       0, 1, 0, 0,
-      0, 0, 0, 0,
+      0, 0, -1, 0,
       0, 0, stereo::scrdist, 1
       };
     glhr::set_modelview(glhr::as_glmatrix(mat));
@@ -299,14 +299,17 @@ void gldraw(int useV, const transmatrix& V, const vector<glvertex>& v, int ps, i
       glEnable(GL_STENCIL_TEST);
 
       glColorMask( GL_FALSE,GL_FALSE,GL_FALSE,GL_FALSE );
+      glhr::set_depthtest(false);
       glStencilOp( GL_INVERT, GL_INVERT, GL_INVERT);
       glStencilFunc( GL_ALWAYS, 0x1, 0x1 );
       glhr::color2(0xFFFFFFFF);
       glDrawArrays(tinf ? GL_TRIANGLES : GL_TRIANGLE_FAN, ps, pq);
       
+      stereo::set_mask(ed);
+      glhr::color2(col);
+      glhr::set_depthtest(model_needs_depth());
+
       if(flags & POLY_INVERSE) {
-        stereo::set_mask(ed);
-        glhr::color2(col);
         glStencilOp( GL_ZERO, GL_ZERO, GL_ZERO);
         glStencilFunc( GL_NOTEQUAL, 1, 1);
         GLfloat xx = vid.xres;
@@ -324,8 +327,6 @@ void gldraw(int useV, const transmatrix& V, const vector<glvertex>& v, int ps, i
         setmatrix(useV, V);
         }
       else { 
-        stereo::set_mask(ed);
-        glhr::color2(col);
         glStencilOp( GL_ZERO, GL_ZERO, GL_ZERO);
         glStencilFunc( GL_EQUAL, 1, 1);
         glDrawArrays(tinf ? GL_TRIANGLES : GL_TRIANGLE_FAN, ps, pq);
@@ -337,6 +338,7 @@ void gldraw(int useV, const transmatrix& V, const vector<glvertex>& v, int ps, i
     
     if(outline) {
       glhr::color2(outline);
+      glhr::set_depthtest(model_needs_depth());
       glDrawArrays(GL_LINE_STRIP, ps, pq);
       }
     }
diff --git a/shaders.cpp b/shaders.cpp
index b981b624..798a34a2 100644
--- a/shaders.cpp
+++ b/shaders.cpp
@@ -670,4 +670,13 @@ void store_in_buffer(vector<glvertex>& v) {
 #endif
   }
 
+bool current_depthtest;
+
+void set_depthtest(bool b) {
+  if(b != current_depthtest) {
+    current_depthtest = b;
+    if(b) glEnable(GL_DEPTH_TEST);
+    else glDisable(GL_DEPTH_TEST);
+    }
+  }
 }
diff --git a/textures.cpp b/textures.cpp
index 91891039..285359ec 100644
--- a/textures.cpp
+++ b/textures.cpp
@@ -483,6 +483,7 @@ void texture_config::drawRawTexture() {
     }
   glhr::set_modelview(glhr::translate(0, 0, stereo::scrdist));
   glhr::prepare(rtver);
+  glhr::set_depthtest(false);
   glDrawArrays(GL_TRIANGLE_FAN, 0, 4);
   }