epix 1.2.14-1 / usr / share / doc / epix / examples / log.xp

/* -*-ePiX-*- */
#include "epix.h"
using namespace ePiX;

// October 28, 2006
//
// This file depicts two sheets of the Riemann surface of log, together
// with the log function over the unit circle. Surface and path elements
// must be drawn together. However, path-like mesh elements must be drawn
// after surface elements that lie at the same location in the scene,
// since otherwise surface elements' edges occlude the path.
//
// A mesh_elt class stores the necessary P data as well as a flag marking
// the element as a segment or a surface patch. Elements are stored in a
// std::vector (global to the scene) and sorted by distance. To ensure that
// surface patches are printed first, their true distance is increased by a
// "fudge" amount (0.25), making the By_distance class see them as farther
// than they really are.
//
// Path elements are drawn in green; surface patches are shaded gray
// according to the angle between the incoming light (from a point source)
// and the surface normal.
//
// Surface patches are drawn as octagons to make the mesh appear smoother
// than a quadrilateral mesh. The vertices of a patch are the images of the
// corners and midpoints of a coordinate quadrilateral, "retracted" inward
// by a small "gap" EPS (possibly zero). A positive gap simulates surface
// transparency.
//
// (u,v+dv)            (u+0.5*du,v+dv)          (u+du,v+dv)
//    +----------------------+-----------------------+
//    | *--------------------*---------------------* |
//    | |                                          | |
//    | |                                          | |
//    | |                                          | |
//    + * (u+EPS,v+0.5*dv)                         * + (u+du,v+0.5*dv)
//    | |                                          | |
//    | |                                          | |
//    | | (u+EPS,v+EPS)                            | |
//    | *--------------------*---------------------* |
//    +----------------------+-----------------------+
//  (u,v)               (u+0.5*du,v)              (u+du,v)


const P LIGHT(2, 2, 0); // location of light, for shading
const P VIEWPT(15,-10,6);

// surface and path mesh fineness
const int N1(18);
const int N2(80);

const int N3(120);

const double du(4.5/N1);
const double dv(6.0/N2);

const double dt(4.0/N3);

// "gap size" between surface mesh elements
const double EPS(0); // (0.002);

// visual styles
void  path_color() { green(0.8); }
void label_color() { yellow(0.5); }
void   dot_color() { red(); }
void  path_width() { pen(1.5); }

// class can represent either a surface element or a path element
class mesh_elt {
private:
  P pt1, pt2, pt3, pt4, pt5, pt6, pt7, pt8, center;

  double fudge; // artificial increment to distance
  static bool last_was_seg;
  bool is_segment;

public:
  double how_far() const
  {
    return fudge + norm(center -  camera.viewpt());
  }

  mesh_elt(P f(double u, double v), double u0, double v0)
    : pt1(f(u0+EPS,v0+EPS)),       pt2(f(u0+0.5*du,v0+EPS)),
      pt3(f(u0+du-EPS,v0+EPS)),    pt4(f(u0+du-EPS,v0+0.5*dv)),
      pt5(f(u0+du-EPS,v0+dv-EPS)), pt6(f(u0+0.5*du,v0+dv-EPS)),
      pt7(f(u0+EPS,v0+dv-EPS)),    pt8(f(u0+EPS,v0+0.5*dv)),
      center(0.25*(pt1 + (pt3 + (pt5 + pt7)))),
      fudge(0.25), is_segment(false) { }

  mesh_elt(P f(double), double t0)
    : pt1(f(t0)), pt2(f(t0+0.25*dt)), pt3(f(t0+0.5*dt)),
      pt4(f(t0+0.75*dt)), pt5(f(t0+dt)), pt6(), pt7(), pt8(),
      center(0.333*(pt1 + (pt3 + pt5))),
      fudge(0), is_segment(true) { }

  void draw() const
  { 
    if (!is_segment) {
      P normal((pt2 - pt1)*(pt4 - pt1));
      normal *= 1/norm(normal);

      double dens(0.5*(1-((normal|LIGHT)/norm(LIGHT))));

      if (mesh_elt::last_was_seg)
	{
	  mesh_elt::last_was_seg=false;
	  plain(); // reset pen width
	}

      black();
      gray(dens);

      std::vector<P> bd;
      bd.push_back(pt1);
      bd.push_back(pt2);
      bd.push_back(pt3);
      bd.push_back(pt4);
      bd.push_back(pt5);
      bd.push_back(pt6);
      bd.push_back(pt7);
      bd.push_back(pt8);

      path temp(bd, true, true); // closed and filled
      temp.draw();
    }
    else { // segment
      if (!mesh_elt::last_was_seg)
	{
	  mesh_elt::last_was_seg=true;
	  path_width();
	  path_color();
	}

      std::vector<P> bd;
      bd.push_back(pt1);
      bd.push_back(pt2);
      bd.push_back(pt3);
      bd.push_back(pt4);
      bd.push_back(pt5);

      path temp(bd, false, false);
      temp.draw();
    }

  } // end of mesh_elt::draw()

}; // end of class mesh_elt

// initialize static data
bool mesh_elt::last_was_seg = false;

class By_distance {
public:
  bool operator() (const mesh_elt arg1, const mesh_elt arg2)
  {
    return (arg1.how_far()) > (arg2.how_far());
  }
};

// the maps to be plotted
P C_log(double u, double v)
{
  return polar(exp(u), M_PI*v) + P(0, 0, u+M_PI*v);
}

P C_log1(double t)
{
  return C_log(0,t);
}

int main() {
  bounding_box(P(-6, -12),P(6,12));
  unitlength("1in");
  picture(P(4,8));

  begin();
  fill();

  degrees();
  //  label_angle(90);
  label(P(0, ymin()), P(0,4), 
	"$z=\\mathrm{Re}\\,\\log(x+iy) + \\mathrm{Im}\\,\\log(x+iy)$", t);
  radians();
  //  label_angle(0);

  viewpoint(VIEWPT);
  camera.range(20);

  std::vector<mesh_elt> mesh_data;

  // surface elements
  for (int i=0; i<N1; ++i)
    for (int j=0; j<N2; ++j)
      mesh_data.push_back(mesh_elt(C_log, -3+du*i, -3+dv*j));

  // path elements
  for (int i=0; i<N3; ++i)
    mesh_data.push_back(mesh_elt(C_log1, -2+i*dt));

  stable_sort(mesh_data.begin(), mesh_data.end(), By_distance());

  for (int i=0; i<mesh_data.size(); ++i)
    mesh_data.at(i).draw();

  dot_color();
  marker(C_log(0,-2), BOX);
  marker(C_log(0, 0), BOX);
  marker(C_log(0, 2), BOX);

  label_color();
  label(C_log(0,-2), P(6, 0), "$-2\\pi i$", r);
  label(C_log(0, 0), P(6, 0), "$0$", r);
  label(C_log(0, 2), P(-6, 0), "$2\\pi i$", l);

  end();
}