[Question]
  [
An [epicycloid](http://en.wikipedia.org/wiki/Epicycloid) is the curve a point on a circle makes as it rolls around another circle. A [cyclogon](http://demonstrations.wolfram.com/Cyclogons/) is the shape a point on a [regular polygon](http://en.wikipedia.org/wiki/Regular_polygon) makes as it rolls across a plane. An *epicyclogon* is the curve traced by a point on one regular polygon as it rolls around another.
Write a program that draws an epicyclogon given `r`, `r1`, `r2`, `n1`, `n2`:
```
r = number of clockwise revolutions rolling polygon makes around stationary polygon (any real number as limited by float values) 
r1 = distance from center of stationary polygon to each of its vertices (positive real number)
r2 = distance from center of rolling polygon to each of its vertices (positive real number)
n1 = number of sides stationary polygon has (integer greater than 2)
n2 = number of sides rolling polygon has (integer greater than 2)
```
### Notes
* When `r` is negative the roller should go **counterclockwise**.
* For `r`, one revolution occurs when the line connecting the centroids of the two shapes sweeps out a full 360 degrees. This notion is expanded to include all values of `r`. (So in a quarter revolution the line connecting the centroids sweeps out 90 degrees.)
* These arguments should come from the command line or your program should prompt for them (e.g. with Python's `input()`).
* `r1` and `r2` are relative to each other, not the dimensions of the image. So you can set one "unit" to be any number of actual pixels.
The point you must trace out is one of the vertices of the rolling shape. The shapes must start with this vertex touching a stationary vertex and two sides adjacent:
![epicyclogon example](https://i.stack.imgur.com/igPEtm.png)
The exact starting vertices and the angle of the stationary polygon do not matter.
# Output
The output should go to an image that is at least 600x600 pixels (or some variable dimension than can be set to 600). It must show the entire epicyclogon curve specified by the parameters, well framed in the image.
The rolling and stationary polygons must also be drawn (with the roller in it's final state). The two shapes and the epicyclogon should be three noticeably different colors.
There must also be a simple way to *not* draw the polygons (a change of `true` to `false` in the code suffices).
Please show us at least 2 output images. It's ok to shrink them if necessary.
# Scoring
The shortest code that produces valid output images wins.
**Bonuses** 
* Minus 50 bytes if the output is an animated gif (or similar) of the curve being drawn.
* Minus 150 bytes if you let `n1` and `n2` take the value 2 so the shapes become line segments of length `2 * r1` (or `r2`), "rolling" around each other. How you handle `r` when `n1` and `n2` are 2 is up to you since the centroids don't revolve around each other the way they do in other cases. (Not "rolling" at all does not count as handling it.)
Since I am quite eager to see this novel idea executed well (and it's not exactly a cakewalk), **I am going to award 150 bounty rep** to the winner. The contest will end the same day the bounty runs out.
The bounty will not be awarded to the winner if it is clear that they simply rewrote most of the code from another submission.
Library functions that already happen to do this (if there are any) are not allowed.
*Note: This came from my [leftover questions](http://meta.codegolf.stackexchange.com/a/2007/26997) that anyone is free to post. But if no one else posts them there's a good chance I will in time. :P*
  
[Answer]
# Java - ~~2,726~~ 2,634 - 200 = 2434 characters
Improved from 3800 ish bytes
Thanks to all for your suggestions (especially pseudonym117), here is the new version.
I added a class P which is the point class and a class L which extends ArrayList
I also added some minor logic changes. 
Here is the main class (not golfed):
```
import java.awt.*;
import java.awt.geom.*;
import javax.swing.*;
public class Polygons2 extends JPanel{
    public static void main(String[] args) throws InterruptedException{new Polygons2(args);}
    double q=Math.PI*2;
    int d=1;
    public Polygons2(String[] args) throws InterruptedException{
        double revolutions=Double.valueOf(args[0])*q;
        double stationaryRadius = Double.valueOf(args[1]);
        double rollingRadius = Double.valueOf(args[2]);
        int stationarySides = Integer.valueOf(args[3]);
        int rollingSides = Integer.valueOf(args[4]);    
        double dist = stationaryRadius+rollingRadius+70;
        P sp = new P(dist,dist);
        P rp = new P(sp.x,sp.y-rollingRadius-stationaryRadius);
        //get points for rolling polygon and stationary polygon
        int key=0;
        for(double stationaryAngle=-q/4;stationaryAngle<q-q/4;stationaryAngle+=q/stationarySides){
            P p=new P(Math.cos(stationaryAngle)*stationaryRadius+sp.x,Math.sin(stationaryAngle)*stationaryRadius+sp.y);
            p.k=key;key++;
            stationaryPoints.add(p);
        }
        for(double rollingAngle=q/4;rollingAngle<q+q/4;rollingAngle+=q/rollingSides){
            P p=new P(Math.cos(rollingAngle)*rollingRadius+rp.x,Math.sin(rollingAngle)*rollingRadius + rp.y);
            p.k=key;key++;
            rollingPoints.add(p);
        }
        double g=(q/2)-((q/2-(q/rollingSides))/2) - ((q/2-(q/stationarySides))/2)-.05;
        for(P p:rollingPoints){p.r(getPoint(0), g);}
        //set up JFrame
        JFrame f = new JFrame();
        f.add(this);
        f.setSize((int)dist*2+60,(int)dist*2+60);
        f.setVisible(true);
        int[] pKeys= new int[]{stationaryPoints.get(0).k,rollingPoints.get(0).k};
        int index=1;
        P rc = rollingPoints.c();
        P sc =stationaryPoints.c();
        double currentRadian=Math.atan2(rc.y-sc.y,rc.x-sc.x);
        double totalRadian = 0;
        while(Math.abs(totalRadian)<revolutions){
            P rc2 = rollingPoints.c();
            P sc2 =stationaryPoints.c();
            double angle = Math.atan2(rc2.y-sc2.y,rc2.x-sc2.x);
            if(currentRadian-angle<2){totalRadian+=(angle-currentRadian);}
            currentRadian=angle;
            L clone=(L)path.clone();
            clone.add(new P(rollingPoints.get(1).x,rollingPoints.get(1).y));
            path = clone;
            for(P p:rollingPoints){
                p.r(getPoint(pKeys[index]),.01);
                int size = stationaryPoints.size();
                for(int i=0;i<size;i++){
                    P stationaryPointAtI = stationaryPoints.get(i);
                    P nextPoint=null;
                    if(i==size-1){nextPoint=stationaryPoints.get(0);}
                    else{nextPoint=stationaryPoints.get(i+1);}
                    if(p.b(stationaryPointAtI, nextPoint)==1&&containsKey(pKeys,p.k)==0){
                        //rolling point is between 2 stationary points
                        if(index==1){index=0;}else{index=1;}
                        pKeys[index]=p.k;
                    }
                    int size2=rollingPoints.size();
                    for(int h=0;h<size2;h++){
                        P nextPoint2=null;
                        if(h==size2-1){nextPoint2=rollingPoints.get(0);}
                        else{nextPoint2=rollingPoints.get(h+1);}
                        if(stationaryPointAtI.b(rollingPoints.get(h), nextPoint2)==1&&containsKey(pKeys,stationaryPointAtI.k)==0){
                            //stationary point is between 2 rolling points
                            if(index==1){index=0;}else{index=1;}
                            pKeys[index]=stationaryPointAtI.k;
                        }
                    }
                }
            }
            repaint();
            Thread.sleep(5);
        }
    }
    volatile L path = new L();
    L rollingPoints = new L();
    L stationaryPoints = new L();
    P getPoint(int key){
        for(P p:rollingPoints){if(p.k==key){return p;}}
        for(P p:stationaryPoints){if(p.k==key){return p;}}
        return null;
    }
    int containsKey(int[] keys,int key){
        for(int i:keys){if(key==i){return 1;}}
        return 0;
    }
    @Override
    public void paintComponent(Graphics g){
        Path2D.Double sPath = new Path2D.Double();
        sPath.moveTo(stationaryPoints.get(0).x, stationaryPoints.get(0).y);
        for(P p:stationaryPoints){
            sPath.lineTo(p.x, p.y);
        }
        sPath.closePath();
        Path2D.Double rPath = new Path2D.Double();
        rPath.moveTo(rollingPoints.get(0).x, rollingPoints.get(0).y);
        for(P p:rollingPoints){
            rPath.lineTo(p.x, p.y);
        }
        rPath.closePath();
        g.setColor(Color.white);
        g.fillRect(0,0,getWidth(),getHeight());
        Graphics2D t = (Graphics2D)g;
        if(d==1){
        t.setColor(Color.black);
        t.draw(sPath);
        t.setColor(Color.blue);
        t.draw(rPath);
        }
        g.setColor(Color.green);
        for(P p:path){g.fillOval((int)p.x-1, (int)p.y-1, 2, 2);}
    }
}
```
And the golfed version:
```
import java.awt.*;import java.awt.geom.*;import javax.swing.*;import static java.lang.Math.*;class Polygons2Golfed extends JPanel{public static void main(String[]a)throws Exception{new Polygons2Golfed(a);}double q=PI*2;int d=1;public Polygons2Golfed(String[]a)throws Exception{double b,c,f;b=Double.valueOf(a[1]);c=Double.valueOf(a[2]);int d,e;d=Integer.valueOf(a[3]);e=Integer.valueOf(a[4]);f=b+c+100;P o=new P(f,f);P r=new P(o.x,o.y-c-b);int s=0;for(double u=-q/4;u<q-q/4;u+=q/d){P p=new P(cos(u)*b+o.x,sin(u)*b+o.y);p.k=s;s++;l.add(p);}for(double u=q/4;u<q+q/4;u+=q/e){P p=new P(cos(u)*c+r.x,sin(u)*c+r.y);p.k=s;s++;k.add(p);}double g=q/e/2+q/d/2-.05;for(P p:k){p.r(v(0),g);}JFrame j=new JFrame();j.add(this);j.setSize((int)f*2+60,(int)f*2+60);j.setVisible(true);m=new int[]{l.get(0).k,k.get(0).k};int ad=1;P rc=k.c();P sc=l.c();double ab,ac;ab=atan2(rc.y-sc.y,rc.x-sc.x);ac=0;while(abs(ac)<Double.valueOf(a[0])*q){P rc2=k.c();P sc2=l.c();double ah=atan2(rc2.y-sc2.y,rc2.x-sc2.x);if(ab-ah<2)ac+=(ah-ab);ab=ah;L ag=(L)n.clone();ag.add(new P(k.get(1).x,k.get(1).y));n=ag;for(P p:k){p.r(v(m[ad]),.01);int af=l.size();for(int i=0;i<af;i++){P aa=l.get(i);P w=null;if(i==af-1){w=l.get(0);}else{w=l.get(i+1);}if(p.b(aa, w)==1&&w(p.k)==0){if(ad==1)ad=0;else ad=1;m[ad]=p.k;}int ae=k.size();for(int h=0;h<ae;h++){P u=null;if(h==ae-1)u=k.get(0);else u=k.get(h+1);if(aa.b(k.get(h),u)==1&&w(aa.k)==0){if(ad==1)ad=0;else ad=1;m[ad]=aa.k;}}}}repaint();Thread.sleep(5);}}L n=new L();L k=new L();L l=new L();P v(int key){for(P p:k){if(p.k==key)return p;}for(P p:l){if(p.k==key)return p;}return null;}int[]m;int w(int key){for(int i:m){if(key==i)return 1;}return 0;}@Override public void paintComponent(Graphics g){Path2D.Double aq=new Path2D.Double();aq.moveTo(l.get(0).x,l.get(0).y);for(P p:l){aq.lineTo(p.x, p.y);}aq.closePath();Path2D.Double aw=new Path2D.Double();aw.moveTo(k.get(0).x, k.get(0).y);for(P p:k){aw.lineTo(p.x, p.y);}aw.closePath();g.setColor(Color.white);g.fillRect(0,0,getWidth(),getHeight());Graphics2D t=(Graphics2D)g;if(d==1){t.setColor(Color.black);t.draw(aq);t.setColor(Color.blue);t.draw(aw);}g.setColor(Color.green);for(P p:n){g.fillOval((int)p.x-1,(int)p.y-1,2,2);}}}
```
As well as classes P:
```
import java.awt.geom.*;class P{double x,y;public P(double a,double b){x=a;y=b;}int k;void r(P c,double g){double a,r;a=Math.atan2(y-c.y,x-c.x)+g;r=Math.sqrt((c.x-x)*(c.x-x)+(c.y-y)*(c.y-y));x=Math.cos(a)*r+c.x;y=Math.sin(a)*r+c.y;}public int b(P a,P b){if(Line2D.ptSegDist(a.x,a.y,b.x,b.y,x,y)<.5)return 1;return 0;}}
```
And L:
```
import java.util.*;public class L extends ArrayList<P>{public P c(){double x,y;x=0;y=0;for(P p:this){x+=p.x;y+=p.y;}return new P(x/size(),y/size());}}
```
Change int d to 0 or 1 to show polygons
arguments - 1 100 50 5 2
![enter image description here](https://i.stack.imgur.com/Pdwce.gif)
args - 1.5 100 100 7 3
![enter image description here](https://i.stack.imgur.com/K5ZCA.gif)
args - 2 40 100 3 7
![enter image description here](https://i.stack.imgur.com/d0pd8.gif)
[Answer]
# Javascript, 1284 chars (-200 = 1084 chars)
Minified code is
```
function epi(B,r2,r1,n2,n1){K=Math;function C(t){return K.cos(t)}function S(t){return K.sin(t)}function A(y,x){return K.atan2(y,x)}P=K.PI;v=[[],[]];w=[[],[]];z=[];function Z(x,y,j){c=C(t=f*H+P/2);s=S(t);v[j][n]=c*x-s*y;w[j][n]=s*x+c*y;}function E(i){return{x:r1*S(t=p-i*q),y:r1*C(t)};}function D(x,y,X,Y,t){L=A(m.y,m.x);M=K.sqrt(m.x*m.x+m.y*m.y);N=K.sqrt(X*X+Y*Y);O=~~(t*(M>N?M:N)+1);for(i=J;i<=O;i++){J=1;z[n]=f*H+P+t*i/O;Z(x+M*C(T=L+t*i/O),y+M*S(T),0);Z(x+N*C(T=A(Y,X)+t*i/O),y+N*S(T),1);n++}}function F(x,y,n,r,L,s){I.strokeStyle=s;I.beginPath();for(i=0;i<n;i++)I[i?'lineTo':'moveTo'](x+r*C(t=L+(1-2*i)*P/n),y+r*S(t)*W);I.closePath();I.stroke()}p=P/n1;q=2*p;u=P/n2;H=2*u;s2=r2*S(u);g=f=l=n=J=h=0;R=300;while(l<=(B*2+1)*P/H){o=E(0);m=E(h);m.y-=o.y;m.x-=o.x;if(g<s2){D(g,-r2*C(u),-o.x,-o.y,q);h=(h+1)%n1;g+=2*r1*S(p)}else{m.x+=g-s2;D(s2,-r2*C(u),-o.x+g-s2,-o.y,H);g-=s2*2;f=(f+1)%n2;l++}}return function(e_,t,aa,W_){W=aa?-1:1;I=(e=e_).getContext('2d');I.fillStyle='black';I.fillRect(0,0,600,600);W_&1&&F(R,R,n2,r2,0,'white');T=A(w[1][0],v[1][0]);U=V=0;I.strokeStyle='teal';I.beginPath();I.moveTo(R+v[0][0],R+w[0][0]*W);while(U<t){_=A(w[1][V+1],v[1][V+1]);U+=_-T+(_+1<T?2*P:0);T=_;V++;I.lineTo(R+v[0][V],R+w[0][V]*W)}W_&2&&I.stroke();W_&4&&F(R+v[1][V],R+w[1][V]*W,n1,r1,z[V],'red')}}
```
Full code is
```
function epi( nr, r2, r1, n2, n1 ) {
function C( t )
    { return Math.cos( t ); }
function S( t )
    { return Math.sin( t ); }
function A( dy, dx )
    { return Math.atan2( dy, dx ); }
var iCCW, e, t_, xs = [[],[]], ys = [[],[]], ts = [], n = 0, iArc0 = 0;
function addpt( x, y, iBin ) {
    var c_ = C(t_ = iFrame*t2 + Math.PI/2 ),
        s_ = S(t_);
    xs[iBin][n] = c_*x-s_*y;
    ys[iBin][n] = s_*x+c_*y;
}
function poly1pt( iP )
    { return { x: r1*S(t_ = t1b2-iP*t1), y: r1*C(t_) }; }
function arc1( P_Arc_, xP_, yP_, xC_, yC_, t ) {
    var dx_, dy_, dxC, dyC;
    var t0 = A( dy_ = P_Arc_.y, dx_ = P_Arc_.x ),
        r_ = Math.sqrt( dx_*dx_ + dy_*dy_ ),
        t0C = A( dyC = yC_, dxC = xC_ ),
        rC = Math.sqrt( dxC*dxC + dyC*dyC ),
        nt = ~~(t*(r_>rC?r_:rC)+1);
    for( var i = iArc0; i <= nt; i++ ) {
        iArc0 = 1;
        ts[n] = iFrame*t2 + Math.PI + t*i/nt;
        addpt( xP_ + r_*C(t_ = t0+t*i/nt), yP_ + r_*S(t_), 0 );
        addpt( xP_ + rC*C(t_ = t0C+t*i/nt), yP_ + rC*S(t_), 1 );
        n++;
    }
}
function poly( x,y, n, r, t0, sColor ) {
    var Cx = e.getContext('2d');
    Cx.strokeStyle = sColor;
    Cx.beginPath();
    for( var i = 0; i < n; i++ )
        Cx[i ? 'lineTo' : 'moveTo']( x + r*C(t_ = t0+(1-2*i)*Math.PI/n), y + r*S(t_)*iCCW );
    Cx.closePath();
    Cx.stroke();
}
var t1b2 = Math.PI/n1,
    t1 = 2*t1b2,
    t2b2 = Math.PI/n2,
    t2 = 2*t2b2,
    s1 = 2*r1*S(t1b2),
    s2 = 2*r2*S(t2b2),
    xPivot = 0,
    iPivot = 0,
    iFrame = 0,
    P_Pivot, P_Arc,
    nFrame = 0;
while( nFrame <= (nr*2+1)*Math.PI/t2 ) {
    P_Pivot = poly1pt( 0 );
    P_Arc = poly1pt( iPivot );
    if( xPivot < s2/2 ) {
        P_Arc.x -= P_Pivot.x;
        P_Arc.y -= P_Pivot.y;
        arc1( P_Arc, xPivot, -r2*C(t2b2), -P_Pivot.x, -P_Pivot.y, t1 );
        iPivot = (iPivot+1) %n1;
        xPivot += s1;
    } else {
        P_Arc.x -= (P_Pivot.x - (xPivot - s2/2));
        P_Arc.y -= P_Pivot.y;
        arc1( P_Arc, s2/2, -r2*C(t2b2), -P_Pivot.x + xPivot - s2/2, -P_Pivot.y, t2 );
        xPivot -= s2;
        iFrame = (iFrame+1) %n2;
        nFrame++;
    }
}
function renderTo( eCanvas, t, isCCW, sWhat ) {
    iCCW = isCCW ? -1 : 1;
    var Cx = (e = eCanvas).getContext('2d');
    Cx.fillStyle = 'black';
    Cx.fillRect( 0,0, 600,600 );
    if( sWhat &1 )
        poly( 300,300, n2, r2, 0, 'white' );
    var tRef = A( ys[1][0], xs[1][0] ),
        tCum = 0,
        i0 = 0;
    Cx.strokeStyle = 'green';
    Cx.beginPath();
    Cx.moveTo( 300+xs[0][0], 300+ys[0][0]*iCCW );
    while( tCum < t ) {
        t_ = A( ys[1][i0+1], xs[1][i0+1] );
        tCum += t_ - tRef + (t_ - tRef < -1 ? 2*Math.PI : 0);
        tRef = t_;
        i0++;
        Cx.lineTo( 300+xs[0][i0], 300+ys[0][i0]*iCCW );
    }
    if( sWhat &2 )
        Cx.stroke();
    if( sWhat &4 )
        poly( 300+xs[1][i0], 300+ys[1][i0]*iCCW, n1, r1, ts[i0], 'red' );
}
return renderTo;
}
```
A fiddle to behold the routine in all its polygony glory (and to demonstrate animation) is found at
<http://jsfiddle.net/7rv751jy/2/embedded/result/>
The script defines a function called `epi` that accepts the five listed parameters in the OP. `epi` returns a function with the signature `(e,t,isCCW,flags)` which accepts arguments:
* `e` - a reference to a 600x600 HTML5 canvas element on which to render
* `t` - the total angle (in radians) that the centroid of the second polygon should sweep around the centroid of the first. The argument passed in should not exceed 2 pi times the number of rotations passed to `epi`.
* `isCCW` - boolean indicating whether the trace should proceed in a counterclockwise direction (as opposed to a clockwise one)
* `flags` - a set of bit flags indicating which elements should be rendered
	+ bit 1 - render polygon 1 if set
	+ bit 2 - render trace if set
	+ bit 3 - render polygon 2 if set
The function can be called any number of times with varying sets of arguments.
**Some Notes:**
* The routine handles the degenerate cases where `n1 = 2` and/or `n2 = 2`. When animating, certain combinations of lengths will cause sudden rapid advances in the trace. This is because the animation frames are indexed by the angle to the centroid of the second polygon, and d theta poly2 / d theta centroid becomes singular in cases where the centroid of 2-sided poly 2 is near a vertex of 2-sided poly 1. This does not affect the trace, however.
* The parameter names in `epi` will seem confusing since throughout development, I referred to polygon 1 as "2", and polygon 2 as "1". When I realized the inconsistency between my convention and that of the OP, rather than swapping all of the indices in the code, I simply swapped the order of the arguments in `epi`.
* The fiddle above imports jQuery, but this is to handle the UI. The `epi` function has no library dependencies.
* The code handles CCW traces by simply inverting the Y axis. This is somewhat inelegant since polygon 2 starts in a Y-inverted position during CCW traces, but nobody said the routine had to be elegant. ;)
[Answer]
# MATLAB: 735 bytes - 200 bonus = 535
My program handles the n=2 case and draws a real-time animation.
There are a few differences between the golfed and ungolfed versions:
The ungolfed version only has an option to save the animation to a file 'g.gif', by setting `savegif = 1` in the code. It is off by default as it can be annoying for a few reasons: 
* Creation of an unwanted file
* Possible lag
* An error is generated if you have multiple monitors and the plot window is not on the right one...
The gif saving had to be dropped in the golfed version as it took about 100 bytes, exceeding the size of the bonus.
The ungolfed version draws a circle on the tracer vertex. It also produces more frames and moves faster (though this can be adjusted in the golfed version by changing the numbers).
## Samples:
`f(11,5,90,2,99,0)` after program termination
![golfed sample](https://i.stack.imgur.com/3iBLb.jpg)
`epic(1.3,4,2,6,6,1)` with gif output
![ungolfed sample](https://i.stack.imgur.com/0BVO8.gif)
## Ungolfed code
```
%epicyclogon animation outputs to 'g.gif' if savegif=1 as well as animating in real time
function[] = epic(r,r1,r2,n1,n2,dispPoly)
savegif = 0;  %set to 1 to write .gif
cs = @(a) [cos(a);sin(a)];
vert = @(r, n, v) r * cs(2*pi*v/n);
polyPt = @(l, s, n, r) vert(r, n, floor(l/s)) + mod(l/s,1)*(vert(r, n, floor(l/s)+1) - vert(r, n, floor(l/s)));
polyPt2 = @(i, f, n, r) vert(r, n, i) + f*(vert(r, n, i+1) - vert(r, n, i));
rotm = @(a) [cos(a) -sin(a);sin(a) cos(a)];
arrpluspt = @(a, p) a + kron(p, ones(1,length(a)));
arg = @(p) atan2(p(2), p(1));
E = 1e-9;
dispPoly = dispPoly / dispPoly;
sgn = sign(-r);
r = abs(r);
s1 = 2*r1*sin(pi/n1);
s2 = 2*r2*sin(pi/n2);
%d1 = (r1*r1 - s1*s1*.25)^.5;
d2 = (r2*r2 - s2*s2*.25)^.5;
plotmax = r1+2*r2;
astep = .05; %determines amount of frames per rotation
delay = .01; % time per frame
l = 0;
lRem = 0;
lr = 0;
P1 = vert(r1, n1, 1:n1+1) * dispPoly; 
trace = [];
first = 1;
while 1
    if lr %exists while rotating about a corner of the stationary
        rotA = 2*pi/n1;
    else
        rotA = 2*pi/n2;
    end
    rotPt = polyPt(l, s1, n1, r1);
    lb = l + lRem;
    side1 = floor(l / s1 - E);
    side1up = side1 + lr;
    p2cen = polyPt2(side1, lb/s1 -side1 - .5 * s2/s1, n1, r1) + d2 * cs(2*pi*(side1+.5)/n1);
    if first
        p2cen0 = p2cen;
        r = r + arg(p2cen0)/(2*pi);
    end
    for a = 0:astep:rotA    
        P2 = vert(r2, n2, 0:n2);
        P2 = rotm( pi +pi/n1 -pi/n2   +2*pi*side1/n1) * P2;
        P2 = arrpluspt(P2, p2cen);
        P2 = arrpluspt(P2, -rotPt);
        P2 = rotm(a) * P2;
        P2 = arrpluspt(P2, rotPt);
        trV = mod(floor(l/s2 + E) + lr, n2) + 1;
        cen = rotm(a) * (p2cen - rotPt) + rotPt;
        trace = [trace,P2(:,trV)]; 
        plot(P1(1,:), sgn*P1(2,:), P2(1,:)*dispPoly, sgn*P2(2,:)*dispPoly, trace(1,:),sgn*trace(2,:),P2(1,trV), sgn*P2(2,trV),'o');
        %plot(P1(1,:), P1(2,:), P2(1,:), P2(2,:), trace(1,:),trace(2,:),...
        %[0,p2cen0(1)],[0,p2cen0(2)],[0,cen(1)],[0,cen(2)], P2(1,trV), P2(2,trV),'o');
        axis([-plotmax,plotmax,-plotmax,plotmax]);
        axis square
        figure(1);
       if savegif
           drawnow
           frame = getframe(1); % plot window must be on same monitor!
           img = frame2im(frame);
           [img1,img2] = rgb2ind(img,256);
       end
       if first
           if savegif
               imwrite(img1,img2,'g','gif','DelayTime',2*delay); %control animation speed(but not really)
           end
           first = 0;
       else
           if savegif
               imwrite(img1,img2,'g','gif','WriteMode','append','DelayTime', 2*delay);
           end
       end
       pause(.01);
        adf = mod(arg(cen) - r*2*pi, 2*pi);
        if adf < astep & l/(n1*s1) + .5 > r
            return
        end
    end
%cleanup for next iteration 
    jump = lRem + ~lr * s2; 
    lnex = l + jump; 
    if floor(lnex / s1 - E) > side1up 
        lnex = s1*(side1up+1);
        lRem = jump - (lnex - l);
        lr = 1;
    else    
        lRem = 0;
        lr = 0;
    end
    l = lnex;
end
```
## Golfed code
```
function[]=f(r,h,H,n,N,d)
P=pi;T=2*P;F=@floor;C=@(a)[cos(a);sin(a)];g=@(i,f,n,r)r*C(T*i/n)*(1-f)+f*r*C(T*(i+1)/n);R=@(a)[C(a),C(a+P/2)];W=@(a,p)[a(1,:)+p(1);a(2,:)+p(2)];b=@(p)atan2(p(2),p(1));E=1e-9;d=d/d;S=1-2*(r>0);r=-r*S;x=2*h*sin(P/n);X=2*H*sin(P/N);M=h+2*H;l=0;z=0;L=0;A=h*C(T*(0:n)/n)*d;t=[];while 1
v=l/x;D=F(v-E);q=g(D,v-D,n,h);Z=D+L;c=g(D,v+z/x-D-.5*X/x,n,h)+H*cos(P/N)*C(T*D/n+P/n);r=r+~(l+L)*b(c)/T;for a=0:.1:T/(L*n+~L*N)
O=@(p)W(R(a)*W(p,-q),q);B=O(W(R(P+P/n-P/N+T*D/n)*H*C(T*(0:N)/N),c));t=[t,B(:,mod(F(l/X+E)+L,N)+1)];plot(A(1,:),S*A(2,:),d*B(1,:),d*S*B(2,:),t(1,:),t(2,:)*S)
axis([-M,M,-M,M],'square');pause(.1);if.1>mod(b(O(c))-r*T,T)&v/n+.5>r
return;end;end;j=z+~L*X;J=l+j;L=F(J/x-E)>Z;l=L*x*(Z+1)+~L*J;z=L*(J-l);end
```
## Instructions:
Save the function to a file with the same name, i.e. `epic.m` or `f.m`.
Run it by calling the function from the Matlab console.
Usage: `epic(r, r1, r2, n1, n2, dispPoly)`
where `dispPoly` is a Boolean variable (zero if false, a nonzero number if true) determining whether to draw the polygons.
Edit: Added bonus of 50 for animated image.
]