Epicycloids  etc. - Animated

Hyperlinks - to pre-computed examples and explanations ( in  OidEG.m )

Definitions with Explanations :

Animated Examples :

The little hyperlinks " o " center the nearest figure for good viewing.

Instructions --- Create Your Own  ---

"Execute" a command group by placing the cursor in the red stuff (anywhere) and [ENTER].

1.  You MUST execute the command group "Initialize".

2.   Select one  of the 4 types by removing the "#" to un-comment your choice, and execute.

3.  i)  Set "N" only for ----cycloid types (always  a = 1,  and  "b", "c" are computed from  N).

ii)  Set "b" also for ----trochoid types (always  a = 1,  and  "c" is computed from  N).

[Since these curves began by studying radius b-circles rolling on (or inside) radius a-circles,]

[usually  b < a (= 1), but interesting effects can be produced with larger values of  b.]

iii)  Execute this command group.

Now, you see the "Static Graph" for your choices. If happy continue else go to step 2 or 3.

4.   Animate your curve by simple executing the next command group.  Some patience is needed.

5.   (optional)  Do "Save As" from F ile.  "Save" is no good since this is a "Read Only" file, for safety.

Initialize - resets Maple environment, loads needed command packages, sets standard form

 > restart: with(plots): with(plottools):

Basic Parametric 2-D Curve  :   all points  (x, y)  with  t  in the domain  [L . . R]

 > G:=[x(t),y(t),t=L..R]: L:=0: R:=2*Pi:

Standard Form   :   Epicycloids, Hypocycloids, Epitrochoids, Hypotrochoids

 > x:=t->A*cos(t)+B*cos(C*t): y:=t->A*sin(t)+B*sin(C*t):

```Warning, the name changecoords has been redefined

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```Warning, the name arrow has been redefined

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Select a Curve Type -- un-comment ONE  of the following

un-assigning  a, b, c  allows not starting completely over to switch between curve types.

 > a:='a': b:='b': c:='c':

Epicycloid -

 > #A:=a+b: B:=-b: C:=(a+b)/b: titl:="Epicycloid": zz:=1:

Hypocycloid -

 > #A:=a-b: B:=b : C:=-(a-b)/b: titl:="Hypocycloid": zz:=2:

Epitrochoid -

 > #A:=a: B:=-b: C:=c: titl:="Epitrochoid": zz:=3:

Hypotrochoid -

 > A:=a: B:=b: C:=-c: titl:="Hypotrochoid": zz:=4:

 > titl, [x, y] =  G;

Set Values for  a, b, c  --  and see Static Plot

N = number of vertices (= a/b for ----cycloids) , you can change "b" if doing  ----trochoids.

 > N:=3: b:=2/3: c:=N-1: a:=1:

 > if (zz<3) then b:=a/N elif (zz=3) then c:=N+1 fi:

 > txN:=cat("  -  N = ",convert(N,string)): titx:=cat(titl,txN):

 > txb:=cat(" ,  b/a = ",convert(b,string)):

 > if (zz>2) then titx:=cat(titl,txN,txb) fi:

 > Opt1:=scaling=constrained,axes=none,numpoints=200,title=titx:

 > Optg:=Opt1,titlefont=[TIMES,BOLD,14],color=black,thickness=2:

 > g:=plot(G,Optg): g; G;

Animated with Rotating Circles --

 > FF:=min(64,8*N): NN:=100+min(100,10*N): Dt:=evalf(2*Pi/FF):

 > OptAg:=Opt1,frames=FF+1,numpoints=NN,color=red,thickness=4:

 > Ag:=animatecurve(G,OptAg):

 > cent:=circle([0,0],a,color=green,thickness=3):

 > Optci:=color=blue,thickness=2,numpoints=32:

 > cir:=t->circle([A*cos(t),A*sin(t)],b,Optci):

 > Optra:=color=coral,thickness=5:

 > ray:=t->plot([[A*cos(t),A*sin(t)],[x(t),y(t)]],Optra):

 > Optpt:=color=maroon,font=[TIMES,BOLD,14]:

 > pt:=t->textplot([x(t),y(t),"P"],Optpt):

 > dc:=t->display(pt(t),cir(t),ray(t)):

 > cs:=seq([dc(i*Dt)],i=0..FF): V:=A+b+.1:

 > Optcs:=scaling=constrained,insequence=true,view=[-V..V,-V..V]:

 > dcs:=display(cs,Optcs): display(Ag,g,dcs,cent); G;

 >

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