Rotations and reflections of geometric objects in the plane - with an emphasis on animating the transformation.

restart: with(plots): with(plottools):with(LinearAlgebra):

Warning, the name changecoords has been redefined

Warning, the assigned name arrow now has a global binding

p := [<1,1>, <-1,1>, <-1,-1>, <1,-1>]; # p := [<2,0>,<3,2>,<1,4>,<0,2>,<-1,3>,<-2,0>,<-2,-2>,<0,-2>];

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

Construct a regular n-gon on the unit circle.

nGon := n -> [seq(<cos((4*i+2-n)*Pi/(2*n)),sin((4*i+2-n)*Pi/(2*n))>,i=0..(n-1))]; p := nGon(6);

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for i from 1 to (nops(p)-1) do l[i] := [p[i],p[i+1]]; end do; l[nops(p)] := [p[nops(p)],p[1]]; nops(l);

The colors list will allow objects with up to 16 defining points.

colors := [green,blue,red,cyan,yellow,magenta,coral,violet,orange,sienna,navy,brown,aquamarine,gold,maroon,khaki]:

Create lighter shades of each color.

lcolors := [COLOR(RGB,.5,1,.5),COLOR(RGB,.5,.5,1),COLOR(RGB,1,.5,.5),COLOR(RGB,.5,1,1),COLOR(RGB,1,1,.5),COLOR(RGB,1,.5,1),COLOR(RGB,1,0.7490196100,.5),COLOR(RGB,0.6549019600,0.5921568650,0.6549019600),COLOR(RGB,.9,0.5980392150,0.5980392150),COLOR(RGB,0.7784313750,0.7098039200,0.5686274500),COLOR(RGB,0.5686274500,0.5686274500,0.7784313750),COLOR(RGB,0.8235294100,0.5823529400,0.5823529400),COLOR(RGB,0.7196078450,0.9294117650,0.7882352950),COLOR(RGB,.9,0.7490196100,0.5980392150),COLOR(RGB,0.7784313750,0.5686274500,0.7098039200),COLOR(RGB,0.8117647050,0.8117647050,0.6862745100)]:

Show the colors with the lighter shades in the order they are used in plots.

for i from 1 to 16 do tp[i] := plot([t,3,t=i..i+.5],0..17,0..10,color=colors[i],thickness=15): tpl[i] := plot([t,2.5,t=i..i+.5],0..17,0..10,color=lcolors[i],thickness=15): end do: display([seq(tp[i],i=1..16),seq(tpl[i],i=1..16)]);

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for i from 1 to nops(p) do lp[i] := pointplot(l[i],color = colors[i], thickness = 5,connect=true): end do:

display([seq(lp[i],i=1..nops(p))],scaling=constrained,axes=none);

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rotmat := alpha -> <<cos(alpha), sin(alpha)> | <-sin(alpha), cos(alpha)>>;

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rot90 := rotmat(Pi/2);

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multmatbylist :=(multmat, listofvecs)-> map((x,y)-> y.x,listofvecs,multmat):

rotateObj := (listoflistofvecs,ang) -> seq(multmatbylist(rotmat(ang),listoflistofvecs[i]),i=1..nops(p));

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ObjRot1:= rotateObj(l,Pi/3); l[1]; ObjRot1[1]; type([ObjRot1],list); [ObjRot1][1]; l[1];

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for i from 1 to nops(p) do ObjRotp[i] := pointplot(ObjRot1[i],color = colors[i], thickness = 5,connect=true): end do: display([seq(ObjRotp[i],i=1..nops(p))],scaling=constrained);

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display([seq(display([seq(pointplot(l[i],color = lcolors[i], thickness = 5,connect=true),i=1..nops(p)),seq(pointplot(multmatbylist(rotmat((counta/18)*Pi/2),l[i]),color=colors[i],connect=true,thickness=5),i=1..nops(p))]), counta=0..18)], insequence = true);

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

print(l); nops(l); l[1];l[2];

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AnimRotate := proc(l,ang) local NumSteps; NumSteps := ceil(36*abs(ang)/Pi): display([seq(display([seq(pointplot(l[i],color = lcolors[i], thickness = 5,connect=true),i=1..nops(p)),seq(pointplot(multmatbylist(rotmat((counta/NumSteps)*ang),l[i]),color=colors[i],connect=true,thickness=5),i=1..nops(p))]), counta=0..NumSteps)],scaling=constrained, insequence = true) end proc;

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

AnimRotate(l,Pi/3);

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

Now work on the reflections. Basic reflection matrix reflects in steps across x-axis, but we represent this as rotation around the x-axis in R3.

refinxmat := alpha -> <<1,0,0>|<0,cos(alpha),sin(alpha)>|<0,-sin(alpha),cos(alpha)>>;

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

refinxmat(Pi);

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

To reflect in the line at angle theta we need to conjugate by the change of basis matrix

chbasis := theta -> <<cos(theta),sin(theta),0>|<-sin(theta),cos(theta),0>|<0,0,1>>; refmat := (theta,alpha) -> chbasis(theta).refinxmat(alpha).(chbasis(theta))^(-1);

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refmat(3*Pi/4,Pi);

To reflect a vector we embed into R3, apply the reflection matrix, and then project back to R2.

projembmultmatbylist :=(multmat, listofvecs)-> map((x,y)-> <(y.<x[1],x[2],0>)[1],(y.<x[1],x[2],0>)[2]>,listofvecs,multmat):

projembmultmatbylist(refmat(Pi/4,Pi),l[1]);

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reflectObj := (listoflistofvecs,ang) -> seq(projembmultmatbylist(refmat(ang,Pi),listoflistofvecs[i]),i=1..nops(p));

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refObj1 := reflectObj(l,Pi/6);

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for i from 1 to nops(p) do refObj1p[i] := pointplot(refObj1[i],color = colors[i], thickness = 5,connect=true): end do: display([seq(refObj1p[i],i=1..nops(p))],scaling=constrained);

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

AnimReflect := proc(l,ang) display([seq(display([plot([cos(ang)*t,sin(ang)*t,t=-1.1..1.1],linestyle=DOT,thickness=2,color=black),seq(pointplot(l[i],color = lcolors[i], thickness = 5,connect=true),i=1..nops(p)),seq(pointplot(projembmultmatbylist(refmat(ang,(counta/18)*Pi),l[i]),color=colors[i],connect=true,thickness=5),i=1..nops(p))]), counta=0..18)], insequence = true,scaling=constrained) end proc;

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AnimReflect(l,Pi/3);

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