GraphingCalculator 4; Window 2 6 852 1412; FontSizes 18; PaneDivider 360; BackgroundType 0; StackPanes 1; Slider 0 1; SliderControlValue 40; SliderMoving 1; 2D.BottomLeft -5.40625 -3.890625; 2D.Axes 0; 2D.GraphPaper 0; 3Dp.View 0.800845072123107 0.3806085836414124 -0.4623681179694616 -0.3296167023318871 0.9247362359389217 0.1903042918207054 0.4999999999999999 0 0.8660254037844387; 3Dp.Speed 0.0448758357137238; 3Dp.Axis 0 0 -1; Text " The 2-D configuration (in the left window) The fixed vertical line (with equation y = 0):"; Color 6; Opacity 0.7; Expr prime(y)=0; Text " Let P be a dragable point"; Color 2; Expr P=-1.96875+0.640625*i; Text "and let"; Color 17; Expr a=Re(P); Color 17; Expr b=Im(P); Text " The initial vector (a, b):"; Color 7; Expr vector(0,0),vector(a,b); Color 7; Expr vector(-0.02,0),vector(a-0.02,b); Color 7; Expr vector(0,-0.02),vector(a,b-0.02); Text " The tip of the initial vector (in the left widow): "; Color 17; Expr vector(0,0),vector(a,b); Text "The initial vector (in the right widow):"; Color 17; Expr A=vector(a,b,c); Text " The origin:"; Color 8; Expr vector(0,0); Text " The first (yellow) mirror:"; Color 6; Expr vector(x,y)=t*vector(-10,0)+[1-t]*vector(10,0); Color 6; Expr vector(x,y)=t*vector(-10+0.02,0)+[1-t]*vector(10+0.02,0); Color 6; Expr vector(x,y)=t*vector(-10,0.02)+[1-t]*vector(10,0.02); Text " The reflection of the initial vector (a, b) in the (yellow) line y = 0:"; Color 6; Expr vector(0,0),vector(a,-b); Color 6; Expr vector(-0.02,0),vector(a-0.02,-b); Color 6; Expr vector(0,-0.02),vector(a,-b-0.02); Text " The tip of the yellow vector:"; Color 17; Expr vector(a,-b); Text " The reflected vector B (in the right window):"; Color 17; Expr B=vector(a,-b,c); Text " The second (light-blue) line mirror: "; Color 5; Expr [sin(pi*k)]*x-([cos(pi*k)]*y)=0; Color 5; Expr [sin(pi*k)]*[x+0.01]-([cos(pi*k)]*[y+0.01])=0; Text " The circle on which the tips of the initial and the reflected vectors are located:"; Color 7; Expr vector(x,y)=vector([a^2+b^2]^(1/2)*cos([2*pi*t]),[a^2+b^2]^(1/2)*sin([2*pi*t])); Color 7; Expr vector(x,y)=vector([a^2+b^2-0.01]^(1/2)*cos([2*pi*t]),[a^2+b^2-0.01]^(1/2)*sin([2*pi*t])); Color 7; Expr vector(x,y)=vector([a^2+b^2+0.01]^(1/2)*cos([2*pi*t]),[a^2+b^2+0.01]^(1/2)*sin([2*pi*t])); Color 7; Expr vector(prime(x),prime(y),prime(z))=vector([a^2+b^2]^(1/2)*cos([2*pi*t]),[a^2+b^2]^(1/2)*sin([2*pi*t]),c),'radius'=0.025; Text "Projection of (a, - b) on the normal to the second (light-blue) line:"; Color 17; Expr vector(0,0),[dot(vector(a,-b),vector(sin(pi*k),-cos(pi*k)))]*vector(sin(pi*k),-cos(pi*k)); Text " Reflection of (a, - b) in the second (light-blue) mirror line: "; Color 5; Expr vector(0,0),vector(a,-b)-(2*[dot(vector(a,-b),vector(sin(pi*k),-cos(pi*k)))]*vector(sin(pi*k),-cos(pi*k))); Color 5; Expr vector(0.03,0),vector(a-0.02,-b)-(2*[dot(vector(a-0.02,-b),vector(sin(pi*k),-cos(pi*k)))]*vector(sin(pi*k),-cos(pi*k))); Color 5; Expr vector(0,0.02),vector(a,-b-0.02)-(2*[dot(vector(a,-b-0.02),vector(sin(pi*k),-cos(pi*k)))]*vector(sin(pi*k),-cos(pi*k))); Text " The reflection of the yellow vector in the blue mirror is given by the vector C: The tip of the light-blue vector:"; Color 17; Expr vector(a,-b)-(2*[dot(vector(a,-b),vector(sin(pi*k),-cos(pi*k)))]*vector(sin(pi*k),-cos(pi*k))); Text ""; Color 17; Expr C=vector(a+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*sin(pi*k),-b+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*[-cos(pi*k)],c); Text " The 3D-correspondence (in the right window): The horizontal (green) plane "; Color 4; Opacity 0.7; Expr prime(z)=0; Text " The initial vector:"; Color 7; Expr vector(prime(x),prime(y),prime(z))=t*vector(a,b,c),'radius'=0.03; Text " has the tip"; Color 2; Grain 0.9833333333333333; Expr [prime(x)-a]^2+[prime(y)-b]^2+[prime(z)-c]^2=1/100; Text " reflected in the yellow plane (y = 0) gives: "; Color 6; Expr vector(prime(x),prime(y),prime(z))=t*vector(a,-b,c),'radius'=0.03; Text "which has the tip"; Color 6; Expr [prime(x)-a]^2+[prime(y)+b]^2+[prime(z)-c]^2=1/81; Text " This vector is then reflected in the light-blue plane. (This is the moving vertical plane with the angle πk relative to the fixed (yellow) vertical plane): "; Color 5; Opacity 0.7; Expr [sin(pi*k)]*prime(x)-([cos(pi*k)]*prime(y))=0; Text " This gives the reflected vector"; Color 5; Expr vector(prime(x),prime(y),prime(z))=t*vector(a+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*sin(pi*k),-b+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*[-cos(pi*k)],c),'radius'=0.03; Text " which has the tip"; Color 5; Expr [prime(x)-[a+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*sin(pi*k)]]^2+[prime(y)-[-b+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*[-cos(pi*k)]]]^2+[prime(z)-c]^2=1/81; Text " The line of intersection between the yellow plane and the light-blue plane is given by: "; Color 7; Expr vector(prime(x),prime(y),prime(z))=vector(0,0,5*[t-(1/2)]),'radius'=0.04; Text "The angle between the light-blue line and the yellow line (in the left window) is given by:"; Color 8; Expr vector(r,theta)=vector(0.45,t*pi*k); Text " Twice this angle can be expressed as:"; Color 2; Expr vector(r,theta)=vector(0.55,t*arg(P)+[1-t]*[arg(P)+2*pi*k]); Text " The angle between the yellow plane and the light-blue plane (in the right window):"; Color 8; Expr vector(prime(x),prime(y),prime(z))=0.45*vector(cos([t*pi*k]),sin([t*pi*k]),0),'radius'=0.015; Text " Testing:"; Color 17; Expr vector(prime(x),prime(y),prime(z))=t*A,'radius'=0.05; Color 17; Expr vector(prime(x),prime(y),prime(z))=t*B,'radius'=0.05; Color 17; Expr vector(prime(x),prime(y),prime(z))=t*vector(a+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*sin(pi*k),-b+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*[-cos(pi*k)],c),'radius'=0.05; Text " The plane that contains both A and C (can't use the symbol for C here): "; Color 17; Opacity 0.7; Expr dot([cross(A,vector(a+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*sin(pi*k),-b+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*[-cos(pi*k)],c))],vector(prime(x),prime(y),prime(z)))=0; Text " Testing:"; Color 17; Expr vector(prime(r),prime(theta),prime(z))=vector(0.45,t*arg(P)+[1-t]*[arg(P)+2*pi*k],0),'radius'=0.04; Text ""; Color 17; Expr vector(prime(x),prime(y),prime(z))=matrix(3,3,a,a+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*sin(pi*k),[b+2*K]*cos(pi*k),b,-b+[-(2*[a*sin(pi*k)+b*cos(pi*k)])]*[-cos(pi*k)],-a+2*K*sin(pi*k),c,c,0)*vector(0,1,0); Text " We introduce the parameter "; Color 17; Expr K=a*sin(pi*k)+b*cos(pi*k); Text "and we get the matrix that transforms the plane z = 0 into the plane that contains the vectors A and C: "; Color 17; Expr M=matrix(3,3,a,a+[-(2*K)]*sin(pi*k),[b+2*K]*cos(pi*k),b,-b+[-(2*K)]*[-cos(pi*k)],-a+2*K*sin(pi*k),c,c,0); Text " The angle between the gray vector and the blue vector is represented by the red circular arc"; Color 2; Expr vector(prime(x),prime(y),prime(z))=R*((t*A+[1-t]*vector(a+[-(2*K)]*sin(pi*k),-b+[-(2*K)]*[-cos(pi*k)],c))/abs(t*A+[1-t]*vector(a+[-(2*K)]*sin(pi*k),-b+[-(2*K)]*[-cos(pi*k)],c))),'radius'=0.015; Text " Changing the angle πk between the two planes:"; Color 17; Expr k=n; Text " Controlling the distance R from the origin"; Color 17; MathPaneSlider 99; Expr R=slider([0,2]); Text " Controlling the height z above the xy-plane: "; Color 3; MathPaneSlider 159; Expr c=slider([-2,2]);