GraphingCalculator 4; Window 187 42 709 936; FontSizes 18; PaneDivider 449; BackgroundType 0; StackPanes 1; Slider 0 1; SliderControlValue 18; SliderOneDirection 1; 2D.Scale 0.2758620689655172 0.2758620689655172 2 2; 2D.BottomLeft -2.4137931034483 -2.1034482758621; 2D.Axes 0; 2D.GraphPaper 0; Text "================================ Matemagi® by Ambjörn Naeve ©Dialectica ================================ Equivalent representations of a 1-blade in 2D. Dragable point P :"; Color 17; Radius 0.1099679129464286; Expr P=cos(2*pi*n)+i*sin(2*pi*n); Color 4; Radius 0.1192801339285714; Expr vector(cos(2*pi*n),sin(2*pi*n)); Text "Length l"; Color 8; MathPaneSlider 182; Expr l=slider([0,4]); Text "The blue point A : "; Color 3; Radius 0.09880719866071429; Expr vector(Re(P)+l*cos(a),Im(P)+l*sin(a)); Text " The blue vector PA : "; Color 3; Radius 0.1015625; Expr vector(Re(P),Im(P)),vector(Re(P)+l*cos(a),Im(P)+l*sin(a)); Text " The black point B :"; Color 8; Radius 0.09612165178571429; Expr vector(Re(P)+2*(l/pi)*cos(a),Im(P)+2*(l/pi)*sin(a)); Text " The red point C :"; Color 2; Radius 0.09270368303571429; Expr vector(Re(P)+l/pi*cos(a),Im(P)+l/pi*sin(a)); Text " The angle a :"; Color 17; MathPaneSlider 28; Expr a=slider([0,2*pi]); Text "The circle with radius l/π around the red point C :"; Color 2; Radius 0.3117621527777778; Expr [x-[Re(P)+l/pi*cos(a)]]^2+[y-[Im(P)+l/pi*sin(a)]]^2=[l/pi]^2; Text " ANALYSIS: The blue vector PA represents a one-blade with length l. The red circle has radius l/π. Therefore the length of its circumference is 2π x l/π = 2l. Hence each of the red half-circles PB that connect the green point P with the black point B have the length l. Moreover, the points P and B lie on the blue vector PA, and each of the two directed half-circles PB have the same direction and orientation as the vector PA. Therefore each of them represents the same one-blade as the vector PA. Moreover, translating the two red half-circles PB along the blue vector PA gives equivalent representations of the underlying one-blade, since the translated half-circles have the same direction, the same orientation, and the same length as the original half-circles."; Color 8; Radius 0.09193638392857142; Expr s=slider([0,1]); Text " "; Color 8; Radius 0.08998325892857142; Expr vector(Re(P)+2*(l/pi)*cos(a),Im(P)+2*(l/pi)*sin(a))+vector(s*cos(a),s*sin(a)); Color 4; Radius 0.1063406808035714; Expr vector(Re(P),Im(P))+vector(s*cos(a),s*sin(a)); Color 2; Radius 0.1269531250000001; Expr [x-[Re(P)+[l/pi+s]*cos(a)]]^2+[y-[Im(P)+[l/pi+s]*sin(a)]]^2=[l/pi]^2; Text " NOTE: Changing the value of either the length l or the angle a changes the one-blade."; Color 17; Expr ?; Text "";