GraphingCalculator 3.5;
Window 69 313 858 1352;
PaneDivider 238;
SignificantDigits 14;
FontSizes 14 12 10;
BackgroundType 0;
BackgroundColor 255 255 255;
T 0 10;
U -10 10;
V -10 10;
3D.Depth 0.6830909742999999;
3D.View 0.3610406386536392 -0.9214264064357528 -0.1436072269198904 0.8578024845983008 0.3885491597899672 -0.3364586879609383 0.3658203869643722 -0.001711376475433019 0.930683897213246;
3D.Speed 0;
Text "Differential geometry of curves.   Frenet Vectors.

To do:
(i) Execute the f'ing osculating circle as a parameterized curve.


Path X.

For now, just a spiral with variable radius and pitch.";
Color 5;
MathPaneSlider 70;
Expr A=slider([0,10]);
Color 6;
MathPaneSlider 49;
Expr p=slider([0,1]);
Color 4;
Expr function(f,x)=A*cos(x),function(g,x)=A*sin(x),function(h,x)=p*x;
Expr function(F,x)=A*cos(x)*[1+a*cos(P*x)],function(G,x)=A*sin(x)*[1+a*cos(P*x)],function(H,x)=p*x;
Expr a=slider([0,0.3]);
Color 3;
Expr P=slider([0,10,10]);
Color 3;
Expr function(X,t)=vector(function(f,t),function(g,t),function(h,t));
Color 2;
Expr function(Y,t)=vector(function(F,t),function(G,t),function(h,t));
Text "Frenet vectors: the unit tangent T, normal N and binormal B (possibly with sign of latter reversed). ";
Color 5;
Expr function(T,t)=1/abs(function(oppartial(t),function(X,t)))*[function(oppartial(t),function(X,t))];
Color 7;
Expr function(N,t)=(function(oppartial(t),function(oppartial(t),function(X,t)))-([dot([function(oppartial(t),function(oppartial(t),function(X,t)))],function(T,t))]*function(T,t)))/abs(function(oppartial(t),function(oppartial(t),function(X,t)))-([dot([function(oppartial(t),function(oppartial(t),function(X,t)))],function(T,t))]*function(T,t)));
Color 8;
Expr function(B,t)=cross(function(T,t),function(N,t))/abs(cross(function(T,t),function(N,t)));
Text "Curvature and radius of curvature:";
Color 2;
Expr function(k,t)=abs(cross([function(oppartial(t),function(X,t))],[function(oppartial(t),function(oppartial(t),function(X,t)))]))/abs(function(oppartial(t),function(X,t)))^3;
Color 4;
Expr function(R,t)=1/function(k,t);
Color 17;
Expr function(R,s);
Text "Standard unit vectors:";
Color 6;
Expr O=vector(0,0,0),I=vector(1,0,0),J=vector(0,1,0),K=vector(0,0,1);
Text "Path:";
Color 6;
Expr function(X,t);
Text "Coordinate frame along curve defined by Frenet vectors:";
Color 7;
MathPaneSlider 16;
Expr s=slider([0,10]);
Grain 0;
Expr function(X,s);
Color 2;
Expr function(X,s),function(X,s)+function(T,s);
Color 3;
Expr function(X,s),function(X,s)+function(N,s);
Color 8;
Expr function(X,s),function(X,s)+function(B,s);
Text "Osculating circle:";
Color 6;
Grain 0;
Expr function(X,s)+function(N,s)*function(R,s);
Text "(Much faster with center of circle only.)";
Text "

Author: David A. Craig <<http://web.lemoyne.edu/~craigda/>";