GraphingCalculator 4;
Window 47 6 865 1211;
PaneDivider 411;
SignificantDigits 14;
FontSizes 14;
BackgroundType 0;
BackgroundColor 255 255 255;
StackPanes 1;
T -50 50;
3D.X -4 4;
3D.Y -4 4;
3D.Z -4 4;
3D.View 0.1929058849159247 0.9696131700517842 -0.1504580341283036 -0.9110554339894328 0.2339352455191676 0.3394882871455603 0.3643697514475641 0.07158632112930584 0.9284988329901723;
3D.Speed 0;
Text "Cylinder – sections with planes

A cylinder is the set of points a fixed distance from some axis.   Alternately, if X is a point on the surface of the cylinder relative to some fixed point on the axis, the projection of X onto a plane perpendicular to the axis has a fixed length.   We'll specify the axis of the cylinder with a unit vector A.   AA is the projection operator onto the direction of the axis; 1-AA is the projection onto a plane perpendicular to A.  The cylinder is then given by C^2= [(1-AA).X]^2=X^2-(A.X)^2";
Color 3;
MathPaneSlider 36;
Expr a=slider([0,1]);
Color 6;
MathPaneSlider 99;
Expr p=slider([0,1]);
Color 2;
MathPaneSlider 101;
Expr R=slider([0,0.05]);
Expr O=vector(0,0,0),I=vector(1,0,0),J=vector(0,1,0),K=vector(0,0,1);
Color 3;
Expr X=vector(x,y,z);
Text "Axis of cylinder";
Color 7;
Expr A=vector(sin(pi*a)*cos(2*pi*p),sin(pi*a)*sin(2*pi*p),cos(pi*a));
Color 2;
Expr O,A,'radius'=R;
Color 5;
Expr t*A;
Text "Cylinder of radius C (and length 2L, if we want ... but plots with somewhat jagged ends)";
Color 4;
MathPaneSlider 200;
Expr C=slider([0,1]);
Color 17;
Grain 1;
Expr C^2=dot(X,X)-[dot(A,X)]^2;
Color 8;
Expr L=2;
Color 3;
Grain 1;
Expr C^2=dot(X,X)-[dot(A,X)]^2,abs(dot(A,X))<L;
Text "Intersection of the cylinder with a plane is an ellipse.  (This is guaranteed because the intersection of the cylinder with e.g. z=0 is a quadratic: C^2=X^2-(A.X)^2 evaluated with z-0.)  ";
Color 5;
MathPaneSlider 30;
Expr d=slider([0,0.2]);
Grain 1;
Expr C^2=dot(X,X)-[dot(A,X)]^2,abs(z)<d;
Text "Math is right.   How to improve the appearance?";
Color 7;
Expr z=0;
Text "Ends";
Color 17;
Expr dot(A,X)=L,dot(X,X)<C^2+L^2;
Color 17;
Expr dot(A,X)=-L,dot(X,X)<C^2+L^2;