Dr. Sarah's Maple File for Spherical Epitrochoid Maple is a computer algebra system, which was first developed in 1980 at the University of Waterloo in Canada.Maple can handle numerical, symbolic, and graphical representations. Exposure to this type of technology is a course goals of Linear Algebra, Calculus II with Analytic Geometry, and this course. Each time you open this document, hit return anywhere in the Maple code of each line, including the following line with the packages: with(Student[VectorCalculus]): with(plots): x:=(1+1/5*cos(2*Pi/3))*cos(t)-1/5*(cos(2*Pi/3)*cos(5*t)*cos(t)-sin(5*t)*sin(t)); y:=(1+1/5*cos(2*Pi/3))*sin(t)-1/5*(cos(2*Pi/3)*cos(5*t)*sin(t)+sin(5*t)*cos(t)); z:=1/5*sin(2*Pi/3)*(1-cos(5*t)); Domain values: I have set this from 0 to 2Pi. t_1:=0; t_2:=2*Pi; Velocity, Acceleration, Jerk and Speed Hit return for this without modification. velocity:= diff([x,y,z],t); acceleration:= diff(diff([x,y,z],t),t); jerk:= diff(diff(diff([x,y,z],t),t),t); speed:= simplify(sqrt(velocity^2+velocity^2+velocity^2)); Arc Length Depending on the parameterization, arc length may take a while. These worked for me for your curve with the parameterizations and domain values. If need be, you can stop it from continuing via the symbol that is an octagon with an exclamation point inside of it and check that you have the correct domain and coordinate functions. arclength:= ArcLength(<x,y,z>,t=t_1..t_2); TNB, Plot of Curve and TNB Animation and Static Plot TNBFrame(<x,y,z>,t); curveplot:=spacecurve([x,y,z],t=t_1..t_2,axes=frame); Sometimes one of these will produce a graph and the other will produce no graph, so first, so first, try them both. If one or both of these give errors and no graph, try changing any command with errors via the frames= to different numbers, like from odd to even or even to odd! Or, for error messages, try modifying the range. You can use specific values like range=.1..2 TNBFrame(<x, y,z>,range=t_1..t_2,output=plot,scaling=constrained, axes=frame,frames=3,curveoptions=[color="Blue", linestyle=dot],tangentoptions=[color=black, width=.05], normaloptions=[width=.05], binormaloptions=[width=.05]); If you can't see the T, N, B vectors well, above or below, change the "width=.05" command to other values, like .1 or similar. Conversly, if the vectors are too wide, make them smaller. TNBFrame(<x, y,z>,range=t_1..t_2,output=animation,scaling=constrained, axes=frame,frames=50,curveoptions=[color="Blue", linestyle=dot],tangentoptions=[color=black, width=.05], normaloptions=[width=.05], binormaloptions=[width=.05]); Curvature, Torsion and Osculating Circle Animation and Static Plot curvature:=simplify(Curvature(<x,y,z>,t),trig); torsion:=simplify(Torsion(<x,y,z>,t),trig); Here are some numerical approximations for curvature and then torsion. You can change the t value to experiment and insert extra execution groups to list interesting numerical values for curvature and torsion. evalf(subs(t=1,curvature)); evalf(subs(t=1,torsion)); Sometimes one of these will produce a graph and the other will produce no graph, so first, try them both. If one or both of these give errors and no graph, try changing any command with errors via the circles= to different numbers, like from odd to even or even to odd! Or, for error messages, try modifying the range. You can use specific values like range=.1..2 RadiusOfCurvature(<x,y,z>,t,range=t_1..t_2, output=plot,scaling=constrained, circles=5); RadiusOfCurvature(<x,y,z>,t,range=t_1..t_2, output=animation,scaling=constrained, circles=50); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv