Maple Lab 1 (Spring 2007 Math 251 Sections 05-07) 

Exercises 

 

Feel free to use Maple's built in functions for curvature, torsion, etc. 

 

Note: Use appropriate colors and plot options to make your plots look nice. The appearance of 

          you lab will be factored into your grade. You may add execution groups if you need more. 

          Please delete unused execution groups. 

> restart; 1; with(VectorCalculus); -1; with(plots); -1
 

1. We can parametrize the ellipse VectorCalculus:-`+`(VectorCalculus:-`*`(x^2, 1/`^`(3, 2)), VectorCalculus:-`*`(y^2, 1/`^`(1, 2))) = 1 using sines and cosines as follows: 

    x(t) = VectorCalculus:-`*`(3, cos(theta)), y(t) = sin(theta), and z(t) = 0 where 0 <= theta and theta <= VectorCalculus:-`*`(2, Pi). 

 

   Define a vector valued function called "r1" so that the graph of r1(t) is this ellipse. 

> r1 := proc (t) options operator, arrow; `<,>`(3*cos(t), sin(t), 0) end proc; -1
 

> r1(t); 1
 

Vector[column](%id = 159839536) 

1(a). Define the arc length function sArc(t) for this parametrization and then use it to 

       compute the arc length of the ellipse. 

> sArc := proc (t) options operator, arrow; int(Norm(diff(r1(u), u)), u = 0 .. t) end proc; -1
 

> simplify(sArc(VectorCalculus:-`*`(2, Pi))); 1
 

12*EllipticE(2/3*2^(1/2)) 

1(b). Compute expressions for T1(t) (the unit tangent), N1(t) (the normal), and B1(t)(the binormal). 

> ellipseFrame := TNBFrame(r1(t), t); -1
 

> T1 := simplify(ellipseFrame[1]); 1
 

Vector[column](%id = 159177352) 

> N1 := simplify(ellipseFrame[2]); 1
 

Vector[column](%id = 160252172) 

> B1 := simplify(ellipseFrame[3]); 1
 

Vector[column](%id = 162331668) 

1(c). Plot the ellipse together with its TNB-frame at the point t = VectorCalculus:-`*`(Pi, 1/2). 

> ellipsePlot := spacecurve(r1(t), t = 0 .. VectorCalculus:-`*`(2, Pi), color = black); -1
 

> TPlot := arrow(r1(VectorCalculus:-`*`(Pi, 1/2)), subs(t = VectorCalculus:-`*`(Pi, 1/2), T1), shape = arrow, color = blue); -1
 

> NPlot := arrow(r1(VectorCalculus:-`*`(Pi, 1/2)), subs(t = VectorCalculus:-`*`(Pi, 1/2), N1), shape = arrow, color = green); -1
 

> BPlot := arrow(r1(VectorCalculus:-`*`(Pi, 1/2)), subs(t = VectorCalculus:-`*`(Pi, 1/2), B1), shape = arrow, color = red); -1
 

> display({BPlot, ellipsePlot, NPlot, TPlot}, scaling = constrained, axes = boxed); 1
 

Plot 

1(d). Graph the curvature function kappa(t) for 0 <= t and t <= VectorCalculus:-`*`(2, Pi). 

> plot(Curvature(r1(t), t), t = 0 .. VectorCalculus:-`*`(2, Pi)); 1
 

Plot 

1(e). Compute the torsion function for the ellipse. Notice that is it zero. So the derivative of  

       the binormal should be the zero vector. Compute this derivative...depending on how  

       you had Maple compute the binormal...Maple may be wrong!!! 

> Torsion(B1, t); 1
 

0 

> diff(B1, t); 1
 

Vector[column](%id = 161440216) 

1(f). Explain why the torsion is zero and then make a conjecture about curves with zero torsion. 

 

The torsion is zero since the ellipse's osculating plane is the same for every value of t -- this means our curve is 

not "twisting" out of the osculating plane. In other words, the torsion is zero because our curve lies in the xy-plane. 

 

We conjecture that any curve lying in a plane will have zero torsion (this is actually a fairly easy theorem to prove). 

 

2. Consider the twisted cubic given by the following parametric equations: 

   x(t) = t, y(t) = t^2, and z(t) = t^3 where 0 <= t and t <= 2. 

 

   Define a vector valued function called "r2" so that the graph of r2(t) is this twisted cubic. 

> r2 := proc (t) options operator, arrow; `<,>`(t, t^2, t^3) end proc; -1
 

> r2(t); 1
 

Vector[column](%id = 162509520) 

2(a). Approximate (using "evalf") the arc length of the twisted cubic. 

> sArc2 := proc (t) options operator, arrow; int(Norm(diff(r2(u), u)), u = 0 .. t) end proc; -1
 

> sArc2(t); 1
 

1/3*(2*t*(-2+I*5^(1/2))^(1/2)+8*t^3*(-2+I*5^(1/2))^(1/2)+18*t^5*(-2+I*5^(1/2))^(1/2)+I*t*(-2+I*5^(1/2))^(1/2)*5^(1/2)+4*I*t^3*(-2+I*5^(1/2))^(1/2)*5^(1/2)+9*I*t^5*(-2+I*5^(1/2))^(1/2)*5^(1/2)+2*I*(1+2...
1/3*(2*t*(-2+I*5^(1/2))^(1/2)+8*t^3*(-2+I*5^(1/2))^(1/2)+18*t^5*(-2+I*5^(1/2))^(1/2)+I*t*(-2+I*5^(1/2))^(1/2)*5^(1/2)+4*I*t^3*(-2+I*5^(1/2))^(1/2)*5^(1/2)+9*I*t^5*(-2+I*5^(1/2))^(1/2)*5^(1/2)+2*I*(1+2...
1/3*(2*t*(-2+I*5^(1/2))^(1/2)+8*t^3*(-2+I*5^(1/2))^(1/2)+18*t^5*(-2+I*5^(1/2))^(1/2)+I*t*(-2+I*5^(1/2))^(1/2)*5^(1/2)+4*I*t^3*(-2+I*5^(1/2))^(1/2)*5^(1/2)+9*I*t^5*(-2+I*5^(1/2))^(1/2)*5^(1/2)+2*I*(1+2...
1/3*(2*t*(-2+I*5^(1/2))^(1/2)+8*t^3*(-2+I*5^(1/2))^(1/2)+18*t^5*(-2+I*5^(1/2))^(1/2)+I*t*(-2+I*5^(1/2))^(1/2)*5^(1/2)+4*I*t^3*(-2+I*5^(1/2))^(1/2)*5^(1/2)+9*I*t^5*(-2+I*5^(1/2))^(1/2)*5^(1/2)+2*I*(1+2...
1/3*(2*t*(-2+I*5^(1/2))^(1/2)+8*t^3*(-2+I*5^(1/2))^(1/2)+18*t^5*(-2+I*5^(1/2))^(1/2)+I*t*(-2+I*5^(1/2))^(1/2)*5^(1/2)+4*I*t^3*(-2+I*5^(1/2))^(1/2)*5^(1/2)+9*I*t^5*(-2+I*5^(1/2))^(1/2)*5^(1/2)+2*I*(1+2...
1/3*(2*t*(-2+I*5^(1/2))^(1/2)+8*t^3*(-2+I*5^(1/2))^(1/2)+18*t^5*(-2+I*5^(1/2))^(1/2)+I*t*(-2+I*5^(1/2))^(1/2)*5^(1/2)+4*I*t^3*(-2+I*5^(1/2))^(1/2)*5^(1/2)+9*I*t^5*(-2+I*5^(1/2))^(1/2)*5^(1/2)+2*I*(1+2...
1/3*(2*t*(-2+I*5^(1/2))^(1/2)+8*t^3*(-2+I*5^(1/2))^(1/2)+18*t^5*(-2+I*5^(1/2))^(1/2)+I*t*(-2+I*5^(1/2))^(1/2)*5^(1/2)+4*I*t^3*(-2+I*5^(1/2))^(1/2)*5^(1/2)+9*I*t^5*(-2+I*5^(1/2))^(1/2)*5^(1/2)+2*I*(1+2... 

> Re(evalf(sArc2(2))); 1
 

9.570572082 

2(b). Graph the curvature and the torsion in the same plot window. 

> plot({Curvature(r2(t), t), Torsion(r2(t), t)}, t = 0 .. 2); 1
 

Plot 

2(c). Graph the twisted cubic along with its osculating circle and TNB-frame at t = 1. 

> cubicFrame := TNBFrame(r2(t), t); -1
 

> T2 := subs(t = 1, cubicFrame[1]); 1
 

Vector[column](%id = 161971012) 

> N2 := subs(t = 1, cubicFrame[2]); 1
 

Vector[column](%id = 161967156)
Vector[column](%id = 161967156) 

> B2 := subs(t = 1, cubicFrame[3]); 1
 

Vector[column](%id = 162567648) 

> kappa2 := subs(t = 1, Curvature(r2(t), t)); 1
 

1/1372*76^(1/2)*49^(1/2)*14^(1/2) 

> OscCircle := VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`+`(r2(1), VectorCalculus:-`*`(1/kappa2, N2)), VectorCalculus:-`*`(VectorCalculus:-`*`(1/kappa2, cos(theta)), N2)), VectorCalculus:...
OscCircle := VectorCalculus:-`+`(VectorCalculus:-`+`(VectorCalculus:-`+`(r2(1), VectorCalculus:-`*`(1/kappa2, N2)), VectorCalculus:-`*`(VectorCalculus:-`*`(1/kappa2, cos(theta)), N2)), VectorCalculus:...
 

Vector[column](%id = 166776704)
Vector[column](%id = 166776704)
Vector[column](%id = 166776704)
Vector[column](%id = 166776704)
Vector[column](%id = 166776704) 

> cubicPlot := spacecurve(r2(t), t = 0 .. 2, color = black); -1
 

> T2Plot := arrow(r2(1), T2, shape = arrow, color = blue, thickness = 2); -1
 

> N2Plot := arrow(r2(1), N2, shape = arrow, color = green, thickness = 2); -1
 

> B2Plot := arrow(r2(1), B2, shape = arrow, color = red, thickness = 2); -1
 

> oscPlot := spacecurve(OscCircle, theta = 0 .. VectorCalculus:-`*`(2, Pi), color = grey); -1
 

> display({B2Plot, oscPlot, N2Plot, cubicPlot, T2Plot}, scaling = constrained, axes = boxed, orientation = ([VectorCalculus:-`-`(6), 62])); 1
display({B2Plot, oscPlot, N2Plot, cubicPlot, T2Plot}, scaling = constrained, axes = boxed, orientation = ([VectorCalculus:-`-`(6), 62])); 1
 

Plot 

2(d). Graph the twisted cubic along with its osculating plane at t = 1. 

> oscPlane := Typesetting:-delayDotProduct(B2, VectorCalculus:-`+`(`<,>`(x, y, z), VectorCalculus:-`-`(r2(1)))) = 0; 1
 

3/266*19^(1/2)*196^(1/2)*(x-1)-3/266*19^(1/2)*196^(1/2)*(y-1)+1/266*19^(1/2)*196^(1/2)*(z-1) = 0
3/266*19^(1/2)*196^(1/2)*(x-1)-3/266*19^(1/2)*196^(1/2)*(y-1)+1/266*19^(1/2)*196^(1/2)*(z-1) = 0 

> oscPlot := plot3d(solve(oscPlane, z), x = 0 .. 2, y = 0 .. 4); -1
 

> cubicPlot := spacecurve(r2(t), t = 0 .. 2, color = black, thickness = 3); -1
 

> display({oscPlot, cubicPlot}, axes = boxed, orientation = ([VectorCalculus:-`-`(150), 70])); 1
 

Plot