Space Curve Student Lab

Notes to Student:

Three-dimensional curves and surfaces are an integral (no pun intended) part of advanced calculus. Visualization of such objects can be tricky. You'll want to rerun the animations frequently and examine the outputs carefully in order to better understand them. The objects can be quite interesting so enjoy.

Introduction:

The use of vectors to model three dimensional curves (paths) is critical to the sciences. Whether we model the flight path of an object or the structure of deoxyribonucleic acid (DNA), vectors and vector-valued functions play a key role in the process. The ability to view how an object changes when we let a parameter vary is paramount in ascertaining accurate graphs and valuable information about the model in general.

Objectives Defined:

The purpose of this lab is to do the following things with regards to space curves and vector-valued functions.

1. View several examples of space curves over specified domains.

2. Determine the appropriate domains of several vector functions and use MAPLE to graph them.

3. Examine the intersection of two surfaces and determine the vector function which represents the curve of the intersection while commenting on any interesting domain observations.

4. Determine if the trajectories of two seperate particles will collide.

5. Provide additional practice pen-and-pencil problems.

Time allocation: 50 minutes.

Viewing Space Curves:(Obj.1) Animated

Below you will find three examples of animated space curves and their respective domains. Examine them carefully paying particular attention to their shapes. You run the animations by clicking on the picture and then using the controls located at the top. Place the cursor in the red " restart " word and press "Enter"to get started.

> restart:with(plots):with(plottools): *Place the cursor in the red restart and press Enter.
n:=50:
for i from 1 to n do a[i]:=spacecurve([cos(4*t),t,sin(4*t)],t=0..i*2*Pi/n,axes=boxed,thickness=4, title="The Domain is [0,2pi]",titlefont=[TIMES,BOLD,14],orientation=[20,60]):
b[i]:=spacecurve([t^3,ln(3-t),sqrt(t)],t=0..i*3/n,axes=boxed,thickness=4,title="The Domain is [0,3]",titlefont=[TIMES,BOLD,14],orientation=[20,60]):
c[i]:=spacecurve([(2+cos(1.5*t))*cos(t),(2+cos(1.5*t))*sin(t),sin(1.5*t)],t=0..i*4*Pi/n,axes=boxed,thickness=4,title="The Domain is [0,4pi]",titlefont=[TIMES,BOLD,14],orientation=[20,60]):
end do:
display(seq(a[i],i=1..n),insequence=true,labels=[x,y,z]);
display(seq(b[i],i=1..n),insequence=true,labels=[x,y,z]);
display(seq(c[i],i=1..n),insequence=true,labels=[x,y,z]);

Warning, the name changecoords has been redefined

You will notice that the last curve, the Trefoil Knot, is difficult to discern if it crosses over itself. MAPLE has a tubeplot option to help you see if the graph does, indeed, cross itself. Position the cursor in the red "restart" code line and press "Enter" to see the tubeplot. You should click on the image to animate and rotate it..

> restart:with(plots):with(plottools):
n:=50:
for i from 1 to n do
d[i]:=tubeplot([(2+cos(1.5*t))*cos(t),(2+cos(1.5*t))*sin(t),sin(1.5*t)],t=0..i*4*Pi/n,radius=0.2,axes=boxed,labels=[x,y,z],title="Tubeplot",titlefont=[TIMES,BOLD,14],orientation=[20,60]):
end do:
display(seq(d[i],i=1..n),insequence=true,labels=[x,y,z]);

Warning, the name changecoords has been redefined

Procede to the next section .

>

Determining Appropriate Domians:(Obj.2)

Space curves and vector-valued functions require their domains to be specified for the parametric/component functions. Find the appropriate domains for the two vector functions shown below and write your solutions in the space provided.

Problem 1 Problem 2

Solutions:

Using your answers found above, enter in your Problem 1 endpoints for "a" and "b" in the code lines.

> restart:with(plots):

> a:= 1: * Enter left-hand endpoint here.
b:= 5: * Enter right-hand endpoint here.

Warning, the name changecoords has been redefined

> spacecurve([t^2,sqrt(t-1),sqrt(5-t)],t=a..b,axes=boxed,thickness=4, title="Space-Curve Problem 1",titlefont=[TIMES,BOLD,14]);

Using your answers found above, enter in your Problem 2 endpoints for "a" and "b" in the code lines.

> a:= -3; * Enter left-hand endpoint here.
b:= 3; * Enter right-hand endpoint here.

> spacecurve([(t-2)/(t+2),sin(t),ln(9-t^2)],t=a..b,axes=boxed,thickness=4, title="Space-Curve Problem 2",titlefont=[TIMES,BOLD,14]);

Procede to the next section.

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Examining Surface Intersections:(Obj.3)

The intersection of two surfaces can be traced by a space curve. The following example below shows the intersection of a cylinder and a plane. The surfaces are defined as follows:

Cylinder: x^2+y^2=1 Plane: y+z=2

We can parameterize their intersection as: x(t) = cos(t), y(t) = sin(t), and z(t) = 2 - sin(t). Following this example you will be asked to determine the parameterization of the intersection of two surfaces and an appropriate domain.Once agian, place the cursor in the red "restart" word and press "Enter".

> restart:with(plots):with(plottools):
p1:=[implicitplot3d({x^2+y^2=1,y+z=2},x=-2..2,y=-2..2,z=-2..5,numpoints=1000,orientation=[30,80],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light3)]:
p2:=[spacecurve([cos(t),sin(t),2-sin(t)],t=0..2*Pi,color=black,thickness=4)]:
display3d(p1,p2,title="Intersection of Surfaces\nCylinder and Plane",titlefont=[TIMES,BOLD,14]);

Warning, the name changecoords has been redefined

Examine the figure by clicking it and rotating it.

>

Problem 3: The code below will produce the intersection of a cone and a plane. The surfaces are defined as follows:

Cone: sqrt(x^2+y^2)=z Plane: y+1=z

> restart:with(plots):with(plottools):
p1:=[implicitplot3d({sqrt(x^2+y^2)=z,y+1=z},x=-4..4,y=-4..4,z=-1..4,numpoints=2000,orientation=[27,82],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:
display3d(p1,title="Intersection of Cone and Plane",titlefont=[TIMES,BOLD,14]);

Warning, the name changecoords has been redefined

>

Determine algebraically the vector component functions for Problem 3's intersection and write them in the space below along with an appropriate domain. Then place your three functions and domain endpoints in the code as directed.

Solution to Problem 3:

> restart:with(plots):with(plottools):
xt:= t: * Enter your x(t) function here.
yt:=(t^2-1)/2: * Enter your y(t) function here.
zt:=(t^2-1)/2+1: * Enter your z(t) function here.
a:=-4 : *Enter your left-endpoint of the domain here.
b:=4 : *Enter your right-endpoint of the domain here. p1:=[implicitplot3d({sqrt(x^2+y^2)=z,y+1=z},x=-4..4,y=-4..4,z=-1..4,numpoints=2000,orientation=[27,82],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:
p2:=[spacecurve([xt,yt,zt],t=a..b,color=black,thickness=4)]:
display3d(p1,p2,title="Intersection of Cone and Plane\nWith Parametric Curve",titlefont=[TIMES,BOLD,14]);

Warning, the name changecoords has been redefined

Question: What do you observe about your domain?

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Problem 4: The code below will produce the intersection of an ellipsoid and a parabolic cylinder. The surfaces are defined as follows:

Ellipsoid: x^2+4*y^2+4*z^2=16 Parabolic Cylinder: y=x^2

> restart:with(plots):with(plottools):
p1:=[implicitplot3d({x^2+4*y^2+4*z^2=16,y=x^2},x=-5..5,y=-5..5,z=-5..5,numpoints=2000,orientation=[44,26],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:
display3d(p1,title="Intersection of Ellipsoid and Parabolic Cylinder",titlefont=[TIMES,BOLD,14]);

Warning, the name changecoords has been redefined

>

Determine the parametric curve for the top-half of the intersection. Write your equations for x(t), y(t), and z(t) along with an appropriate domain interval in the space below. Once done, substitute your answers in the red code lines as directed.

Solution to Problem 4:

> restart:with(plots):with(plottools):
xt:= t: *Enter your x(t) function here.
yt:=t^2: *Enter your y(t) function here.
zt:=sqrt((16-t^2-4*t^4)/4): *Enter your z(t) function here.
a:= -5: *Enter your left-endpoint of the domain here.
b:= 5: *Enter your right-endpoint of the domain here. p1:=[implicitplot3d({x^2+4*y^2+4*z^2=16,y=x^2},x=-5..5,y=-5..5,z=-5..5,numpoints=2000,orientation=[44,26],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:
p2:=[spacecurve([xt,yt,zt],t=a..b,color=black,thickness=5)]:
display3d(p1,p2,title="Intersection of Ellipsoid and Parabolic Cylinder",titlefont=[TIMES,BOLD,14]);

Warning, the name changecoords has been redefined

Question: What do you observe about your domain?

>

Problem 5: The code below will produce the intersection of a sphere and a cone. The surfaces are defined as follows:

Sphere: x^2+y^2+z^2=1 Cone: z^2=x^2+y^2

> restart:with(plots):with(plottools):
p1:=[implicitplot3d({x^2+y^2+z^2=1,z^2=x^2+y^2},x=-1..1,y=-1..1,z=-1..1,numpoints=2000,orientation=[44,26],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:
display3d(p1,title="Intersection of Sphere and Cone",titlefont=[TIMES,BOLD,14]);

Warning, the name changecoords has been redefined

>

Determine the parametric curve for the top-half of the intersection. Write your equations for x(t), y(t), and z(t) along with an appropriate domain interval in the space below. Once done, substitute your answers in the red code lines as directed.

Solution to Problem 5:

> restart:with(plots):with(plottools):
xt:= t: *Enter your x(t) function here.
yt:=sqrt(1/2-t^2): Enter your y(t) function here.
zt:=sqrt(1/2): *Enter your z(t) function here.
a:=-sqrt(0.5) : *Enter your left-endpoint of the domain here.
b:=sqrt(0.5) : *Enter your right-endpoint of the domain here. p1:=[implicitplot3d({x^2+y^2+z^2=1,z^2=x^2+y^2},x=-1..1,y=-1..1,z=-1..1,numpoints=2000,orientation=[44,26],axes=boxed,labels=[x,y,z],style=patchnogrid,lightmodel=light1)]:
p2:=[spacecurve([xt,yt,zt],t=a..b,color=black,thickness=3)]:
display3d(p1,p2,title="Intersection of Sphere and Cone",titlefont=[TIMES,BOLD,14]);

Warning, the name changecoords has been redefined

Question: What do you observe about your domain? Does your parameterization trace the entire intersection?

Proceed to the next section.

>

Determining If Two Particles will collide:(Obj.4) Animated

The following problem seeks to determine if two particles following different space curves will collide. The two paths are defined as follows:

Problem 6: Determine algebraically if the two paths cross for some time(s), t, in the solution space below;

Solution to Problem 6:

Examine the animation of the two space curves. Rotate the figure to see if it appears that they intersect.

> restart:with(plots):with(plottools):
n:=20:
for i from 1 to n do
p1[i]:=spacecurve([t^2,7*t-12,t^2],t=0..5*i/n,color=blue,thickness=3):
p2[i]:=spacecurve([4*t-3,t^2,5*t-6],t=0..5*i/n,color=red,thickness=3):
p3[i]:=display3d(([p1[i],p2[i]])):
end do:
display3d(seq(p3[i],i=1..n),insequence=true,labels=[x,y,z],title="Particle Intersection Problem",titlefont=[TIMES,BOLD,14],axes=framed,orientation=[25,66]);

Warning, the name changecoords has been redefined

Next examine the tubeplot of the same trajectories.

> restart:with(plots):with(plottools):
p1:=[tubeplot([t^2,7*t-12,t^2],t=0.5..5,radius=1.0,color=blue,thickness=2)]:
p2:=[tubeplot([4*t-3,t^2,5*t-6],t=0.5..5,radius =1.0,color=red,thickness=2)]:
display3d(p1,p2,title="Particle Intersection Problem",titlefont=[TIMES,BOLD,14],style=patchnogrid,orientation=[25,66],axes= framed,labels=[x,y,z]);

Warning, the name changecoords has been redefined

Procede to the next section.

>

Extra Practice Problems: (Obj.5)

Below you will find 5 extra problems which can be done with pencil and paper.

#1. Determine an appropriate domain for the given vector function.



#2. Sketch the given vector function.



#3. Find a parameterization for the intersection of the following two surfaces

.

#4. Determine if two particles will collide if their respective paths are given by the space curves

.

#5. Show that the curve with parametric equations:

x(t) = sin t
y(t) = cos t
z(t) = t

lies on the cone . Then sketch the graph.