Additional 3D constructions

Comprehension Questions

Math 250 - Mathematical Computing
Christopher Hanusa

11-1. A helix is a three-dimensional spiral shape that has parametric equations
x(t)=sin(t), y(t)=cos(t), and z(t)=t.  
Create a helix that spirals around multiple times and has some thickness, that looks something like this:

11-2. Create a torus using ParametricPlot3D.  Determine how to increase the width of the donut as well as its thickness.

11-3. Add some spheres that intersect this torus.

Important reminder: When you need to combine multiple Graphics objects, use the Show command.

11-4. Create a bowl using the equation of a paraboloid.

Hint for people who never took Multivariable Calculus:

A paraboloid is a function of the form z=x^2+y^2.

11-5. Create a rectangular prism by using RegionPlot3D.

Hint 1:

Determine the equations of the faces of the solid.

Hint 2:

Equations of faces look like x=a or y=b or z=c.

11-6. What does the 3D region look like that is the intersection of two perpendicular cylinders?  Three perpendicular cylinders?

Hint for those who did not take Multivariable Calculus:

The equations of some solid cylinders are x^2+y^2<=1, x^2+z^2<=1, y^2+z^2<=1.

11-7. Create a block with upper surface defined by f(x,y)=sin(x+y^2)

11-8. Intersect some other interesting 3D objects in different ways.  What is the most interesting object you can create in this way?

Solutions.  Don’t Peek until you’ve tried them!

Question 11-1.

ParametricPlot3D[{Sin[t],Cos[t],t},{t,0,5Pi},PlotStyle->{Blue,Tube[.25]},PlotRange->All]

Question 11-2.

torus=ParametricPlot3D[{Sin[t],Cos[t],0},{t,0,2Pi},PlotStyle->{Yellow,Tube[.25]},PlotRange->All]

Question 11-3.

spherecenters=Table[{Sin[t],Cos[t],0},{t,0,2Pi,2Pi/5}];
spheres=Graphics3D[{Darker[Green],Map[Sphere[#,.4]&,spherecenters]}];

Show[torus,spheres]

Question 11-4.

Did you try it first?  If not, go try it!

Plot3D[x^2+y^2,{x,-3,3},{y,-2,2},Mesh->None,PlotPoints->50,Extrusion->.2]

Question 11-5

RegionPlot3D[1≤x≤4&&1≤y≤3&&2≤z≤4,{x,0,5},{y,0,5},{z,0,5},PlotPoints->30]

Question 11-6

RegionPlot3D[x^2+y^2≤1&&x^2+z^2≤1,{x,-1.5,1.5},{y,-1.5,1.5},{z,-1.5,1.5},PlotPoints->80,Mesh->None,Boxed->False,Axes->False]

RegionPlot3D[x^2+y^2≤1&&x^2+z^2≤1&&y^2+z^2≤1,{x,-1.5,1.5},{y,-1.5,1.5},{z,-1.5,1.5},PlotPoints->80,Mesh->None,Boxed->False,Axes->False]

Question 11-7

RegionPlot3D[Sin[x+y^2]>z,{x,0,3Pi},{y,0,3Pi},{z,-3,3},PlotPoints->30]