Multivariable Calculus
3D Printing Project #2
Maple Commands for Quadric Surfaces
K. Schreck 2016
Using implicitplot3d and plot3d Maple commands
To plot a quadric surface such as the hyperboloid of one sheet given by
2
2
x
y
2
z = 1 in Maple, which is an implicit function (i.e., not solved
4
4
for any variable), we can use the implicitplot3d command as follows.
Note that the equation for this surface is given in terms of x, y, and z.
Thus, this is the rectangular form of the surface.
> restart :
with plots :
with VectorCalculus :
x2
> implicitplot3d
4
y2
9
z2 = 1, x = 50 ..50, y = 50 ..50, z = 10 ..10, numpoints
= 100000, shading = zhue, style = wireframe, axes = normal, scaling = unconstrained,
axes = boxed
If we are given (can compute or look up) a parametrization of the this same
surface in terms of variables u and v such as:
f u, v = 2 cos u
3 v sin u , 2 sin u
3 v cos u , v
then we can use the plot3d command to generate a more realistic and
smoother rendering of this surface:
2 cos u
3 v sin u , 2 sin u
..2, scaling = unconstrained
3 v cos u , v , u = 0 ..2 Pi, v = 2
> plot3d
Finding Traces and Intercepts
Finding the traces and intercepts of a surface helps us to visualize the
surface.
Study the following example for the hyperboloid of one sheet given by
2
x
4
2
y
9
2
z =1.
To find the traces in the coordinate planes yz, xz, or xy, we set
x = 0, y = 0, and z = 0.
To find the traces in a plane parallel to one of the coordinate
planes, we set x = x , y = y , and z = z .
0
0
0
Use the appropriate commands to find the traces for each of the implicitly
defined surfaces to which you were assigned.
> restart :
h
x, y, z
x2
4
y2
9
h
z2
1
x, y, z
1 2
x
4
1 2
y
9
z2
1
(2.1)
> h 0, y, z = 0
When x = 0, the traces are hyperbolas in the yz plane.
1
1 2
y
9
z2 = 0
(2.2)
> h x, 0, z = 0
(2.3)
>
When y = 0, the traces are hypebolas in the xz
1 2
x
4
plane.
z2 = 0
1
(2.3)
> h x, y, 0 = 0
When z = 0, the traces are ellipses in the xy
1 2
x
4
1 2
y
9
plane.
1=0
(2.4)
To find the intercepts (i.e., the points where the surface intersects the
coordinate axes), set two of the coordinates of x, y, and z equal to zero.
> xintercept
h x, 0, 0 = 0;
solve xintercept, x ;
The x intercepts of the surface are the points 2, 0, 0 and
1 2
xintercept
x
4
2, 2
2, 0, 0
2, 0, 0
2, 0, 0 .
1=0
(2.5)
> yintercept
h 0, y, 0 = 0;
solve yintercept, y ;
The y
intercepts of the surface are the points 0, 3, 0 and 0, 3, 0 .
yintercept
1 2
y =0
9
1
3, 3
0, 3, 0
0, 3, 0
(2.6)
> zintercept
h 0, 0, z ;
solve zintercept, z ;
There are no z
intercepts. Why not?
z2
zintercept
1
I, I
Creating an STL file of a Maple plot
To 3D print a surface in Maple, we need to export the plot as an STL file.
(2.7)
Here is an example of the commands required to export your Maple plot as an STL file.
Cut and paste these commands into your Maple worksheet and modify the code as
required.
The number given below the surface is the byte count of the STL file.
> hyperboloid
plot3d 2 cos u
3 v sin u , 2 sin u
3 v cos u , v , u
= 0 ..2 Pi, v = 2 ..2,
style = surfacewireframe, lightmodel = light4, scaling = unconstrained,
axes = none ;
myfile
FileTools:-JoinPath currentdir , "myhyprev.stl" ;
plottools exportplot myfile, hyperboloid ;
myfile
>
"/Users/Kristen/Documents/3D Printing 2016/3D with Maple 2016/myhyprev.stl"
230484
(3.1)