Subsections

• Introduction
• Background
• Exercises

Surfaces

Introduction

The purpose of this lab is to acquaint you with some common three-dimensional shapes.

Background

Three-dimensional curves can be entered as a function (or functions) of two variables or as an expression.

>with(plots):
>f:=(x,y)->25-x^2+1/300*y^4-y^2;
>blob:=z=x^2+3.7*y^2;
>plot3d(f(x,y),x=-10..10,y=-15..15,axes=boxed);
>implicitplot3d({f(x,y)=z,blob},x=-10..10,y=-15..15,z=-200..200,axes=boxed,
numpoints=2000,style=wireframe,color=aquamarine);

Remember the definition of a function when entering your shape. For example, the sphere can be entered as two functions or as one implicit expression.

>f:=(x,y)->sqrt(-x^2-y^2+1);g:=(x,y)->-sqrt(-x^2-y^2+1);
>plot3d({f(x,y),g(x,y)},x=-1..1,y=-1..1,numpoints=15000,scaling=constrained,
style=patchnogrid,axes=boxed);
>sphere:=x^2+y^2+z^2=1;
>implicitplot3d(sphere,x=-1..1,y=-1..1,z=-1..1,axes=boxed);

To look at the cross-section of the sphere you cut the sphere along a plane - i.e. you hold a variable constant. So the intersection of the sphere and the $z=\frac{1}{2}$ plane is:

>sph_at_half:=x^2+y^2+1/2^2=1;
>implicitplot(sph_at_half,x=-1..1,y=-1..1);

Notice that the plot is a two-dimensional circle. To intersect vertical planes hold the $x$ or $y$ constant. Notice this can be done using the function or expression.

>plot({f(1/3,y),g(1/3,y),f(-2/3,y),g(-2/3,y)},y=-1..1,labels=[y,z]);
>sph_y1:=x^2+0.6^2+z^2=1;sph_y2:=x^2+0.8^2+z^2=1;
>implicitplot({sph_y1,sph_y2},x=-1..1,z=-1..1);

Other three-dimensional shapes can be made from known conic sections. A few of these will be analyzed in the exercises.

Exercises

(Note: In all plots include the option scaling=constrained).

1. Given $z^2=70-3.25x^2-7.4y^2$

   A)

   Plot the three-dimensional shape over the intervals $-5 \leq x \leq 5$, $-4 \leq y \leq 4$, and $-10 \leq z \leq 10$.

   B)

   Is the given equation a function?

   C)

   Plot the intersections of this shape and two planes perpendicular to the z-axis. What two-dimensional shapes are graphed?

   D)

   Plot the intersections of this shape and two planes perpendicular to the y-axis. What two-dimensional shapes are graphed?

   E)

   Plot the intersections of this shape and two planes perpendicular to the x-axis. What two-dimensional shapes are graphed?

   F)

   What three-dimensional shape is the equation (a sphere, cylinder, cone, elliptic cone, paraboloid, elliptic paraboloid, ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, or hyperbolic paraboloid (saddle))?

2. Given $z=\frac{x^2}{4}-\frac{y^2}{20}$

   A)

   Plot the three-dimensional shape over the intervals $-10 \leq x \leq 10$, $-10 \leq y \leq 10$, and $-5 \leq z \leq 10$.

   B)

   Is the given equation a function?

   C)

   Plot the intersections of this shape and two planes perpendicular to the z-axis. What two-dimensional shapes are graphed?

   D)

   Plot the intersections of this shape and two planes perpendicular to the y-axis. What two-dimensional shapes are graphed?

   E)

   Plot the intersections of this shape and two planes perpendicular to the x-axis. What two-dimensional shapes are graphed?

   F)

   What three-dimensional shape is the equation (a sphere, cylinder, cone, elliptic cone, paraboloid, elliptic paraboloid, ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, or hyperbolic paraboloid (saddle))?