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 an expression.

>with(plots):
>blob:=z=x^2+3.7*y^2;
>implicitplot3d(blob,x=-15..15,y=-10..10,z=0..200,axes=boxed,
numpoints=5000);

The sphere can be entered as one implicit expression.

>g:=x^2+y^2+z^2=1;
>implicitplot3d(g,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:

>implicitplot(subs(z=0.5,g,x=-1..1,y=-1..1,labels=[x,y]);

Notice that the plot is a two-dimensional circle. To intersect vertical planes hold the $x$ or $y$ constant.

>implicitplot({subs(y=0.6,g),subs(y=-0.8,g)},x=-1..1,z=-1..1,labels=[x,z]);

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, a paraboloid, an ellipsoid, an hyperboloid, or an 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, a paraboloid, an ellipsoid, an hyperboloid, or an hyperbolic paraboloid (saddle))?