Subsections

# 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 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 or 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
A)
Plot the three-dimensional shape over the intervals , , and .
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
A)
Plot the three-dimensional shape over the intervals , , and .
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))?