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 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 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 or 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
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 parallel to the z-axis. What two-dimensional shapes are graphed?
D)
Plot the intersections of this shape and two planes parallel to the y-axis. What two-dimensional shapes are graphed?
E)
Plot the intersections of this shape and two planes parallel 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, a parabolic saddle, or an hyperbolic 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 parallel to the z-axis. What two-dimensional shapes are graphed?
D)
Plot the intersections of this shape and two planes parallel to the y-axis. What two-dimensional shapes are graphed?
E)
Plot the intersections of this shape and two planes parallel 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, a parabolic saddle, or an hyperbolic saddle)?
3. Given
A)
Plot the three-dimensional shape over the interval for all three variables..
B)
Is the given equation a function?
C)
Plot the intersections of this shape and two planes parallel to the z-axis. What two-dimensional shapes are graphed?
D)
Plot the intersections of this shape and two planes parallel to the y-axis. What two-dimensional shapes are graphed?
E)
Plot the intersections of this shape and two planes parallel 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, a parabolic saddle, or an hyperbolic saddle)?