McCE Math Paper

7/30/2019 McCE Math Paper

http://slidepdf.com/reader/full/mcce-math-paper 1/19

Egan M,

The Creation of Spheres

Introduction

The purpose of this paper is for people to understand the creation of the three dimensionalfigures; specifically this paper will deal with the topic of spheres. We will first try to understand the equation of a circle because circles and spheres are so closely related. After understandingcircles we can compare that to the equation of a sphere.

How to Make a Circle

The simplest way to understand how a sphere is formed is to first look at the equation of a circle.A circle is like a sphere but n two dimensions. A circle’s equation contains 3 parts: x

2, y

2, and the

radius2. We can use 3 as an example for the radius, so the equation would look like x2 + y2 = 32.If we were to draw this circle, it would look something like this: `



Now we can see how the x-variable and the y-variable affect the size of the circle. If we look along the x-axis we see the circle intersects the axis at 3 and -3. Those coordinates are (-3,0), and (3,0). When we solve the equation with that point we get (-3)

2+ 0

2= 3

2or 3

2+ 0

3= 3

2. The same

works if we look at the intersection of the circle along the y-axis: (-3,0), and (3,0).

Here is a point, (√ 5,√ 4) or (2.236,2). When it is put in the equation for the circle, (√ 5)2 +(√ 4)

2= 3

2, the simplified equation is 9 = 9, therefore we know that this point exists on the circle.

This is intimately related to the Pythagorean Theorem; this will come in handy later whenunderstanding a sphere. Hopefully now we understand that this circle is a representation of allthe points a distance of 3 away from the origin.



How to Make a Sphere

The way a sphere is works in a similar manner to how a circle is written. One of the definitionsof a sphere (according to Merriam Webster) is: “a solid that is bounded by a surface consisting of all points at a given distance from a point constituting its center.” Spheres are circles in a third dimension so we can just add another variable to make the equation of a circle the equation of asphere: x2+y2+z2=r 2. The variable z denotes the depth of a point from the xy plane. If we were tothink of a table’s surface as the xy plane the z plane would tell us the distance (up and down)away from the table.



Now we are going to use the Pythagorean Theorem in 3D space just like we did in the lastsection in 2D space. The point of this will be to show how we can confirm points on the graph of a sphere. We will start with point (1,2,2) lying on a sphere of radius 3.

The two red legs (1 and 2) make the blue hypotenuse (√ 5), which also happens to be one of the

blue triangles legs. Those two legs (with √ 5

and 2) end up making the green hypotenuse (with√ 9 or 3).



Transforming a Circle

Next there are a total of six ways of transforming the graph to another image. We can translateany image: up/down, left/right, and stretch/shrink. (x-3)2 + (y-3)2 = 32 will give us a circle that isshifted to the right three and up three. However for our purposes we want to see how the generalshape of things change and moving up/down and left/right doesn’t change the overall structure just where the object is. So we will focus on stretching and shrinking.

Now to understand the shrinking and stretching we are going to compare our original x2+y2=32 and x

2+(3y)

2=3

2.

The graph represented here is the latter equation and these are now called ellipses; in here theintersections of the x-axis are the same as our normal circle, but the intersections of the y-axisare instead at points (0,1) and (0,-1).



In this image we have both images overlapped with one another. We can see that y-axisintersections of the latter circle is one-third of the original circle.



Just to illustrate how changes in the equation also show changes in the image we will graph theequation (3x)

2+y

2=3

2. We can expect the graph to look like x

2+(3y)

2=3

2, but the shrink will be on

the x-axis.



Transforming a Sphere

We can now apply the same transformations to the equation of a sphere. We can take theequation of the sphere, x2+y2+z2=32, and transform it in various ways.



We can apply the changes now in three directions and it will affect the x-variable, y-variable, and the z-variable. Our first change will be x-variable; we will make the new equation:(3x)

2+y

2+z

2=3

2.

With this change in the equation we can see that the x-axis intersections have changed from(3,0,0) and (-3,0,0) to (1,0,0) to (-1,0,0). This is a similar idea to when we shrunk the graph of acircle. Just as a note, objects that are no longer called spheres are called ellipsiods and that iswhat we will be changing now. We can now make a generalization saying that the 3 in theequation graph is responsible for shrinking the x-coordinates by a factor of 3 because that is whatit looks like on that graph.



We can now try out another equation to see how that equation affects the graph; we will be

graphing the equation: (

x)2+y2+ z2=32. We can make a guess here that this will stretch the image

by a factor of 3 for the x-coordinates.

We know this works because if you take certain points on the graph and you plug them into theequation they should equal 32.

The same thing works if we were to edit y and z variables in the equations. For example if we

were to graph this: (

x)

2+(

y)

2+ (5z)

2=3

2, we can make a guess about what we think the ellipse

will look like. The x-coordinates will be stretched by a factor of three, the y-coordinates will bestretched by a factor of two and the z-coordinates will be shrunk by a factor of five.



This is not the best image to be look at but it does convey the guesstimate that he made.

Now we can make a full generalization about spheres and its equation when regardingtransformations. In the equation:

(ax)2+(by)

2+ (cz)

2=1

2

The variable “a” is currently shrinking all the x-coordinates by a factor of “a”, the variable “b” iscurrently shrinking all the y-coordinates by a factor of “b”, and the variable “c” is currentlyshrinking all the z-coordinates by a factor of “z”.



Another interesting part of the equation of a sphere is when we look across a multitude of equations and their graphs, specifically if we edit the “a” variable making it closer to zero.

First let us look at the “a” equaling

, so (

x)

2+y

2+z

2=1

2.

This gives us an ellipsoid that is slightly stretched in the x-direction. Next look at the graph

where “a” equals

: (

x)2+y2+z2=12. We already have this graphed on a previous page, but here it

is again so it is easier to compare it.

Now the ellipsoid is even more skewed in the x-direction. Now when the graph “a” equaling.0001 we can guess that the ellipsoid will be really stretched out.



(.0001x)2+y2+z2=12

We might be wondering why this image was graphed when we expected a stretched ellipsoid.We can understand why it looks like this if we look at “a” in the context of the whole equation.(0.0001x)2+y2+z2=12, in here “a” is practically zero so the new equation would look y2+z2=12. If we were to look at the graph directly face on the y-axis and the z-axis it would be a circle of

radius 1.



Post-Script

As I was exploring the mathematical ideas presented with this paper I noticed many things and not all of them I would be able to share in the context of this paper this is why I shall just briefly provide some of the images reflecting what I noticed and hopefully further research can be doneabout this topic.

2D:

x2-y2=12

-x2+y

2=1

2



3D:

-x2+y

2+z

2=1

2

x2-y2+z2=12



x2+y2-z2=12

-x2-y

2+z

2=1

2



-x2+y2-z2=32

x2-y

2-z

2=3

2



(0.0001)x2-y2+z2=12

(0.0001)x2-y

2-z

2=1

2

The first six different graphs are the 3 dimensional versions of the graphs that I mentioned in the2D section.



References and Resources

http://www.merriam-webster.com/dictionary/sphere

http://wolframalpha.com/

Documents

McCE Math Paper