3/2/2012
Ellipses with the Human Eye
By:
3/13/11
I mean they look soo tight!!!
We should learn about how
they work ☺
Why the Human Eye?
• The complexity of the Human eye goes way
beyond Biology… The math that is involved is
High Level Calculus, and Physics… I will explain
why later ☺
• Before my research I knew about the Concave
and Convex shapes of lens in our eyes and I
thought may be it had to do with Ellipses.
• The Human eye is incredibly important to our
everyday life. Learning about its characteristics
can make us further think about the complexity
of the Human body.
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3/2/2012
Why the Human Eye Cont…
• Learning about the Human eye can show us
how important Math is to our everyday bodily
functions.
• The Human eye has a Small Lens that is inside
of it that directs every spec of light that travels
through our eyes to our retina
• This lens is called the Crystalline Lens
The Crystalline Lens
• When light enters our eyes it goes through a
very sensitive and special lens.
• The Lens is called the Crystalline Lens
• The Crystalline lens is a 3 dimension (3-D),
Ellipse shaped lens that allows light to pass
through it.
• This lens shape can be controlled… Next slide
☺
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This is where it gets crazy…
• The Crystalline Lens is extremely flexible it can
take various different shapes.
• There is a small muscle called the Ciliary
Muscle on the Crystalline Lens that expands
and contracts making the Ellipse of the
Crystalline lens change its definition of the
Ellipse.
This is where it gets crazy cont…
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When light comes in…
When light and or an image comes in the from the source the Ciliary Muscle determines
How much to move the Crystalline Lens.
Ciliary Muscle
If you notice carefully on the diagram that when the Crystalline Lens changes
shape the path of light that is going in changes a little and this because the
Focus points are changing. This is VERY IMPORTANT!
The Shapes that the Lens makes…
• The Ellipse of the Lens can take nearly any type of ellipse.
• At this very second your eyes constistanly adjusting their lenses to
Focus on this Powerpoint!!
The flattened Ellipse allows the Eye to see further, while the taller one lets you see
things up close.
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The Math behind this…
Center (h, k)
Y
Minor Axis
“b” value
Important!!
X
Focus Point
“c” value
When the Focus points are
Spread out more away from the
Center the Ellipse becomes longer
Like soo…
Focus Point
“c” value
Major Axis
“a” value
The Math behind continue…
If the Ellipse is Horizontal the Equation is…
(x-h)^2 + (y-k)^2 = 1
--------- ---------a^2
b^2
This is where the a^2 is the Major Axis on the X-Axis
If the Ellipse is Vertical the Equation is…
To find the Foci you use…
c^2 = a^2 – b^2
(x-h)^2 + (y-k)^2 = 1
-------- ----------b^2
a^2
This is where the
a^2 is the Major
Axis on the Y-Axis
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3/2/2012
The Scenario
We will be using the Horizontal Equation because the Crystalline lens looks like this.
The Major Axis is the longest one so we use the
Horizontal one because the X-Axis is the largest.
Ok ☺, lets say you have two Equations…
x^2 + y^2 = 1
----- -----36
9
x^2 + y^2 = 1
----- -----9
4
Lets figure out what these look like!!…
Next Slide…….
x^2 + y^2 = 1
1. ----- -----a^2 b^2
First Equation Solving…
8.
x^2 + y^2 = 1
2. ----- -----36
9
3. c^2 = ?
a^2 = 36
b^2 = 9
4.
a^2 = 36
5. b^2 = 9
c^2 = 27
6.
a^2 = 36
a^2*sqrt = sqrt(36)
a=6
7.
b^2 = 9
b^2*sqrt = sqrt(9)
b=3
c^2 = 27
c^2*sqrt = sqrt(27)
c = 5.2
The ^2*sqrt cancel
in each of steps 6,7,8
c^2 = a^2 – b^2
c^2 = 36 – 9
c^2 = 27
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Second Equation Solving…
1.
2.
x^2 + y^2 = 1
----- ----a^2 b^2
5.
a^2 = 9
b^2 = 4
c^2 = 5
x^2 + y^2 = 1
------ -----9
4
6.
a^2 = 9
a^2*sqrt = sqrt(9)
a=3
3.
c^2 = ?
a^2 = 9
b^2 = 4
4.
c^2 = a^2 – b^2
7.
8.
c^2 = 5
c^2*sqrt = sqrt(5)
c = 2.2
The ^2*sqrt cancel
in each of steps 6,7,8
b^2 = 4
b^2*sqrt = sqrt(4)
b=2
c^2 = 9 – 4
c^2 = 5
Ahh ha Moment
• In the first equation the “c” value was 5.2
• In the second equation the “c” value was 2.2
• What this means is that the First equation
Focus Points are spread out more than the
Focus point of the Second Equation.
• Lets take a closer look… ☺
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c = 2.2
c = 5.2
c = 2.2
c = 5.2
What have we learned?...
• We have learned that when the Focus Points
spread further out the further away we can see
objects clearly.
• Which means that when the Focus Points are
closer together that means that we can see
objects more clearly close to us.
• Soo yeah, just think that every time you are
focusing on the board or a Powerpoint your “c”
value is increasing and decreasing all of the time!
• Which means your Focus points in your Lens are
consistently changing.
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Why is this important?...
• This is important to understand that even in this
Pre-Calculus class learning about Conics is only
fractions of what Conics really do for our
everyday lives.
• Without the properties of Ellipses it would be
impossible to focus correctly on objects and see
clearly.
• This proves that Conics play a vital role in our
ability to see with our eyes. And it proves that
Math is everywhere!
Thanks for Listening to my
Powerpoint!
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