March 15, 2018 Thursday
Bedtime Story
Beauty of Conic Sections
I was trying to explain you the beauty of
conic sections and how easy it is to visually grasp its three main constituents:
hyperbola, parabola and ellipse.
Imagine that you placed an order for a cake
that has a shape of cone, just like the cone of an ice-cream, the only difference
being that your cake cone will have to be inverted with its tip being on top
and its circular base at the bottom.
Also imagine further that the cake of your
specifications has been delivered at your door step and you have placed it on
your dining table.
Having placed this cake on your dining
table, you need to have a sharp knife in hand that than slice through the cake
with utmost ease without slipping or getting engaged in the substance of it.
Now if you were to take your sharp knife
and go through the cake parallel to the table, the shape you would be left on
top of the remaining mass would be a circle.
That does not take a lot of imagination I
think and any average ape should be easily able to visualize it mentally.
Now if you were to incline your knife and
angle it slightly down and cut through the cake, the shape of the top that you
will end up with would be an ellipse.
Mathematically, a circle is considered a
special case of ellipse and some even consider it as the fourth type of conic
section.
You will get either a circle or an ellipse while
cutting this cake such that your knife does not hit the base and you get a 360
degrees closed curve.
Now when you take your knife and cut it
steeply such that the plane of your cut is parallel to exactly one generating
line of the cone, which essentially means that the slope of your knife is
parallel to the slope of the cone opposite to your cut, the shape your cake end
up is parabola.
Any other cut that of the cake that does
not obey the above rule and slices the base of the cake would give you a hyperbola.
I will see if I can fetch a sketch for everything
that I have written above for after all, as the great English-language idiom
goes – “a picture is worth a thousand words”.
Now that you have a more intuitive idea
about a hyperbola, we can proceed further and see what it was that
Saint-Vincent was studying.
Saint-Vincent in all likely hood studied
many things in mathematics, but what he is most known for is studying a hyperbola
drawn on a graph over x y axis and trying to find the area under a specific
part of the curve.
Such an area below a hyperbolic curve is
known as quadrature of the hyperbola.
Such a curve is given by the equation xy =
k
The area of such a quadrature starting from
the point 1 on the x axis to any other point a on x axis to the right is simply
given as ln(a) or natural logarithm of a.
This may not strike to most apes as
anything of significance but mathematicians find very fascinating when two very
unrelated entities such as hyperbola and logarithm end up in some way connected.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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