the lemniscate of bernoulli
DESCRIPTION
The Lemniscate of Bernoulli. The cool way to write infinity By Jonathan O’Connell. Who was Bernoulli?. Bernoulli was born in The Netherlands in 1700. Part of a predominate math family. Worked in areas of statistics, physics, and mathmatics. Developed fluid dynamics equations. How it works. - PowerPoint PPT PresentationTRANSCRIPT
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The cool way to write infinityBy Jonathan O’Connell
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Bernoulli was born in The Netherlands in 1700.
Part of a predominate math family.
Worked in areas of statistics, physics, and mathmatics.
Developed fluid dynamics equations.
![Page 3: The Lemniscate of Bernoulli](https://reader035.vdocument.in/reader035/viewer/2022062217/56812c55550346895d90dee2/html5/thumbnails/3.jpg)
1 2OP QQ
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Cartesian:
Polar:
Parametric:
2 2 cos 2r a
2
2
cos
1 sinsin cos
1 sin
a tx
ta t t
yt
2 2 2 2 2( ) ( )x y x y
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OC = CA = OP’ is parallel to
HF’ OQ = PP’ 2QC = OH 2PQ = HH’ Then is
similar to Now 2PC = OH’
= a
12
2a
'H HOPQC
My cool geogebra file
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COQ Let be and OQ be r
Segment OH =
OA = sinOA
2a
sin 2 sinOA a
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It’s a trig identity!
2 2 2 2' 'r HH OH OH 2 2 22 sina a
2 cos 2a
0.2
0.4
0.6
0.8
1
30
210
60
240
90
270
120
300
150
330
180 0
Matlab generated polar graph
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2 2 cos 2r a
cos
sin
x r
y r
2
2
2
cos 2
cos cos 2
sin cos 2
r a
x a
y a
4 4
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
x axis
yaxi
s
r2=cos(2*theta) on [-pi/4,pi/4] and [3*pi/4,5*pi/4]
Matlab generated graph
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=
Delicious math
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http://en.wikipedia.org/wiki/Daniel_Bernoulli
Lee, Xah. "Lemniscate of Bernoulli." Visual Dictionary Of Special Plane Curves. 23 Nov. 2008 <http://xahlee.org/specialplanecurves_dir/lemniscateofbernoulli _dir/lemniscateofbernoulli.html>.
A Book of Curves by E. H. Lockwood