Conic Sections
Yu Cheng
Lucas Mastro
Brandon Nowakowski
Definitions
a.
A conic section is a curve
obtained as the intersection of
the surface of the cone and a
plane.
Definitions
b. Textbook:
Apollonius’ definition
“From a point a straight line is joined to the circumference of a circle
which is not in the same plane as the point , and the line extended in both
directions, and if with the point remaining fixed, the straight line is rotated
about the circumference of the circle...then the generated surface
composed of the two surfaces lying vertically opposite one another… is a
conic surface. ”(Page 115)
Development
a.
Menaechmus (380-320 BC)
i. Greek mathematician
ii. Came up with the definition of conic
sections while solving the Delian
problem(doubling the cube), the
problem is, to find the side length of
the cube after doubling its volume.
iii. The works were lost.
The Delian problem
Long time ago, In the ancient Greek city Delos, there was a plague. It was sent by Apollo who is
the god of healing. The oracle said that Apollo was angry because the altar(perfect square
shape) in his temple is too small, and it needs to be twice big of its current one.
People started making the side length twice long, and the plague was still going on.
Until gets close to the right answer, the plague eventually stopped.
Menaechmus’s method
If x, y are two mean proportional between straight lines a, b
a/x=x/y=y/b
then x2 = ay, y2 = bx , and xy = ab
Therefore each two pair of these three curves gives a solution to Delian problem.
1.
x2 = ay, and xy = ab
2.
y2 = bx , and xy = ab
3.
x2 = ay, and y2 = bx
Development
c. Euclid
i. Studied it and had few books written about it.
ii. His work didn’t survive neither.
d. Apollonius
iii. Studied systematically.
Most work about conic sections are done by Apollonius.
Applications
Can you think of any?
Applications
Most frequently used in physics, astronomy and
aerospace.
a. Planet orbits
b. Parabolic mirror
c. Optical telescopes
d. Satellites
e. Solar oven
Types of conic sections
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Circles
○ Equation
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Parabola
○ Equation
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Ellipse
○ Equation
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Hyperbola
○ Equation
Equations for ellipse
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Cartesian form
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Polar form
Apollonius
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Born in Perga, Pamphylia (sometime between 247 and 222 B.C.)
Went to Alexandria to study with successors of Euclid
Conics
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7 of the 8 books remain
○
○
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4 in original Greek
3 in Arabic translation
No evidence of algebra
Generalized notion of a cone
Given: point and circle are NOT in the same plane
Defining the 3 conic sections
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●
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Ellipse--EG intersects both AB and AC
Parabola--EG parallel to AB or AC
Hyperbola--EG intersects side beyond A
Derivation of symptoms of a parabola
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●
L is on the parabola
Pass a plane through L parallel to the base circle
○
●
●
●
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Circle has diameter PR
M: intersection of plane with EG
LM PR⇒LM2=PM*MR
Through similarity: PM*MR = EH*EM
It follows that LM2=EH*EM
y=LM, x=EM, p=EH
Obtain y2=px
Application of Parabolas: Burning Mirrors
Defining Asymptotic Behavior
Construction Postulate
Johannes Kepler (1571-1630)
-Studied Theology, Mathematics and Philosophy.
-Revolutionized Astronomy, the impact he had there is still
felt today.
-He had two wives, and eleven children, six of whom died in
childhood.
-He has 17 works attributed to him, much of this is, today,
considered almost nonsensical. However, some of his
discoveries were truly groundbreaking.
Copernican model of astronomy
The following are four fundamental concepts of Copernican astronomy.
i) The Earth’s orbit is the center of the solar system.
ii) The planes of the planets’ orbits oscillate in space
iii) Planetary motion is uniform
iv) Planetary motion is circular.
-
Kepler’s most relevant discovery was aimed at dismissing these four
statements and replacing them with what are known as Kepler’s Three Laws.
Kepler’s Three Laws, background
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-
Kepler noticed that he could not resolve a difference of eight minutes of an
arc from a model assuming a circular of Mars and the observed orbit.
Kepler took three observed points from the orbit of Mars and constructed a
circle along them, then he took three other observed points and constructed
another circle.
These circles were distinct, and therefore he realized that the orbit of Mars
was not circular.
This discovery sent him to try to find the true nature of the orbit of Mars and
therefore (as Newton later proved for the first two laws) of all planets in the
solar system, a search that took several years.
Kepler’s Three Laws
i) The planet describes an ellipse, the sun being in one focus.
ii) The straight line joining the planet to the sun sweeps out equal areas in any two
equal intervals of time
iii) The squares of the periodic times are proportional to the cubes of the mean
distances
The Third Law
“The squares of the periodic times are proportional to the cubes of the mean
distances”
Periodic Time:= Time taken to complete a single orbit.
Mean distances:= Semi-major axis of the ellipse of the planet’s orbit.
This law was developed several years after the first two, and is perhaps the most
interesting.
Sources
Apollonius of Perga and Sir Thomas Little Heath. "Treatise on Conic Sections." New York: Barnes & Noble, 1961. Print.
Bryant, Walter W. “Kepler” Royal Observatory, Greenwich 1920
Katz, Victor J. A History of Mathematics. 3rd ed. Boston: Addison-Wesley, 2009. Print.
Leyden, Michael B. “The Elliptical Johannes Kepler.” The Science Teacher, vol. 51, no. 8, 1984, pp. 52–56., www.jstor.org/stable/24142046.
Stapel, Elizabeth. "Conics: Ellipses: Introduction." Purplemath. Available from
http://www.purplemath.com/modules/ellipse.htm