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Conic Sections

From Apollonius to radiotelescopes

A short history of mathematics

Before the Greeksn Babylonian and Egyptian civilizations had arithmetic

of integers and fractions, including positional notation, the beginnings of algebra, and some empirical formulasin geometry. There was almost no symbolism, no abstractions, no formulation of general methodology, and no concept of proof. There was, in fact, no conception of any kind of theoretical science.

n Mathematics in the two civilizations was not a distinct discipline. It was a tool in the form of disconnected, simple rules which answered questions arising in the daily life of the people


n Before the GreeksIt’s is natural to compare the two

civilizations with the Greek, which succeeded them: the Egyptians and Babylonians were crude carpenters, whereas the Greeks were magnificent architects.


n The Greeks enter the picture: 600 BCIn the history of civilization the Greeks are

preeminent, and in the history of mathematics the Greeks are the supreme event.

One of the great problems of the history of civilization is how to account for the brilliance and creativity of the ancient Greeks


n The Classical Greek Mathematics

“This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth”

Proclus


n The Classical Greek Mathematics“You are aware that students of geometry, arithmetic

and the kindred sciences assume the odd and the evenand the figures and three kinds of angles and the likein their several branches of science; these are theirhypotheses, which they and everybody are supposed toknow, and therefore they do not deign to give anyaccount of them either to themselves or others; butthey begin with them, and go on until they arrive at last, and in a consistent manner, at their conclusion”

Plato, The Republic


n The major school of classical period¡ The Ionian school (Thales 640bc-546bc)¡ The Pythagoreans

n the recognitions that mathematics deals with abstractions may be attributed to Pythagoreans

n The proof that the square root of 2 is incommensurable with 1 was given by them

¡ The Eleatic school (Zeno 480BC)¡ The Sophist school ¡ The Platonic school¡ …

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The compass and straightedge constructionDefinition:

n A compass is a tool which can be used for drawing circles, or arcs thereof, whose radii are sufficiently small, and formeasuring lenghts.

n A straightedge is a tool which can be used for drawinglines, or segments thereof.

Consequence:By means of these two tools, one was restricted to only five mathematical operations (to be performed on rational numbers): addition, subtraction, multiplication, division, and taking square roots.


The compass and straightedge constructionExplanation of the restriction:

n The straight line and the circle were, in the Greek view, the basic figures, and the straightedge and compass are their physical analogues.

n Plato objected to other mechanical instruments because they involved too much of the world of the senses rather than the world of ideas, which he regarded as primary

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The 3 unsolved problem of Greek mathematics

n Doubling the cube

n Squaring the circle

n Trisecting the angle


Doubling the cube: history and legend.

n In 430 BC a great plague struck Athens and its great leader was killed.n The Athenians sought guidance from the Oracle at Delos who said that the

god Apollo was angry and if his cubical altar was doubled in size the plaque would be abolished.

n The Athenians built a new altar doubling the width, height, and depth.n The plague worsened so they went to the Oracle and found that Apollo

wanted the volume doubled whereas the Athenians had octupled it.n The plague continued until 423 BC; however this problem was not solved

before the plague ended.

Doubling the cube: unrestricted solution.

n Today it is easy to solve this problem, it is just an algebraic equation: a3=2x3 where a is the side length of an arbitrary cube and x is the side length of the new doubled cube as shown in the diagram.

n With a little algebraic manipulation we get:

which is the side length of the original cube with its volume doubled.


3 2ax =

Squaring the circle

n The area of a circle was generalized to be equal to its radius squaredmultiplied by π. π, however, was not defined exactly whichpresented a problem: π is irrational, that is to say it cannot beexpressed as a ratio of two rational numbers.

Having known the area of a circle and a square, a simplerelationship was noticed: the length of the side of the sought aftersquare is the radius of the given circle multiplied by a factor of the square root of π.

n As a result of Plato's restrictions, a solution was (and still is) impossible by geometric means.

n Archimedes, amongst many others, toiled over this problem, not realizing that π is a transcendental number.

n Unrestricted Solution: For a square to have the area of a circle, its side needs to be the square root of pi times the circle's radius.


πrl =

Trisecting the anglen Problem: find an angle which measures exactly one third

of a given arbitrary angle.n Restricted Solution: The ancient Greeks, having the

capability of bisecting angles and trisecting line segments, had difficulties dividing an arbitrary angle into three equalparts.

n Granted, the Greeks were capable of trisecting a select few special angles like 90 and 180 degrees, a general method could not be formulated.

n Though Hippocrates and Archimedes (and a few others) didfind a way to trisect an angle, it was not within Greekconstruction under Plato's confines.


Solutions to the 3 classical problems

n It was only until the 19th century that it was proven that no solution was a solution to the classic problems under Plato's restrictions.

n There are two important realizations that arose from these threeeasily understood, but unsolvable problems of antiquity:

1. the acceptance of the scientific community that no solution is

itself a solution.

2. Realizing the truth is much easier than justifying the truth.


The compass and straightedge construction.One notable case: the golden spiral


The golden rectangle


Parthenon Leonardo da Vinci, The Last Supper


The Vetruvian Man by Leonardo Da Vinci

Composition with Gray and Light brown by PietMondrian 1918

The Classical Greek MathematicsEuclid (fl. 300 BC), also known as Euclid of Alexandria, was a

Greek mathematician, often referred to as the "Father of Geometry".

His Elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century.

In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms.


The Classical Greek Mathematics

Apollonius of Perga (ca. 262 BC – ca. 190 BC) was a Greek geometer and astronomer noted for his writings on conic sections.

n His mathematical powers were so extraordinary that he became known in his time and thereafter as "the Great Geometer"

n The conic sections, as we know, were studied long before Apollonius' time. Apollonius, however, stripped the knowledge of all irrelevancies and fashioned it systematically. Besides being comprehensive, his ConicSections contains highly original material and is ingenious, extremelyadroit, and excellently organized.

n As an achievement it is so monumental that it practically closed the subjectto later thinkers, at least from the purely geometrical standpoint. It maytruly be regarded as the culmination of classical Greek geometry.


Conic Sections

•Ellipse

•Parabola

•Hyperbola

Exploration

Conic Sections

The curves can also be defined using a straight lineand a point (called the directrix and focus).

If you measure the distance from the focus to a point on the curve, and perpendicularly from the directrix to that point the two distances will alwaysbe the same ratio.•For an ellipse, the ratio is less than 1 •For a parabola, the ratio is 1, so the two distances are equal. •For a hyperbola, the ratio is greater than 1

Conic Sections

That ratio above is called the "eccentricity", so we can say that any conic section is:

“All points whose distance to the focus is equal to the eccentricity times the distance to the directrix”

•For eccentricity < 1 we get an ellipse, •for eccentricity = 1 a parabola, and •for eccentricity > 1 a hyperbola.

A circle has an eccentricity of zero, so the eccentricity shows you how "un-circular" the curve is. The bigger the eccentricity, the lesscurved it is

Hyperbola

•The section between a double cone and a plane gives a hyperbola when the cutting plane is inclined at an angle greater than the exterior angle of the cone

•Hyperbola is also the locus of all pointsin a plane such that the difference oftheir distances from two fixed points isconstant

Hyperbola

Construction and propertiesn Construction from definitionn Optical propertiesn Construction as envelopen Let’s work!

Hyperbola in real life

Satellites

Satellite systems make heavy use of hyperbolas and hyperbolic functions. When scientists launch a satellite into space, they must first use mathematical equations to predict its path. Because of the gravity influences of objects with heavy mass, the path of the satellite is skewed even though it may initially launch in a straight path. Using hyperbolas, astronomers can predict the path of the satellite to make adjustments so that the satellite gets to its destination.


Radio

Radio systems' signals employ hyperbolic functions. One important radio system, LORAN (Long Range Navigation), identifies geographic positions using hyperbolas.

The diagram show how the of the LORAN work: the difference between the time of reception of synchronized signals from radio stations A and B is constant along each hyperbolic curve; when demarcated on a map, such curves are known as "TD lines“ where "TD" stands for "Time Difference". LORAN allows people to locate objects over a wide area and played an important role in World War II.


Lenses and Mirrors

Objects designed for use with our eyes make heavy use of hyperbolas. These objects include microscopes, telescopes and televisions. Before you can see a clear image of something, you need to focus on it. Your eyes have a natural focus point that does not allow you to see things too far away or close up. To view such things as planets or bacteria, scientists have designed objects that focus light into a single point. The designs of these use hyperbolas to reflect light to the focal point. When using a telescope or microscope, you are placing your eye in the focal point.


Hyperboloidal GearsGears shaped like hyperboloids can be used to build a transmission between skew axles. With a given "skew angle" (the angle between the axles), the eccentricity of two hyperboloids whose axes of symmetry coincide with these axles can be found such that there is a constant line of contact between them. These hyperboloids are created by rotating a straight line that is skew to the axle, at an angle that is half that of the skew angle. These lines then act as the gear teeth. If this is done correctly, the teeth on the gears will be in parallel orientations while in contact with one another, thus creating a smooth transmission.

Hyperbola in architecture

Saint Louis Science Centre


Gazprom Building in St. Petersburg (Project)


Brasilia Cathedral - Brazil


Pengrowth Saddledome in Calgary, Alberta (Canada)

Ellipse

•An ellipse (from Greek ἔλλειψιςelleipsis, a "falling short") is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve

•Ellipse is also the locus of all points in a plane such that the sum of theirdistances from two fixed points isconstant

Ellipse

Construction and propertiesn Construction from definitionn The gardener’s ellipsen Optical propertiesn Construction as envelope

Ellipse in real life

Shape of Planetary Orbits

“All planets move about the Sun in elliptical orbits, having the Sun as one of the foci” (Kepler’s first Law)


Shape of Planetary Orbits-2

Agora is a 2009 Spanish English-language historical drama film directed by Alejandro Amenábar. The biopic stars Rachel Weisz as Hypatia, a female mathematician, philosopher and astronomer in late 4th century Roman Egypt, who investigates the flaws of the geocentric Ptolemaic system and the heliocentric model that challenges it. Surrounded by religious turmoil and social unrest, Hypatia struggles to save the knowledge of classical antiquity from destruction. See the clip


Lithotripsy

Lithotripsy is a method for destroying a kidney stone. The patient lies in an elliptical tub. The kidney stone is aligned to one of the focii of the ellipse. Shockwaves emanating from the other focus concentrate on the kidney stone, destroying it. The debris is as small as sand and can pass through the body without discomfort. In this treatment, no surgery is needed.


Whispering Gallery

An ellipse has the property that if light or a sound wave emanates from one focus it will be be reflected to the other focus. This property is used to create whispering galleries, which are structures that allow someone who is whispering in one area to be heard clearly by someone in another area but not by anyone else. Famous examples of whispering galleries include the United States Statuary Capitol Hall and London's St. Paul's Cathedral.


Elliptic trainer

An elliptical training machine simulates the motion of running or climbing to provide the user with a healthful exercise without any impact on the joints. The foot of a user describes the shape of an ellipse as the machine is used. The elliptical trainer provides a stationary exercise for those who wish to avoid joint injury as a result of running.

Ellipse in architecture

Ellipse 1501 – single family house - Rome


Ellipse Building – Swansea (UK)


Tycho Brahe Planetarium – Kobenhavn (Denmark)


Ellipse Building

Brussels

(Belgium)

Ellipse in architectureElliptical Arch

Arches have been used in architecture from the classical period and have seen extensivevariation throughout the regions of the globe. In contrast to the traditional semi-circulararch, a number of structures and differentdesigns have used elliptical arches, eachpresenting benefits (aesthetic appeal) as well asdrawbacks (it does not offer the same strengthas the semicircle).Some of the more common sites for ellipticalarches are in bridge construction and structural entrances.

Parabola

•The section between a double cone and a plane gives a parabola when the cutting plane is inclined at an angle equal to the exterior angle of the cone

•Parabola is also the locus of all pointsin a plane such that its distance from a fixed point is equal to its distance from a fixed line

Parabola

Construction and propertiesn Construction from definitionn Optical-physical propertiesn Construction as envelopen Let’s work!

Parabola in real life

Parabolic motionIf a particle is thrown obliquely near the earth's surface then it moves along a curved path under the action of gravity. If there is only one vertical force applied (gravity) then the trajectory is a parabola.


Rotating liquidWhen liquid is rotated, the forces of gravity meet the centrifugal force, which results in the liquid forming a parabola-like shape. The most common example is when you stir up orange juice in a glass by rotating it round its axis. The juice level rises round the edges while falling slightly in the center of the glass (the axis).

Parabola in real lifeReflector

Parabola is also used in satellite dishes and flashlights. In satellite dishes it helps reflect signals that then go to a receiver, which interprets the signals and shows satellite-transmitted channels on your TV. In flashlights, automobile headlights and spotlights, parabolic shape helps reflect light.


Reflector - RadiotelescopeA radio telescope is a form of directional radio antenna used in radio astronomy. In their astronomical role they differ from optical telescopes in that they operate in the radio frequency portion of the electromagnetic spectrum where they can detect and collect data on radio sources. Radio telescopes are typically large parabolic ("dish") antennas used singly or in an array.See Chromoscope.net

Parabola in architecture

Sydney opera house

Parabolas in architectureNothing to say…

Parabolas in architectureSaint Louis Gateway is not a parabola

Parabolas in architectureDubai Quill

Parabolas in designGaudì parabolic structure (chair) by Studio Bram Geenen

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