APOLLONIUS OF PERGA CONICS PDF

Apollonius of Perga , born c. Most of his other treatises are now lost, although their titles and a general indication of their contents were passed on by later writers, especially Pappus of Alexandria fl. As a youth, Apollonius studied in Alexandria under the pupils of Euclid, according to Pappus and subsequently taught at the university there. He visited both Ephesus and Pergamum , the latter being the capital of a Hellenistic kingdom in western Anatolia , where a university and library similar to the Library of Alexandria had recently been built.

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We use cookies to give you a better experience. The ancient Greeks loved the simplicity and elegance of the line and the circle. In three-dimensional space, combining a circle with a fixed point not in the plane of the circle gives a cone, and it was by slicing this cone that Apollonius studied what were to become some of the most important curves in mathematics: the conic sections.

He lived in Perga, which is in modern day Turkey, and wrote a series of books on conic sections, including the parabola, ellipse and hyperbola. Many of the facts discovered by him would surprise modern high school students for their elegance and richness. The ancient Greeks regarded the line and the circle as the most fundamental and beautiful of all mathematical objects, and if you connect a circle and a point in three-dimensional space with lines, you get a cone.

You might be thinking of a cone like an ice-cream cone, but Apollonius realised it is useful to think of the lines of the cone as extending in both directions — a two sided cone.

Now if we slice such a cone with a plane, we get generally three different kinds of curves, along with a few more special cases.

It all depends on the relation between the slicing plane and the cone. The hyperbola is the case when the plane meets both top and bottom of the cone, the ellipse is the case when we meet only one half of the cone in a finite or bounded region.

The parabola is poised exactly in between the ellipse and the hyperbola; it occurs when the slicing plane is parallel to a tangent plane of the cone. What is a tangent plane? Imagine taking the top half of the cone and rolling it along a plane but not unrolling it!

The plane along which the cone is rolled always forms a tangent plane to the cone and touches the cone along only a single line. Q1 E : An everyday example of a conic section is the shadow from a lamp with a cylindrical lampshade onto a flat surface. What conic section would you get from the shadow formed on the floor?

How about on the wall? Q2 M : Actually there are a few other, more degenerate possibilities for what happens when a plane slices a cone.

Can you describe these? The conic sections are quite symmetrical objects. The ellipse and the hyperbola both have a distinguished point of symmetry, called naturally enough the centre. If we reflect any point on the curve in this centre, we get another point on the curve.

Through the centre of an ellipse or a hyperbola are two special lines of symmetry called the axes of the conic. If we reflect any point on the curve in such a line, then we get another point on the curve. However the parabola is quite different in this respect: it does not have a centre, but it does have a line of symmetry — but only one!

So we speak of the axis of a parabola, since it is unique. Note the terminology here: axis is the singular, and axes is the plural. A chord of a conic is a line that passes through two points on that conic, and a diameter is a chord that passes through the centre of a conic. Try it yourself! So diameters of an ellipse come in pairs, called conjugate diameters.

Apollonius knew that if you took any conic section, then that curve determined a beautiful and remarkable correspondence, often called polarity , between points and lines in the plane. There is a correspondence between a point on a curve and the tangent line to the curve.

The concept of polarity generalises this correspondence to points which are not on the curve. Refer to the figure of the ellipse below. Now take those four points of intersection, and construct the lines passing through each pair of these points. Apollonius did not know everything there was to know about conics.

He was also an astronomer, and interested in the ancient problem of describing the motion of the visible planets in the night sky. If the plane passes through the apex point of the cone, then we could get a pair of lines, or a single line, or just the single point of the apex itself. So these are also in some sense degenerate conics. The top-left gold point is the centre of the ellipse, the other gold point is the centre of the hyperbola, and the line not through either of those is the axis of the parabola.

Yes, the hyperbola also has conjugate diameters. You can verify this yourself if have a picture of a hyperbola to play with. Explore and solve encrypted maths puzzles, in which numbers are replaced by letters or symbols, with this free online course. Included in Unlimited. Discover the science behind nuclear energy and its role in energy provision in the past, present and future. Discover big data: work with airline data to learn the fundamentals of the R platform.

Search term Search. Want to keep learning? Join the course to learn more. View course. Maths for Humans: Linear and UNSW Sydney. In this step, we will see how Apollonius defined the conic sections, or conics learn about several beautiful properties of conics that have been known for over years.

Slicing a cone Now if we slice such a cone with a plane, we get generally three different kinds of curves, along with a few more special cases. Apollonius knew all kinds of lovely facts about the conic sections. For example… Symmetry, centres and axes of ellipses and hyperbolas The conic sections are quite symmetrical objects. Q3 E : Can you find the centre and axes of the conics pictured? Conjugate diameters of an ellipse A chord of a conic is a line that passes through two points on that conic, and a diameter is a chord that passes through the centre of a conic.

Q4 C Does a hyperbola also have a notion of conjugate diameters? The polarity defined by a conic advanced topic Apollonius knew that if you took any conic section, then that curve determined a beautiful and remarkable correspondence, often called polarity , between points and lines in the plane.

Answers A1. Get a taste of this course Find out what this course is like by previewing some of the course steps before you join:. Australian bush tucker. The Cartesian plane and the beauty of graph paper. Quadratics from Apollonius to Bezier. Galileo's ball. More courses you might like Learners who joined this course have also enjoyed these courses. Weizmann Institute of Science. Find out more. The Open University. The Science of Nuclear Energy. Introduction to R for Data Science.

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Apollonius of Perga

His works had a very great influence on the development of mathematics and his famous book Conics introduced the terms parabola, ellipse and hyperbola. View one larger picture. Little is known of his life but his works have had a very great influence on the development of mathematics, in particular his famous book Conics introduced terms which are familiar to us today such as parabola , ellipse and hyperbola. Apollonius of Perga should not be confused with other Greek scholars called Apollonius, for it was a common name. In [ 1 ] details of others with the name of Apollonius are given: Apollonius of Rhodes, born about BC, a Greek poet and grammarian, a pupil of Callimachus who was a teacher of Eratosthenes ; Apollonius of Tralles, 2 nd century BC, a Greek sculptor; Apollonius the Athenian, 1 st century BC, a sculptor; Apollonius of Tyana, 1 st century AD, a member of the society founded by Pythagoras; Apollonius Dyscolus, 2 nd century AD, a Greek grammarian who was reputedly the founder of the systematic study of grammar; and Apollonius of Tyre who is a literary character. Perga was a centre of culture at this time and it was the place of worship of Queen Artemis, a nature goddess.

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Apollonius and conic sections

We use cookies to give you a better experience. The ancient Greeks loved the simplicity and elegance of the line and the circle. In three-dimensional space, combining a circle with a fixed point not in the plane of the circle gives a cone, and it was by slicing this cone that Apollonius studied what were to become some of the most important curves in mathematics: the conic sections. He lived in Perga, which is in modern day Turkey, and wrote a series of books on conic sections, including the parabola, ellipse and hyperbola. Many of the facts discovered by him would surprise modern high school students for their elegance and richness. The ancient Greeks regarded the line and the circle as the most fundamental and beautiful of all mathematical objects, and if you connect a circle and a point in three-dimensional space with lines, you get a cone. You might be thinking of a cone like an ice-cream cone, but Apollonius realised it is useful to think of the lines of the cone as extending in both directions — a two sided cone.

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