Apollonius says that he intended to cover "the properties having to do with the diameters and axes and also the asymptotes and other things The elements mentioned are those that specify the shape and generation of the figures. Tangents are covered at the end of the book. Apollonius claims original discovery for theorems "of use for the construction of solid loci

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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. 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. 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.

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. It all depends on the relation between the slicing plane and the cone. 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? 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.

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. 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.

If you choose any line through the centre of an ellipse, for example in the figure below, and take the midpoints of all chords which are parallel to that diameter, then you get another diameter of the ellipse, namely.

And if you repeat the process with the second diameter, you get back the first diameter. Try it yourself! So diameters of an ellipse come in pairs, called conjugate diameters. 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.

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. Given a point , choose any two lines that pass through and intersect the ellipse at two points.

Now take those four points of intersection, and construct the lines passing through each pair of these points. These lines will meet each other, defining two other diagonal points, and. The dual or polar of point is then the line through those two diagonal points the purple line in the diagram. Remarkably this dual line is independent of the choices of lines through , as Apollonius realised. He was also an astronomer, and interested in the ancient problem of describing the motion of the visible planets in the night sky.

Answers A1. 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.

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## Treatise on conic sections

Definition The black boundaries of the colored regions are conic sections. Not shown is the other half of the hyperbola, which is on the unshown other half of the double cone. A conic is the curve obtained as the intersection of a plane , called the cutting plane, with the surface of a double cone a cone with two nappes. It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics and some authors do not consider them to be conics at all.

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## Apollonius of Perga

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## Apollonius and conic sections

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