Conic Sections

A Cone can be described as the Locus of all lines joining points on the circumference of a ‘Base Circle’ to a point, or ‘Apex’, above it. If the Apex lies perpendicularly above the centre of the base circle it is known as a ‘right cone’ and if not, it is an ‘oblique cone’.

A right circular cone is a Solid of Revolution and can be produced by rotating a right angle triangle around one of the sides opposite its hypotenuse. The hypotenuse generates the surface of a cone as it sweeps round the central ‘Axis’. Any line joining the Apex to the Base Circle is known as a ‘Generator’. Generators can continue through the Apex to produce a second cone opposite to the first, creating a ‘Double Cone,’ and each Cone could extend beyond he Base Circles to infinity.

A right circular cone can be sliced across in various ways to produce a number of Conic Sections. These are curves with particular mathematical properties and which are used in engineering design - e.g. cam profiles and arch bridge design.

Ellipse	The curve produced when a secton is taken across a right circular cone at an angle to the base circle, and intersecting the generators on both sides. An Ellipse is also the result of taking a section across a Cylinder at at angle to its base.
Parabola	The curve produced when a secton is taken through a right circular cone parallel to the generator
Hyperbola	The curve produced when a secton is taken along a right circular cone parallel to the central Axis - note that with a Double Cone, two Hyperbolas are created.
Circle	The curve produced when a secton is taken across a right circular cone parallel to the base circle. The circle can be considered as a fourth type of conic section (as it was by Apollonius) or as a special case of ellipse. The part of a cone (or pyramid) lying between two parallel planes such as this is called the Frustrum
Triangle	Included for completeness, triangles are created when a section is taken through the Apex and across the Base Cicle diameter. Either an Isosceles or Equilaterial Triangle is revealed when a Right Cone is halved in this way. Other triangles would result from similar sectioning of an Oblique Cone.