Description
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.
Constructions
| Ellipse
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- 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.
- Ellipses are a closed curve with a Major Axis and a Minor Axis.
- An Ellipse is also the result of taking a section across a Cylinder at at angle to its base.
- Ellipses describe the orbits of planets and the curve has been used to design Arch Bridges.
- Ellipses can be drawn as Loci with an Eccentricity of less than 1 (i.e. the Locus of all points where the ratio of distances from its Focus and Directrix are a constant which is less than 1)
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| Parabola
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- The curve produced when a secton is taken through a right circular cone parallel to the generator.
- A Parabola describes the path taken by a object thrown into the air and the curve has been used to design Arch Bridges.
- Parabolas can be drawn as Loci with an Eccentricity equal to 1 (i.e. the Locus of all points where the ratio of distances from its Focus and Directrix are a constant which is equal to 1)
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| Hyperbola
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- The curve produced when a secton is taken which cuts both parts of a Double Cone but does not pass through the Apex (e.g. parallel to Axis as shown) - note that with a Double Cone, two Hyperbolae are created.
- Hyperbolae describe the paths taken by non-orbiting comets or the shadow cast on a sun dial for example.
- An Hyperbola can be drawn as a Locus with an Eccentricity greater than 1 (i.e. the Locus of all points where the ratio of distances from its Focus and Directrix are a constant which is greater than 1)
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| Circle
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- 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
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| Triangle
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- 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.
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