Conic Sections: Difference between revisions
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https://en.wikipedia.org/wiki/ | A [https://en.wikipedia.org/wiki/Cone '''Cone'''] can be described as the '''[[Loci|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 [https://en.wikipedia.org/wiki/Solid_of_revolution '''Solid of Revolution'''] and can be produced by rotating a right angle triangle around one of the sides opposite its [https://en.wikipedia.org/wiki/Hypotenuse hypotenuse]. | A '''right circular cone''' is a [https://en.wikipedia.org/wiki/Solid_of_revolution '''Solid of Revolution'''] and can be produced by rotating a right angle triangle around one of the sides opposite its [https://en.wikipedia.org/wiki/Hypotenuse 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 [https://en.wikipedia.org/wiki/Conic_section '''Conic Sections''']. These are curves with particular mathematical properties and which are used in engineering design - e.g. [[Displacement Diagram|cam profiles]] and [[Bridge Types|arch bridge]] design. | A right circular cone can be sliced across in various ways to produce a number of [https://en.wikipedia.org/wiki/Conic_section '''Conic Sections''']. These are curves with particular mathematical properties and which are used in engineering design - e.g. [[Displacement Diagram|cam profiles]] and [[Bridge Types|arch bridge]] design. | ||
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| '''[[Ellipse]]''' | | '''[[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. | |||
| [[File:Conics-ElipseTrans.png|350px|right]] | |||
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| '''[[Parabola]]''' | |||
| <span style="color:#B00000"> | |||
* The curve produced when a secton is taken through a right circular cone parallel to the generator | |||
| [[File:Conics-ParabolaTrans.png |350px|right]] | |||
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| '''[[Hyperbola]]''' | |||
| <span style="color:#B00000"> | |||
* 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. | |||
| [[File:Conics-HyperbolaTrans.png|350px|right]] | |||
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| '''[[Circle]]''' | |||
| <span style="color:#B00000"> | |||
* The curve produced when a secton is taken across a right circular cone parallel to the base circle. | |||
| [[File:Conics-Circle.png |350px|right]] | |||
|- | |- | ||
| | | '''[[Triangle]]''' | ||
| | | <span style="color:#B00000"> | ||
* Included for completeness, triangles are created when a section is taken through the Apex and parallel to the central Axis. | |||
| [[File:Conics-TriangleTrans.png|350px|right]] | |||
|} | |} | ||
Revision as of 18:48, 3 April 2015
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 |
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| Parabola |
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| Hyperbola |
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| Circle |
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| Triangle |
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