Conic Sections - Revision history

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	[[File:ConicSections.jpg\|right]]		[[File:ConicSections.jpg\|right]]
			__TOC__


			=====Description=====

	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 [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’'''''.
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			=====Constructions=====
	{\|		{\|
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	\| [[File:Conics-TriangleTrans.png\|300px\|right]]		\| [[File:Conics-TriangleTrans.png\|300px\|right]]
	\|}		\|}


			{{Drawing Instruments Buyers Guide}}


	[[Category:Secondary]]		[[Category:Secondary]]
	[[Category:Geometry]]		[[Category:Geometry]]

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	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 [[Arch Variations\|arch bridge]] design.

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	\| [[File:Conics-HyperbolaTrans.png\|300px\|right]]		\| [[File:Conics-HyperbolaTrans.png\|300px\|right]]
	\|-		\|-
	\| ~~[https://en.wikipedia.org/wiki/Circle~~ '''Circle''']		\| '''[[Circles\|Circle]]'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* The curve produced when a secton is taken across a right circular cone parallel to the base circle.		* The curve produced when a secton is taken across a right circular cone parallel to the base circle.

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2015-04-05T09:17:34Z

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	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* 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.		* 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 [https://en.wikipedia.org/wiki/Cylinder_%28geometry%29 '''Cylinder'''] at at angle to its base.		* An Ellipse is also the result of taking a section across a [https://en.wikipedia.org/wiki/Cylinder_%28geometry%29 '''Cylinder'''] at at angle to its base.
			* Ellipses describe the orbits of planets and the curve has been used to design [https://en.wikipedia.org/wiki/Arch Arch Bridges].
	\| [[File:Conics-ElipseTrans.png\|~~350px~~\|right]]		* Ellipses can be drawn as '''[[Loci]]''' with an '''[[Conics\|Eccentricity]]''' of less than 1 ''(i.e. the Locus of all points where the ratio of distances from its '''[[Conics\|Focus]]''' and '''[[Conics\|Directrix]]''' are a constant which is less than 1)''
			\| [[File:Conics-ElipseTrans.png\|300px\|right]]
	\|-		\|-
	\| '''[[Parabola]]'''		\| '''[[Parabola]]'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* The curve produced when a secton is taken through a right circular cone parallel to the generator		* The curve produced when a secton is taken through a right circular cone parallel to the generator.
	\| [[File:Conics-ParabolaTrans.png \|~~350px~~\|right]]		* A Parabola describes the path taken by a object thrown into the air and the curve has been used to design [https://en.wikipedia.org/wiki/Arch Arch Bridges].
			* Parabolas can be drawn as '''[[Loci]]''' with an '''[[Conics\|Eccentricity]]''' equal to 1 ''(i.e. the Locus of all points where the ratio of distances from its '''[[Conics\|Focus]]''' and '''[[Conics\|Directrix]]''' are a constant which is equal to 1)''
			\| [[File:Conics-ParabolaTrans.png \|300px\|right]]
	\|-		\|-
	\| '''[[Hyperbola]]'''		\| [https://en.wikipedia.org/wiki/Hyperbola '''Hyperbola''']
	\| <span style="color:#B00000">		\| <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.		* 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.
	\| [[File:Conics-HyperbolaTrans.png\|~~350px~~\|right]]		* 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 '''[[Loci\|Locus]]''' with an '''[[Conics\|Eccentricity]]''' greater than 1 ''(i.e. the Locus of all points where the ratio of distances from its '''[[Conics\|Focus]]''' and '''[[Conics\|Directrix]]''' are a constant which is greater than 1)''
			\| [[File:Conics-HyperbolaTrans.png\|300px\|right]]
	\|-		\|-
	\| '''[[Circle]]'''		\| [https://en.wikipedia.org/wiki/Circle '''Circle''']
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* The curve produced when a secton is taken across a right circular cone parallel to the base 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 [https://en.wikipedia.org/wiki/Apollonius_of_Perga '''Apollonius'''])'' or as a special case of ellipse.		* The circle can be considered as a fourth type of conic section ''(as it was by [https://en.wikipedia.org/wiki/Apollonius_of_Perga '''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 [https://en.wikipedia.org/wiki/Frustum '''Frustrum''']		* The part of a cone ''(or [https://en.wikipedia.org/wiki/Pyramid_%28geometry%29 '''Pyramid'''])'' lying between two parallel planes such as this is called the [https://en.wikipedia.org/wiki/Frustum '''Frustrum''']
	\| [[File:Conics-Circle.png \|~~350px~~\|right]]		\| [[File:Conics-Circle.png \|300px\|right]]
	\|-		\|-
	\| '''[[Triangle]]'''		\| [https://en.wikipedia.org/wiki/Triangle '''Triangle''']
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Included for completeness, triangles are created when a section is taken through the Apex and across the Base Cicle diameter.		* Included for completeness, triangles are created when a section is taken through the Apex and across the Base Cicle diameter.
	* Either an [https://en.wikipedia.org/wiki/Isosceles_triangle '''Isosceles'''] or [https://en.wikipedia.org/wiki/Equilateral_triangle '''Equilaterial Triangle'''] is revealed when a Right Cone is halved in this way. Other triangles would result from similar sectioning of an Oblique Cone.		* Either an [https://en.wikipedia.org/wiki/Isosceles_triangle '''Isosceles'''] or [https://en.wikipedia.org/wiki/Equilateral_triangle '''Equilaterial Triangle'''] is revealed when a Right Cone is halved in this way. Other triangles would result from similar sectioning of an Oblique Cone.
	\| [[File:Conics-TriangleTrans.png\|~~350px~~\|right]]		\| [[File:Conics-TriangleTrans.png\|300px\|right]]
	\|}		\|}

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	\| '''[[Triangle]]'''		\| '''[[Triangle]]'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Included for completeness, triangles are created when a section is taken through the Apex and ~~parallel to~~ the ~~central Axis~~.		* Included for completeness, triangles are created when a section is taken through the Apex and across the Base Cicle diameter.
			* Either an [https://en.wikipedia.org/wiki/Isosceles_triangle '''Isosceles'''] or [https://en.wikipedia.org/wiki/Equilateral_triangle '''Equilaterial Triangle'''] is revealed when a Right Cone is halved in this way. Other triangles would result from similar sectioning of an Oblique Cone.
	\| [[File:Conics-TriangleTrans.png\|350px\|right]]		\| [[File:Conics-TriangleTrans.png\|350px\|right]]
	\|}		\|}

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2015-04-04T14:00:30Z

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	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* The curve produced when a secton is taken across a right circular cone parallel to the base 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 [https://en.wikipedia.org/wiki/Apollonius_of_Perga '''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 [https://en.wikipedia.org/wiki/Frustum '''Frustrum''']
	\| [[File:Conics-Circle.png \|350px\|right]]		\| [[File:Conics-Circle.png \|350px\|right]]
	\|-		\|-

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	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* 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.		* 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.		* An Ellipse is also the result of taking a section across a [https://en.wikipedia.org/wiki/Cylinder_%28geometry%29 '''Cylinder'''] at at angle to its base.

	\| [[File:Conics-ElipseTrans.png\|350px\|right]]		\| [[File:Conics-ElipseTrans.png\|350px\|right]]

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2015-04-04T08:49:16Z

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	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* 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.		* 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.

	\| [[File:Conics-ElipseTrans.png\|350px\|right]]		\| [[File:Conics-ElipseTrans.png\|350px\|right]]
	\|-		\|-

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2015-04-03T17:48:55Z

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	[[File:ConicSections.jpg\|right~~\|thumb~~]]		[[File:ConicSections.jpg\|right]]

	https://en.wikipedia.org/wiki/~~Conic_section~~ '''~~Conic Sections~~'''		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 [https://en.wikipedia.org/wiki/Cone ''' can be described as the '''[[Loci\|Locus]]''' of all lines joining points on the circumference of a base circle to a point above it. If the point 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]]'''
	\| ~~- 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		\| <span style="color:#B00000">
			* 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]]
			\|-
			\| '''[[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]]
			\|-
			\| '''[[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]]
			\|-
			\| '''[[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]]
	\|-		\|-
	\| C		\| '''[[Triangle]]'''
	\| D		\| <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]]
	\|}		\|}

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 https://en.wikipedia.org/wiki/Conic_section '''Conic Sections'''
-A [https://en.wikipedia.org/wiki/Cone '''Cone'''] can be described as the '''[[Loci|Locus]]''' of all lines joining points on the circumfernce of a circle to a point above it.
+A [https://en.wikipedia.org/wiki/Cone ''' can be described as the '''[[Loci|Locus]]''' of all lines joining points on the circumference of a base circle to a point above it.  If the point lies perpendicularly above the centre of the base circle it is known as a '''''‘right cone’''''' and if not, it is an '''''‘oblique cone’'''''.
 [[Category:Secondary]]
 [[Category:Geometry]]