Cycloid: Difference between revisions
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=====Description===== | |||
An [https://en.wikipedia.org/wiki/Cycloid '''Cycloid'''] is the path traced by a point on the circumference of a circle as it rolls without slipping along a straight line. The curve has been used to design [https://en.wikipedia.org/wiki/Cycloid_gear gear teeth] and has sometimes been used by architects to design the shape of a [http://en.wikipedia.org/wiki/Kimbell_Art_Museum curved roof]. The term ''‘cycloid’'' is used also to describe the shape of some [http://en.wikipedia.org/wiki/Fish_scale fish scales]. | An [https://en.wikipedia.org/wiki/Cycloid '''Cycloid'''] is the path traced by a point on the circumference of a circle as it rolls without slipping along a straight line. The curve has been used to design [https://en.wikipedia.org/wiki/Cycloid_gear gear teeth] and has sometimes been used by architects to design the shape of a [http://en.wikipedia.org/wiki/Kimbell_Art_Museum curved roof]. The term ''‘cycloid’'' is used also to describe the shape of some [http://en.wikipedia.org/wiki/Fish_scale fish scales]. | ||
A line produced by a point following a set of rules in this way is known as the '''[[Loci|locus of a point]]''' ''(plural is '''[ [loci]]''')''. | A line produced by a point following a set of rules in this way is known as the '''[[Loci|locus of a point]]''' ''(plural is '''[[loci]]''')''. | ||
=====Constructions===== | |||
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| <span style="color:#B00000"> | | <span style="color:#B00000"> | ||
* Draw a starting circle and divide into equal segments ''(say, 12)'' | * Draw a starting circle and divide into equal segments ''(say, 12)'' | ||
* Draw a horizontal line for the circle to roll along, of length equal to the circle circumference, and divide this into the same number of equal parts. ''( | * Draw a horizontal line for the circle to roll along, of length equal to the circle circumference, and divide this into the same number of equal parts. ''(Constructions used to create '''[[Scales]]''' are convenient ways of doing this)''. | ||
* Project lines parallel to the base line at heights representing the heights reached by the points on the circle as it rolls round. | * Project lines parallel to the base line at heights representing the heights reached by the points on the circle as it rolls round. | ||
* Start at 0 then describe a circle from the second centre position on to the first line above the baseline, which shows the height the point now reaches. | * Start at 0 then describe a circle from the second centre position on to the first line above the baseline, which shows the height the point now reaches. | ||
* Repeat this procedure for each progressive centre position as shown. | * Repeat this procedure for each progressive centre position as shown. | ||
* Sketch a fair curve through the points ''(or use [https://en.wikipedia.org/wiki/French_curve '''French Curves'''] or a [https://en.wikipedia.org/wiki/Flat_spline '''flexible strip'''] for example)''.</span> | * Sketch a fair curve through the points ''(or use [https://en.wikipedia.org/wiki/French_curve '''French Curves'''] or a [https://en.wikipedia.org/wiki/Flat_spline '''flexible strip'''] for example)''.</span> | ||
| [[ File: | | [[ File:CycloidConstruction.png|350px|right]] | ||
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The cycloid is a special case of [https://en.wikipedia.org/wiki/Trochoid '''Trochoid'''] in which the point traced is on the circumference of the rolling circle. Using a similar technique, '''''Superior''''' and '''''Inferior''''' Trochoids can also be generated by plotting points either outside ''(as the tip of a paddle wheel as a boat is propelled by it)'' or inside the rolling circle ''(as if marked on a rolling disc for example)'' | The cycloid is a special case of [https://en.wikipedia.org/wiki/Trochoid '''Trochoid'''] in which the point traced is on the circumference of the rolling circle. Using a similar technique, '''''Superior''''' and '''''Inferior''''' Trochoids can also be generated by plotting points either outside ''(as the tip of a paddle wheel as a boat is propelled by it)'' or inside the rolling circle ''(as if marked on a rolling disc for example)'' | ||
. | . | ||
Also, by similar trechniques, the '''base line''' can be replaced by a '''base circle''' and the rolling circle rolled around either the inside or the outside of it to generate [https://en.wikipedia.org/wiki/ | Also, by similar trechniques, the '''base line''' can be replaced by a '''base circle''' and the rolling circle rolled around either the inside or the outside of it to generate [https://en.wikipedia.org/wiki/Hypocy4cloid '''Hypocycloids'''] or [https://en.wikipedia.org/wiki/Epicycloid '''Epicycloids'''] - as used in [https://en.wikipedia.org/wiki/Epicyclic_gearing '''Epicyclic Gearing'''] for example , | ||
</span> | </span> | ||
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{{Drawing Instruments Buyers Guide}} | |||
[[Category:Secondary]] | [[Category:Secondary]] | ||
[[Category:Geometry]] | [[Category:Geometry]] | ||
Latest revision as of 20:00, 2 June 2016
Description
An Cycloid is the path traced by a point on the circumference of a circle as it rolls without slipping along a straight line. The curve has been used to design gear teeth and has sometimes been used by architects to design the shape of a curved roof. The term ‘cycloid’ is used also to describe the shape of some fish scales.
A line produced by a point following a set of rules in this way is known as the locus of a point (plural is loci).
Constructions
| Drawing a Cycloid |
|
 |
Note: The cycloid is a special case of Trochoid in which the point traced is on the circumference of the rolling circle. Using a similar technique, Superior and Inferior Trochoids can also be generated by plotting points either outside (as the tip of a paddle wheel as a boat is propelled by it) or inside the rolling circle (as if marked on a rolling disc for example) . Also, by similar trechniques, the base line can be replaced by a base circle and the rolling circle rolled around either the inside or the outside of it to generate Hypocycloids or Epicycloids - as used in Epicyclic Gearing for example ,
