Technical Drawing in Photonics

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Technical Drawing in Photonics

Lesson 6

Drawing of different cycloids.

TAMOP-4.1.1.C-12/1/KONV-2012-0005 project

„Preparation of the concerned sectors for educational and R&D activities related to the Hungarian ELI project”

Dr. Zsolt István Benkő

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Technical drawing in Photonics Lesson 6

A cycloid is a trajectory of a point which is fixed to the perimeter of a rolling circle. (The circle rolls on a line.)

If the point is not on the perimeter but outside the circle the curve traced out is a prolate cycloid.

If the point is not on the perimeter but inside the circle the curve traced out is a curtate cycloid.

All the previously described curves can be referred as trochoids.

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If the point is fixed to the perimeter of a circle which rolls inside of an other circle then the trajectory is called hypocycloid.

If the fixed point is not on the perimeter of a circle which rolls inside of an other circle then the trajectory is called hypotrochoid.

If the point is fixed to the perimeter of a circle which rolls outside of an other circle then the trajectory is called epicycloid.

If the fixed point is not on the perimeter of a circle which rolls outside of an other circle then the trajectory is called epitrochoid.

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Drawing of a cycloid.

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Cycloid

3

1 y t( ) y2 t( )

14

1 x t( ) x2 t ( )

0 2 4 6 8 10 12 14

2

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Prolate cycloid

3

1 y t( ) y2 t( )

14

1 x t( ) x2 t ( )

0 2 4 6 8 10 12 14

2

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Curtate cycloid

3

1 y t( ) y2 t( )

14

1 x t( ) x2 t ( )

0 2 4 6 8 10 12 14

2

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Epicycloid

12

12 y t( ) y2 t( )

12 12 x t( ) x2 t ( )

10 0 10

10 0

10 The ratio of the

radii is integer.

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12

12 y t( ) y2 t( )

12

12 x t( ) x2 t ( )

10 0 10

Epicycloid

The ratio of the radii is a rational number.

The curve is closing.

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12

12 y t( ) y2 t( )

12 12 x t( ) x2 t ( )

10 0 10

Epicycloid

The ratio of the

radii is an irrational number.

The curve is never closing.

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12

12 y t( ) y2 t( )

12 12 x t( ) x2 t ( )

10 0 10

Hypocycloid

The ratio of the radii is integer.

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References

1. Ocskó Gy., Seres F.: Gépipari szakrajz, Skandi-Wald Könyvkiadó, Budapest, 2004

2. Lőrincz P., Petrich G.: Ábrázoló geometria, Nemzeti Tankönyvkiadó Rt., Budapest, 1998

3. Pintér M.: AutoCAD tankönyv és példatár, ComputerBooks, Budapest, 2006