How to Draw an Ellipse Using the Piece-of-String-Looped-Over-Two-Nails Method

[hollo]

Homework Statement

i need to draw an ellipse with a circumference of 59cm and a minor axis of 12cm. does anyone know how to draw one using the piece-of-string-looped-over-two-nails method, ie how long would the string need to be & how far apart should the nails be?

i'll also need to know the length of the major axis but can derive it from the spacing of the nails & the length of the string loop

Homework Equations

The Attempt at a Solution

haven't studied maths formally for 10 years so thanks to anyone who can help

 

[1994Bhaskar]

circumference of ellipse is PI * SquareRoot of 2 * ((1/2 major axis)squared + (1/2 minor axis)squared). You he necessary data.Find length of major axis.Then proceed drawing it.

 

[hollo]

thx.

 

[1994Bhaskar]

Find eccentricity of ellipse from major and minor axis.Then find distance of foci of ellipse(length of major axis * eccentricity) from center of ellipse.Fix two nails there and loop pen/pencil over there such tat it touches end of minor axis.

 

[hollo]

The distance from the center C to either focus is f = (major axis)*(eccentricity), which can be expressed in terms of the major and minor radii-
f = sqrt[(major radius)^2 - (minor radius)^2]

 

[Filip Larsen]

circumference of ellipse is PI * SquareRoot of 2 * ((1/2 major axis)squared + (1/2 minor axis)squared).

This does not look correct. The exact circumference of an ellipse cannot be written up as a simple algebraic expression. There are various approximations, but the above expression does not seem to be one of those.

 

[Filip Larsen]

i need to draw an ellipse with a circumference of 59cm and a minor axis of 12cm. does anyone know how to draw one using the piece-of-string-looped-over-two-nails method, ie how long would the string need to be & how far apart should the nails be?

If this is homework you should spend some time dig out relevant geometrical equations for an ellipse, either in your textbook or on the net. Since it may be a tad difficult to solve these if you want a precise answer, you can get an approximation using only a rough approximation for the circumference of an ellipse. If you want better precision you can use a numerical approach where you take a high precision approximation for the circumference and then iteratively (using for instance a spread sheet and bisection or similar) try out values of a until you get close enough.

If this is not homework but something you want to construct, then following above method I get a fairly eccentric ellipse close to your specified values of b and C by having a loop of around 46 cm and a separation of foci of around 21 cm (I've rounded the values in case it is homework so you can check your own results without "spoiling" it for you).

 

[hollo]

hi it isn't homework it's for a vent I'm making for an extractor fan in my bathroom

 

[1994Bhaskar]

This does not look correct. The exact circumference of an ellipse cannot be written up as a simple algebraic expression. There are various approximations, but the above expression does not seem to be one of those.

This is correct expression.Put major axis=minor axis=a (then it will become a circle). You will get the value as 2*pi*a.Same as circumference of circle.

 

[Filip Larsen]

hi it isn't homework it's for a vent I'm making for an extractor fan in my bathroom

Ok, in that case I calculate a loop length of 45.8 cm and foci separation of 21.3 cm. In case you are interested, the major-axis is 24.5 cm and eccentricity is 0.872.

Please note that this is a "theoretical" ellipse. For one, that loop length is assuming your pins and pencil are very small, so if you use thick pins and pencil you may have to add a bit more string (I'm guessing around pi*d, assuming both the pins and the pencil has diameter d). Also, there may be other practical issues when installing such a vent that you have to consider before you start making holes in the wall, so please make sure the numbers make sense in your case.

This is correct expression.Put major axis=minor axis=a (then it will become a circle). You will get the value as 2*pi*a.Same as circumference of circle.

Ok, my bad, found the approximation you mentioned on Mathworld [1], equation (69). The above numbers where calculated using the Ramanujan approximations, equation (71).

[1] http://mathworld.wolfram.com/Ellipse.html

 

[LCKurtz]

hi it isn't homework it's for a vent I'm making for an extractor fan in my bathroom

How do you know it is an ellipse instead of some arbitrary "oval"? And apparently you can't just trace the the outline of the hardware with a pencil, eh?

 

[vela]

This is correct expression.Put major axis=minor axis=a (then it will become a circle). You will get the value as 2*pi*a.Same as circumference of circle.

I think Filip has a point. You made it sound like that's how to calculate the perimeter of an ellipse exactly. It's not. It's only an approximation.