Gardener construction
The gardener design is a method to form a circle or an ellipse with a rope, to draw one or two nails and a pen. The name apparently goes back to the fact that a bed in the shape of an ellipse can be created in the garden with simple tools.
A circle is defined by its center and radius, an ellipse by its two focal points and the length of the major semi-axis (usually referred to as a ) or the lengths of the major (a) and small (b) semi-axis.
Required material
- a " nail " (or a thumbtack or a wooden stake) for a circle, two nails for an ellipse,
- a thread (or rope ), as well
- a pen .
Action
circle
- Tie the pin and the nail to the ends of the thread, preferably with a loop ;
- Put the nail in the center of the circle to be created;
- Pull the pen vertically and with the thread constantly taut around the center .
Ellipse with known focal point position
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- Drive the two nails into the focal points;
- Tie one end of each thread with twice the length of the major semi-axis (= 2a) to each of the nails;
- Use the pen to tighten the thread and place the pen vertically;
- With the thread taut, first draw one half of the ellipse and after repeating the other half.
In the descriptions of the elliptical construction, a thread of length 2a is tied with the ends to a nail. It is also possible to tie a thread of length 2a + 2e in a loop and put it around both nails. This requires a much longer thread, but has the advantage that the entire ellipse can be drawn without being separated. The length of the thread 2a + 2e is easy to grasp by stretching the rope from point B (see picture below) around F1 and back again and knotting it together.
Ellipse with known major and minor axis length
- Define center point M and construct a right-angled cross in the future position of the ellipse;
- Using a rope of length b (small semi-axis), determine point C from M ;
- A rope of length a (main axis of the ellipse) from C , determine the focal points F1 and F2 and drive them into these nails;
- Now proceed as with the construction with known focal position.
Remarks
- ↑ Learning Environment - Construction of the Ellipse , accessed December 7, 2015.