Nicomedes concoid

Conic of Nicomedes . It is the concoid of the line , called “base”. Curve that in its extension constantly approaches the line without ever touching it.

Summary

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• 1 Story
• 2 Definition
• 3 equations
• 4 Construction
• 5 See also
• 6 Sources

History

The name of concoide is due to its resemblance to the shells that were so abundant on Greek beaches. Its creator is Nicomedes , a Greek geometer (280 – 210 BC), who investigated it along with 16 other mathematical curves .

Nicomedes wanted to use the concoid to solve the problems of cube duplication and angle resection, and in fact the curve serves to trisect any angle.

Definition

It is the locus of the points M for which: OM = OP + l or OM = OP-l

Three cases are distinguished: a> l, a = l, a <l

Equations

• Cartesian equation: (x – b) ² (x ² + y ² ) – a ² x ² = 0
• Polar equation:
• Parametric equations

Building

To build the Nicomedes Concoid we need to carry out the following steps:

1. Let k be a positive real number.
2. Draw a horizontal linem (directrix).
3. Let O and T be two points on the line m such that the distance (O, T)> 2k.
4. Draw the circumferenceC _Owith center O and radius k.
5. Let B terminal point C _O.
6. Plot the OB ray.
7. Draw the circumference C _Twith center at T and radius k.
8. Draw the line TB.
9. Let M and N be the intersections of the circumferenceC _Tand the line TB.
10. The locus generated by points M and N when T moves over linem is the Conic of Nicomedes.