Conchoid of Nicomedes
Slide a ruler through a fixed point and mark a fixed distance b beyond a line: the shell-shaped curve Nicomedes used to trisect angles.
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Formulas and key values
- Polar equation
- Cartesian equation
- Asymptote
The pole O is the origin, the fixed line is x = a (a = 1 in the graph), and b is the distance marked off along each ray.
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Start the video01 · Definition
What is the Conchoid of Nicomedes?
Fix a point O and a line at distance a from it. Draw any line through O; where it crosses the fixed line, measure a distance b along it on both sides. The marked points trace the conchoid (Greek konche, shell). Nicomedes, around 200 BC, built a mechanical instrument to draw it and used it to trisect angles and double the cube.
The two branches share the line x = a as an asymptote. The ratio of b to a decides the inner branch: for b < a it is a smooth arch, for b = a it has a cusp at O, and for b > a it forms a loop through O.
02 · Properties
Key properties
- Polar form with the pole at O: r = a·sec θ ± b.
- Cartesian form: (x − a)²(x² + y²) = b²x².
- Both branches approach the asymptote x = a.
- Inner branch: smooth for b < a, a cusp at O for b = a, a loop through O for b > a.
- Nicomedes’ neusis construction with it trisects any angle.
03 · Questions
Conchoid of Nicomedes: common questions
- What is the equation of the Conchoid of Nicomedes?
- Polar equation: r = a sec θ ± b; Cartesian equation: (x - a)^2(x^2 + y^2) = b^2x^2. The pole O is the origin, the fixed line is x = a (a = 1 in the graph), and b is the distance marked off along each ray.
04 · Keep exploring