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KIEROID
Curve studied by Kiernan in 1945, hence the name (see
Yates).
P. J. Kiernan: ?? 
Cartesian equation: ,
i.e. .
Rational quartic (singular point at O) located in the strip with the line x = a as an asymptote if , and bounded otherwise. Polar equation: . 
Given two points N and P with the same ordinate, respectively describing the lines x = a and x = b, the kieroid is the locus of the intersection points between the circle with centre P and radius c and the line (ON). 
When (i.e. when the circle is tangent to the line x = a) , the kieroid can be decomposed into a right rational circular cubic and its asymptote x = a, and all the right rational circular cubics can be obtained this way, hence a new construction of these curves.
When a = b+c, the cubic can be written
or with
the special cases:
a = 2b = 2c: cissoid of Diocles  a = c, b = 0: right strophoid  2a = 2b = c: MacLaurin trisectrix 



When a = b  c, the cubic can be written or ; example: the visiera. 
When a = b, we get the conchoids
of lines:
or . 
When b = c, the kieroid has a cuspidal point at O; when, additionally, a >> 2b, the kieroid gets closer to the piriform quartic 


Compare to the Rosillo
curves.
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© Robert FERRÉOL
2017