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KIEROID
Curve studied by Kiernan in 1945, hence the name (see
Yates).
P. J. Kiernan: ?? |
Cartesian equation: Rational quartic (singular point at O) located in the strip 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). | ![]() |
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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: Mac-Laurin trisectrix |
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When a = b - c, the cubic can be written |
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When a = b, we get the conchoids
of lines:
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When b = c, the kieroid has a cuspidal point at
O; when, additionally, a >> 2b, the kieroid |
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Compare to the Rosillo
curves.
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© Robert FERRÉOL
2017