Two-horned curve

A two-horned curve , also known as a triangular hemisphere due to its similarity to a two-headed hemisphere , is a fourth-degree rational curve defined by the equation

y^{2}(a^{2}-x^{2})=(x^{2}+2ay-a^{2})^{2}.

{\ displaystyle y ^ {2} (a ^ {2} -x ^ {2}) = (x ^ {2} + 2ay-a ^ {2}) ^ {2}.}

{\ displaystyle y ^ {2} (a ^ {2} -x ^ {2}) = (x ^ {2} + 2ay-a ^ {2}) ^ {2}.}

The curve has two cusps and is symmetrical about the y axis.

Content

History

In 1864, James Joseph Sylvester studied the curve

y^{4}-xy^{3}-8xy^{2}+36x^{2}y+16x^{2}-27x^{3}=0

{\ displaystyle y ^ {4} -xy ^ {3} -8xy ^ {2} + 36x ^ {2} y + 16x ^ {2} -27x ^ {3} = 0}

in connection with the classification of . He called the curve two-horn because of the presence of two casps. This curve was later studied by Arthur Cayley in 1867.

Properties

Converted two-horned curve with a = 1

A two-horned curve is a fourth-degree flat algebraic curve of zero genus . The curve has two cusp features in the real plane and a double point in the complex projective plane for x = 0, z = 0. If we transfer x = 0 and z = 0 to the origin and carry out imaginary rotation in x by substituting ix / z for x and 1 / z for y, we get

(x^{2}-2az+a^{2}z^{2})^{2}=x^{2}+a^{2}z^{2}.\,

{\ displaystyle (x ^ {2} -2az + a ^ {2} z ^ {2}) ^ {2} = x ^ {2} + a ^ {2} z ^ {2}. \,}

This curve, a snail of Pascal , has an ordinary double point at the origin and two points of intersection with the axes at the points x = ± i and z = 1.

The parametric equation of the two-horned curve:

$x=a\sin(\theta )$ ${\ displaystyle x = a \ sin (\ theta)}$ and $y={\frac {\cos ^{2}(\theta )\left(2+\cos(\theta )\right)}{3+\sin ^{2}(\theta )}}$ ${\ displaystyle y = {\ frac {\ cos ^ {2} (\ theta) \ left (2+ \ cos (\ theta) \ right)} {3+ \ sin ^ {2} (\ theta)}}}$ with $-\pi \leq \theta \leq \pi$ ${\ displaystyle - \ pi \ leq \ theta \ leq \ pi}$

Notes

Literature

J. Dennis Lawrence. A catalog of special plane curves. - Dover Publications, 1972. - ISBN 0-486-60288-5 .
"Bicorn" at The MacTutor History of Mathematics archive
Weisstein, Eric W. Bicorn on Wolfram MathWorld .
"Bicorne" at "mathcurve"
The Collected Mathematical Papers of James Joseph Sylvester. Vol. II Cambridge (1908) p. 468 ( online )