
CUSP






The cusp or spinnode is another type of singularity which is, too, a double point. It is characterized by two branches of a curve meeting where the tangents are equal to one another. This cusp has the equation x
^{3}y
^{2}=0.



TACNODE






The curve is called a tacnode, it is also a double point like the cusp. The tacnode is a point on a graph where the two, or possibly more, osculating circles meet at a tangent. The tacnode below is at the origin. The osculating part of the graph comes from the latin circulum osculans, which means " kissing circles ".
It's equation is 2x
^{4}3x
^{2}y+y
^{2}2y
^{3}+y
^{4}=0.



RAMPHOID






A ramphoid is also a type of cusp. It comes from the greek "ramphos" which means "the crooked beak of birds, especially birds of prey, " and that is what the curve looks like. Also ramphoids are generally curves that have both branches one one side of the tangent. The equation is x
^{4 }+x
^{2 }y
^{2 }2x
^{2 }yxy
^{2 }+y
^{2 }=0.



TRIPLE






The origin above is is an ordinary triple point it is represented by (x
^{2}+y
^{2})
^{2}+3x
^{2}yy
^{3}=0




CLOVER






The origin above is is an ordinary quadruple point and it has multiplticty four. It's tangents coincide in pairs. And it is represented by (x
^{2}+y
^{2})
^{3}4x
^{2}y
^{2}=0.



FINAL






Here is another singular point with higher multiplicity. It is represented by x
^{6}=x
^{2}y
^{3}y
^{5}=0. and the origin has one triple tangent and two simple tangents. 


Reference 

Robert John Walker, "Singular Points" in Algebraic Curves, Princeton University Press, 1950, pp. 5658. 

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This page was contributed by
Oksana Maeva 2008.
