I'm working on a project and I need to decide whether an airfoil has a cusped or pointed trailing edge given the x and y coordinates of the airfoil.
What is the difference between the two trailing edges?
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Both cusped and conventional airfoils have pointed trailing edges. The lower trailing surface is concave on cusped airfoils, so plot it and get your straightedge. 
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In geometry a cusp is formed when two lines converge to the same tangent, or, as Wikipedia puts it:
a point on a curve where a moving point on the curve must start to move backward
For airfoils with an infinitesimally thin trailing edge, the two options are either a point or a cusp:
In real life, an airfoil will have neither, as the trailing edge will have a finite thickness, but for design and analysis purposes, a point or cusp are simpler to work with.
The issues with cusps were well known to early aerodynamicists, as the Zhukovsky transformation produces cusped airfoils, which are impossible to construct in reality, as the converging tangents at the trailing edge also demand infinitesimal thickness.
