You can explore ordinary and rhamphoid cusps further with an optional demonstration and exercises.

Demonstration 9: Ordinary and Rhamphoid Cusps

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The equation for the curve is X(t) =(3+t)(t^m ,tⁿ) , and the values of m and n determine the nature of the cusp. For example, the values m=2 , n=3 yield an ordinary cusp, and the values m=2 , n=4 yield a rhamphoid cusp.

This demo displays a curve that has either an ordinary cusp or a rhamphoid cusp at t=0 .

What relationship do the values of m and n have to the quadrant in which the curve lies?

Try to establish a rule about m and n that determines whether or not this curve has an ordinary cusp, a rhamphoid cusp or neither. In particular, look at high values of m and n.