Cubic surface parametrized by (s,t,s^{2}+s^{2}t+t^{3}) with parabolic curve

*The contour curve is first empty, and becomes a "lip" curve with 2 cusps. *

*The lip event happens when the contour is an isolated point. *

Parabolic surface of the example surface above

*The parabolic surface is ruled by the unique principal tangents at the parabolic points of a given surface. *

Cubic surface parametrized by (s,t,s^{2}-s^{2}t+t^{3}) with parabolic curve

*Two cusps come together and form a tacnode, which then breaks into 2 smooth branches. *

*The beak-to-beak event happens exactly at the tacnode. *

Parabolic surface of the example surface above

*The parabolic surface is ruled by the unique principal tangents at the parabolic points of a given surface. *

Quartic surface parametrized by (s,t,s^{2}+3st+t^{4}) with flecnodal curve and 2 flecnodal lines at a point

*The contour is first smooth, and becomes a "swallowtail" curve with 2 cusps and a node.*

*The swallowtail event happens when the 2 cusps and the node come together to a higher order singularity. *

Flecnodal surface of the example surface above

*The flecnodal surface is ruled by the flecnodal lines of a given surface. *