Page "Envelope (mathematics)" Paragraph 49

This is a cubic in t and as such has at least one real solution.

It follows that at least one tangent line to γ must pass through any given point in the plane.

If and then each point ( x, y ) has exactly one tangent line to γ passing through it.

The same is true if.

If and then each point ( x, y ) has exactly three distinct tangent lines to γ passing through it.

The same is true if and.

If and then each point ( x, y ) has exactly two tangent lines to γ passing through it ( this corresponds to the cubic having one ordinary root and one repeated root ).

The same is true if and.

If and, i. e.,, then this point has a single tangent line to γ passing through it ( this corresponds to the cubic having one real root of multiplicity 3 ).

It follows that