region filled by the tangent lines is the whole plane. The boundary E3 is therefore the empty set. Indeed, consider a point in the plane, say (x0,y0). This point lies on a tangent line if and only if there exists a t such that 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 Example 2 In string art it is common to cross-connect two lines of equally spaced pins. What curve is formed? For simplicity, set the pins on the x- and y-axes; a non-orthogonal layout is a rotation and scaling away. A general straight-line thread connects the two points (0, k−t) and (t, 0), where k is an arbitrary scaling constant, and the family of lines is generated by varying the parameter t. From simple geometry, the equation of this straight line is y = −(k − t)x/t + k − t. Rearranging and casting in the form F(x,y,t) = 0 gives: Now differentiate F(x,y,t) with respect to t and set the result equal to zero, to get These two equations jointly define the equation of the envelope. From (2) we have: Substituting this value of t into (1) and simplifying gives an equation for the envelope: Or, rearranging into a more elegant form that shows the symmetry between x and y: We can take a rotation of the axes where the b axis is the line y=x oriented northeast and the a axis is the line y=−x oriented southeast. These new axes are related to the original x-y axes by and . We obtain, after substitution into (4)