Gaussian curvature of a ruled surface ($K=-[{\mathbf{a}}^{\prime }\cdot (\mathbf{b}\times {\mathbf{b}}^{\prime }){]}^2/|{\sigma }_x\times {\sigma }_y{|}^4$)

For the regular $C^2$ ruled surface $\sigma (x,y)=\mathbf{a}(x)+y\mathbf{b}(x)$, put $W=|({\mathbf{a}}^{\prime }+y{\mathbf{b}}^{\prime })\times \mathbf{b}|>0$. The second-form coefficient in the ruling direction is zero, and the mixed coefficient is ${\mathbf{a}}^{\prime }\cdot (\mathbf{b}\times {\mathbf{b}}^{\prime })/W$. The determinant formula for Gaussian curvature therefore gives the displayed nonpositive expression. Its vanishing is equivalent to the zero scalar triple product, only at regular points.