= Gaussian curvature of a ruled surface
{c}
{title2=$K=-[\mathbf a'\cdot(\mathbf b\times\mathbf b')]^2/|\sigma_x\times\sigma_y|^4$}
For the regular $C^2$ <ruled surface> $\sigma(x,y)=\mathbf a(x)+y\mathbf b(x)$, put $W=|(\mathbf a'+y\mathbf b')\times\mathbf b|>0$. The second-form coefficient in the ruling direction is zero, and the mixed coefficient is $\mathbf a'\cdot(\mathbf b\times\mathbf b')/W$. 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.
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