Ruled surface

A ruled surface is swept out by a smoothly varying one-parameter family of affine lines.

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- Gaussian curvature of a ruled surface Ruled surface

Gaussian curvature of a ruled surface ( $K = - [a^{'} \cdot (b \times b^{'})]^{2} /∣ σ_{x} \times σ_{y} ∣^{4}$ )

For the regular

C^{2}

σ (x, y) = a (x) + y b (x)

, put

W = ∣ (a^{'} + y b^{'}) \times b ∣ > 0

. The second-form coefficient in the ruling direction is zero, and the mixed coefficient is

a^{'} \cdot (b \times 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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A **ruled surface** is a type of surface in three-dimensional space that can be generated by moving a straight line (the ruling) continuously along a path. In a more technical sense, a ruled surface can be defined as a surface that can be represented as the locus of a line segment in space, meaning that for every point on the surface, there exists at least one straight line that lies entirely on that surface.

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