Distribution Parameter

The distribution parameter of a Noncylindrical Ruled Surface parameterized by

$${\bf x}(u,v)=\boldsymbol{\sigma}(u)+v\boldsymbol{\delta}(u),$$ (1)

where $\boldsymbol{\sigma}$ is the Striction Curve and $\boldsymbol{\delta}$ the Director Curve, is the function $p$ defined by

$$p={\mathop{\rm det}(\boldsymbol{\sigma}' \boldsymbol{\delta}... ...\delta}')\over \boldsymbol{\delta}'\cdot\boldsymbol{\delta}'}.$$ (2)

The Gaussian Curvature of a Ruled Surface is given in terms of its distribution parameter by

$$K=-{[p(u)]^2\over\{[p(u)]^2+v^2\}^2}.$$ (3)

See also Noncylindrical Ruled Surface, Striction Curve

References

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 347-348, 1993.