▁ruled ▁surface ▁is ▁one ▁which ▁can ▁be ▁generated ▁by ▁the ▁motion ▁of ▁a ▁straight ▁line ▁in ▁. ▁Cho osing ▁a ▁direct rix ▁on ▁the ▁surface , ▁i . e . ▁a ▁smooth ▁unit ▁speed ▁curve ▁ ▁orthogonal ▁to ▁the ▁straight ▁lines , ▁and ▁then ▁choosing ▁ ▁to ▁be ▁unit ▁vectors ▁along ▁the ▁curve ▁in ▁the ▁direction ▁of ▁the ▁lines , ▁the ▁velocity ▁vector ▁ ▁and ▁ ▁satisfy ▁ ▁The ▁surface ▁consists ▁of ▁points ▁ ▁as ▁ ▁and ▁ ▁vary . ▁ ▁Then , ▁if ▁ ▁the ▁Gaussian ▁and ▁mean ▁curv ature ▁are ▁given ▁by ▁ ▁The ▁Gaussian ▁curv ature ▁of ▁the ▁ruled ▁surface ▁van ishes ▁if ▁and ▁only ▁if ▁ ▁and ▁ ▁are ▁proportional , ▁This ▁condition ▁is ▁equivalent ▁to ▁the ▁surface ▁being ▁the ▁en velope ▁of ▁the ▁plan es ▁along ▁the ▁curve ▁containing ▁the ▁tang ent ▁vector ▁ ▁and ▁the ▁orthogonal ▁vector ▁, ▁i . e . ▁to ▁the ▁surface ▁being ▁develop able ▁along ▁the ▁curve . ▁More ▁generally ▁a ▁surface ▁in ▁ ▁has ▁van ishing ▁Gaussian ▁curv ature ▁near ▁a ▁point ▁if ▁and ▁only ▁if ▁it ▁is ▁develop able ▁near ▁that ▁point . ▁( An ▁equivalent ▁condition ▁is ▁given ▁below ▁in ▁terms ▁of ▁the ▁metric .) ▁ ▁Min imal ▁surfaces ▁ ▁In ▁ 1 7 6 0 ▁Lag range ▁extended ▁E uler ' s ▁results ▁on ▁the ▁calculus ▁of ▁variations ▁involving ▁integr als ▁in ▁one ▁variable ▁to ▁two ▁variables . ▁He ▁had ▁in ▁mind ▁the ▁following ▁problem : ▁ ▁Such ▁a ▁surface ▁is ▁called ▁a ▁minimal ▁surface . ▁ ▁In ▁