▁is ▁given ▁by ▁ ▁and ▁the ▁mean ▁curv ature ▁by ▁ ▁Since ▁ ▁and ▁ ▁are ▁invariant ▁under ▁is omet ries ▁of ▁, ▁in ▁general ▁ ▁and ▁ ▁where ▁the ▁derivatives ▁at ▁the ▁point ▁are ▁given ▁by ▁ ▁For ▁every ▁orient ed ▁embedded ▁surface ▁the ▁G auss ▁map ▁is ▁the ▁map ▁into ▁the ▁unit ▁sphere ▁sending ▁each ▁point ▁to ▁the ▁( out ward ▁pointing ) ▁unit ▁normal ▁vector ▁to ▁the ▁orient ed ▁tang ent ▁plane ▁at ▁the ▁point . ▁In ▁coordinates ▁the ▁map ▁sends ▁ ▁to ▁ ▁Direct ▁computation ▁shows ▁that : ▁the ▁Gaussian ▁curv ature ▁is ▁the ▁Jacob ian ▁of ▁the ▁G auss ▁map . ▁ ▁Ex amples ▁ ▁Sur faces ▁of ▁revolution ▁ ▁A ▁surface ▁of ▁revolution ▁can ▁be ▁obtained ▁by ▁rot ating ▁a ▁curve ▁in ▁the ▁- plane ▁about ▁the ▁- axis , ▁assuming ▁the ▁curve ▁does ▁not ▁intersect ▁the ▁- axis . ▁Suppose ▁that ▁the ▁curve ▁is ▁given ▁by ▁ ▁with ▁ ▁lies ▁in ▁, ▁and ▁is ▁paramet riz ed ▁by ▁ar cl ength , ▁so ▁that ▁ ▁Then ▁the ▁surface ▁of ▁revolution ▁is ▁the ▁point ▁set ▁ ▁The ▁Gaussian ▁curv ature ▁and ▁mean ▁curv ature ▁are ▁given ▁by ▁ ▁Ge odes ics ▁on ▁a ▁surface ▁of ▁revolution ▁are ▁govern ed ▁by ▁Cla ira ut ' s ▁relation . ▁ ▁Qu ad ric ▁surfaces ▁ ▁Consider ▁the ▁quad ric ▁surface ▁defined ▁by ▁ ▁This ▁surface ▁adm its ▁a ▁paramet riz ation ▁ ▁The ▁Gaussian ▁curv ature ▁and ▁mean ▁curv ature ▁are ▁given ▁by ▁ ▁Ru led ▁surfaces ▁ ▁A