১৮২৭ খ্রীষ্টাব্দে কার্ল ফ্রিড‌রিশ গাউস তার "Disquisitiones generales circa superficies curvas" ("বক্রতলের সাধারণ অনুসন্ধান") considered a surface parametrically, with the কার্তেসীয় স্থানাংক ব্যবস্থা , , and of points on the surface depending on দুটি সহায়ক চলরাশি and . Thus a parametric surface is (in today's terms) a সদিক রাশি বা ভেক্টর রাশি

depending on an ordered pair of real variables , and defined in an open set in the -plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface.

One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.

Canonical measure and volume form
ভূমিকা
Metric in coordinates
দৈর্ঘ এবং কোণ