Definition of surface: A surface is defined as the locus of a point whose Cartesian coordinates (x, y, z) are functions of two independent parameters u and v.
Thus we can write
x = f (u, v) y = g(u, v) z = h(u, v)
or, x =x(u, v) y = y(u, v) z = z(u, v)
Parametric equation of a surface (vector form): A surface is defined as the locus of a point whose position vector x can be expressed as functions of two independent parameters. Thus an equation of the form
x = x (u, v) ———————————- (i)
represents a surface.
The above equation is known as the Gaussian form of the surface. The parameters u and v are called curvilinear coordinates or surface coordinates of the current point on the surface.
Monge’s form of the surface: If the equation of a surface can be written in the form z = f (x, y), then it is called Monge’s form of the equation of the surface. In this form x and y themselves can be regarded as independent parameters.
i.e., x = x, y = y, z = f (x, y)
In vector form x = (x, y, z) = (x, y, f (x, y)) is the equation of a surface with parameters x, y.