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Tuesday , August 22 2017

Surface

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.

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