Consider the paraboloid z=x2+y2. the plane 4x−8y+z−10=0 cuts the paraboloid, its intersection being a curve. find "the natural" parametrization of this curve.

The curve parameter is indeed a map r(t) = <x(t), y(t)> of the interval R= [a, b] parameter to a plane. Co - ordinate functions are considered the functions x(t), y(t). It defines a curve, a surface, etc. with one or even more variables that could be used in a specified range, and the calculation can be defined as follows:

Consider the paraboloid , and the plane  

Calculating the intersection curve of these curves:

Comapring the above equation:

Putting the x and  y value in the above equation then:

Hence the pararnetric equations of the intersecting curve are:

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Complete the square to get

Now let

and the intersection can be expressed as a function of ,

with .

i think it’s a i believe