Question

need word file please :

What is a parametrized surface? How does one describe a surface this way, and why is it useful? What are the trade-offs? That is, how does studying parametrized surfaces simplify things, and what complications do we have to deal with? What is regularity in this context, and why is it important?

Answer 1

ANSWER :-

parametrized surface:-

A parametric surface is a surface in the Euclidean space  which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric in variants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

importance of parameterized surface:-

A rational surface is a surface that admits parameterizations by a rational function. A rational surface is an algebraic surface. Given an algebraic surface, it is commonly easier to decide if it is rational than to compute its rational parameterization

TRADE OFF

A trade-off (or trade off) is a situational decision that involves diminishing or losing one quality, quantity or property of a set or design in return for gains in other aspects. In simple terms, a trade off is where one thing increases and another must decrease. Trade offs stem from limitations of many origins, including simple physics - for instance, only a certain volume of objects can fit into a given space, so a full container must remove some items in order to accept any more, and vessels can carry a few large items or multiple small items. Trade offs also commonly refer to different configurations of a single item, such as the tuning of strings on a guitar to enable different notes to be played, as well as allocation of time and attention towards different tasks.

IMPORTANCE :-

Making decisions requires trading off one item against another. In economics, the term Making decisions requires trading off one item against another. In economics, the term trade-off is often expressed as an opportunity cost, which is the most preferred possible alternative. A trade-off involves a sacrifice that must be made to get a certain product or experience. is often expressed as an opportunity cost, which is the most preferred possible alternative. A trade-off involves a sacrifice that must be made to get a certain product or experience.