CONFORMAL MAPPING

Conformal mapping can be explained as a mathematical equation that preserves the angle. It could be explained as a mathematical equation that preserves the angle between the oriented curve in the magnitude and the orientation, i.e., direction. It is also called an angle preserving map. Conformal mapping plays an active part in the scientific field. It is used for evaluating various values like density, irrotational flow, etc., in fluid dynamics. It is also used in various transformations like the Laplace equation and Maxwell’s equation. The most common use of conformal mapping is in marine navigation.

WHAT IS CONFORMAL MAPPING?

A mathematical equation can be termed conformal if it preserves the angle between the oriented curves in magnitude and orientation. Therefore, a conformal mapping can be defined as a map which is a function that preserves the angles. It is stated that any analytical function can be said to be conformal at any instant point if that point has a non zero derivative. Considering an infinitely small figure, apart from considering the size, the conformal map preserves both angles and the shape of that figure. Mathematically, the conformal mapping can be expressed as,

w=f(z)

According to the above expression, a map will be called conformal mapping if it preserves the angle between the curve through z0 and the orientation, i.e., the direction. The conformal mapping is explained better by the Riemann mapping theorem. This theorem states that a map is considered a conformal map only if the complex plane is in the term of a Mobius transformation. A Mobius transformation is termed a transformation in which the outcome is a rational function. A conform mapping is also called the conformal transformation or angle preserving transformation.

CONFORMAL MAPPING APPLICATIONS:

Conformal mapping is also termed the angle preserving map, an active part of the scientific world. There are several applications of conformal mapping in the field of science. Several conformal mapping applications are:

• It has been very beneficial in the field of fluid mechanics, electrostatics, various physical situations, as well as in heat-conducting.

• The confirming mapping provided the path for simplifying the Laplace equation.

• Control mapping is also used in the evaluation of a root locus.

• Conformal mapping is used in marine navigation because it has uniquely derived features.

• It is also quite useful in studying various physics, like an electromagnetic field in a gravitational field.

• Conformal mapping is also used in evaluating the solution of Maxwell’s equation.

• Conformal mapping has also played a vital role in determining the aspects of fluid mechanics, like determining the density, irrotational flow, and zero viscosity.

• Conformal mapping also has an effective part in geometry. It is used for solving various nonlinear partial differential equations.

TYPES OF CONFORMAL MAPS:

The confirming mappings can be categorised into two groups. Conformal mapping in two dimensions and conformal mapping in 3 dimensions. A conformal mapping in two dimensions can be termed as a function if the complex plane of the map is holomorphic. A plane is a holomorphic function if there are one or more complex variables. These complex variables can be differentiated by the. Domain present in each subset of the similar plane. According to the Riemann theorem, a map in two dimensions is conformal only if its complex plane is in the form of Mobius transformation. One of the common conformal mapping examples in two dimensions is circle inversions. The conformal maps in three or more dimensions are a complex version of a conformal mapping. In this mapping category, two metrics are considered, which are called conformally equivalent, over a plane manifold. Considering a plane manifold Z with two metrics, a and b. Therefore, conformally equivalent will be a=ub. Here, u is some positive function on the plane Z, and this function will be called a conformal factor. One of the common conformal mapping examples in 3 dimensions is stereographic projection.

CONFORMAL MAPPING EXAMPLES:

Conformal mapping is one of the basic aspects of modern science. It has contributed to various fields of engineering and navigation. Some of the basic conformal mapping examples are:

• Conformal mapping is used in marine navigation. Mercator projection and stereographic projections are conformal functions used in cartography.

• The conformal mapping is used to evaluate boundary value problems in fluid mechanics.

• Conformal mappings are the casual transformations that are used widely in general relativity.

• Joukowsky transform is one of the conformal maps used to evaluate several solutions in fluid dynamics.

• Inversion mapping is a conformal mapping used in geometry to convert discrete root locus into continuous root locus.

CONCLUSION:

Conformal mapping is a mathematical equation whose analytical function is non zero derivatives at any point on a plane manifold. Conformal mapping is also called angle preserving transformation because it preserves the angle. Conformal Map is said to preserve the angle attained between the orientation and the oriented curve in magnitude, i.e., direction. Conformal mapping is categorised into two groups, conformal mapping in two dimensions and conformal mapping in three or more dimensions. Conformal mapping in three or more dimensions is a complex version of conformal mapping in two directions. Basic knowledge of analytical functions, Holomorphic function, Complex-valued function is necessary for understanding the concept of a conformal map.