Conformal mapping is a function in mathematics that deals with planes and complex analysis. This type of function preserves the angle between a curve and the points on a plane. When constructing a conformal map, the curves in the map intersect at an angle. Therefore, a map is referred to as a conformal map if and only if it preserves the angle of orientation and the orientations of the curves. The conformal mapping technique has several applications in mathematics and solving real-life problems. This article aims at discussing some of the typical applications of conformal mapping.
1. Complex analysis
Conformal mapping has a gist application in complex analysis. Under this field, the technique of conformal mapping is used to perform analyses of complex numbers with their related functions. The properties of a complex function are simplified, which eases the process of analyzing complex numbers.
Conformal mapping plays a vital role in cartography. Cartography is the technique of analyzing and making maps. The conformal mapping method is extensively used to check the relationship of areas in the map and decode the map’s information.
Conformal mapping is helpful in the engineering field. The knowledge of conformal mapping has a huge application in solving potential theory calculations and other engineering problems. In addition, this technique is extensively used in structural engineering to compute the elevation angle between two points in a curve.
4. Solving Boundary value problems
The boundary value problem falls in the complex problem class. This type of problem involves a boundary with ordinary differential equations that yields derivatives values and solutions at the same time. Thankfully, the conformal mapping knowledge got you covered while computing such problems. Here, the technique is used to analyze the boundaries.
5. Stereographic projection
In mathematics, one application of conformal mapping is in stereographic projection. In this field, a sphere or any other shape is projected from a point onto a plan. The projections map a conformal map onto the plane; thus, conformal mapping techniques are used in computing the plain-sphere problems.
The conformal mapping works hand in hand with the mapping of physics geometries. The knowledge of conformal mapping eases the process of transforming geometries into other practical geometries. In addition, the technique is vital in solving complex problems associated with physics geometries.
The technique is useful when computing electric fields at a point (charged corner). While using other methods, the problems seem clumsy and impossible to solve due to their complex nature. However, using the conformal mapping technique can ease the computation problem. The technique converts the corners into straight ones, easing the mapping.
8. Brain surface mapping
In medical studies, the knowledge of conformal mapping is used in brain-surface mapping. Here, the human brain is mapped onto a plane. However, the reconstruction of the brain is based on genus zero conformal.
Conformal mapping has numerous applications in the mathematical field. It is used to solve complex and two dimensions problems.