CONFORMAL MAPPING.pptx

Conformal and bilinear transformation

1. CONFORMAL MAPPING SUBMITTED BY Name: Diganta Bhuyan Roll No- 223203 Dipartment –Mathematics PG 2 nd semester

2. OVERVIEW Introduction . Mapping. Conformal mapping. Elementary transformation of conformal mapping . Bilinear transformation. Applications. Conclusion.

3. INTRODUCTION In mathematics a Conformal mapping is function which preserve angle. In 1569 , the flemish cartographer Gerardus Mercator devised a cylindrical map projection that preserve angle. Another map projection known to the ancient Greek is the stereographic projection and both example are Conformal. Conformal map is very important in complex analysis as well as physics and engineering.

4. MAPPING Before we start conformal mapping we need to understand about Mapping. In linear it is a mathematical relation such that each element of a given set is associated with an element of another set . A complex function w=f(z) can be regarded as a mapping or transformation of the points in the z= x+iy plane to the points of the w= u+iv plane.

5. CONFORMAL MAPPING If the transformation u=u( x,y ) and v=v( x,y ) the point ( x₀,y ₀) of xy -plane is mapped into the point ( u₀,v ₀) of uv -plane and the intersecting curve c₁ and c₂ are respectively maped into the curve c₁ ʹ and c₂ ʹ and they intersect at ( u₀,v ₀). if the transformation is such that the angle at ( x₀,y ₀) between c₁ and c₂ is equal to the angle at ( u₀,v ₀) between c₁ ʹ and c₂ ʹ is equal in magnitude as well as sense then the transformation is called conformal maaping or conformal transformation. In Conformal mapping the magnitude of angle is same and the direction of angle is also same. If the direction is opposite between the curve then it is called Isogonal Transformation.

6. ELEMENTARY TRANSFORMATION Translation : : w=z+ α , α is complex number. By this transformation every point in z-plane is displaced in direction of α . Rotation :: w= ze ^ i α , α is real By this figure in z-plane rotate through an angle α in w-plane. Magnification:: w= α z, The figure in w-plane is magnified α -times than the size of z-plane. Inversion:: w=1/z , Reflect opposite of the point of the z-plane. Above transformation is important for understanding some properties of Conformal Mapping.

7. BILINEAR TRANSFORMATION It is a Important topic in conformal mapping. It is a c ombination of translation, rotation, magnification and inversion. The transformation of the form w= az+bz / cz+d is called a Bilinear or Mobius Transformation.where a,b,c,d are complex constant and ad-bc≠0

8. APPLICATION Conformal mapping is an important technique used in complex analysis and has many applications I different physical situation. Flows in complicated Geometries. Numerical Analysis and algorithm development strategies. Flow of Ideal fluid. Conformal mapping can be used in scattering and diffraction problems.

9. CONCLUSION Overall, Conformal map are an important tool in mathematics and physics, and they have a wide range of application in many different fields.