Conformal Mapping - Introduction & Examples

A transformation that preserves angles between the two lines (local angles) is termed as conformal mapping or conformal map.

1. Conformal Mapping Mandar Vijay Datar Department of Engineering Sciences International Institute of Information Technology, I²IT www.isquareit.edu.in

2. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Introduction What is a Transformation Transformation is a complex valued function of a complex variable. It is a map from complex z-plane to w- plane. For example Transformations are studied to see the effect on various geometrical objects. In computer graphics it is useful to see the effect of operations on figures and shpaes .

3. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Conformal Mapping What is a Conformal Mapping Consider a mapping w = f(z) Let curve-1 and curve-2 be two smooth curves in z-plane. Suppose they are intersecting at a point z = z The angle between them is say θ If the images of these curves in w-plane are d-1 and d-2 Then the map f(z) is said to be conformal if the angle θ is preserved.

4. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Conformal Mapping Condition for Conformality A mapping w = f(z) is conformal at each point z where f(z) is analytic and

5. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Linear Transformation Linear Transformation : Definition It is defined as w = f(z) = az + b Where a, b are complex constants and The linear transformation is conformal The condition is essential for conformality . Depending on the values of a, b there are different types of linear transformations exist. These transformations are useful in Graphics.

6. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Linear Transformation Types of Linear Transformation Name Condition Definition Identity Transformation a = 1, b = 0 w = f(z) = z Translation a = 1 w = f(z) = z + b Rotation w = f(z) = Stretching or Scaling a > 1, b = 0 w = f(z) = az Contraction < a < 1, b = 0 w = f(z) = az

7. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Linear Transformation Result :- The linear transformations preserve circles. i.e. A circle in a z-plane under linear transformation maps to a circle in a w-plane. Inversion and Reflection Transformation:- Definition : - Circles are invariant under inversion. Straight lines in the limiting cases maps to circles through origin in w-plane

8. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Example Map the strip under the mapping

9. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Example Continued..

10. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Example Continued..

11. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Example Continued..

12. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Summery Conformal Mapping is a angle preserving map, not only in terms of magnitude but also in terms of sense i.e. clockwise or anti-clockwise. Analytic functions are conformal at a point, provided the first order derivative does not vanish at that point. Linear Transformation is a conformal map Inversion is a conformal map except at origin. Linear transformation maps circles onto circles.

13. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] References Higher Engineering Mathematics, B. V. Ramana Mc- Graw -Hill Publication 2. Advanced Engineering Mathematics, Erwin Kreyszyg Wiely Publication, 9 th Edition

14. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - [phone]/2/3 | Website - www.isquareit.edu.in | Email - [email protected] Thank You International Institute of Information Technology (I²IT) P-14, Rajiv Gandhi Infotech Park, MIDC Phase – 1, Hinjawadi , Pune – 411057, India http://www.isquareit.edu.in/ [email protected]