What is Conformal mapping?

Welcome back MechanicaLEi, did you know that in cartography the property distortion of shapes to make them as small as desired is achieved using conformal mapping? This makes us wonder, what is Conformal mapping? Before we jump in check out the previous part of this series to learn about what Harmonic functions are? Now, consider an analytic mapping f which maps the set capital D into the set capital E and is given by w equals f of z, where z equals x plus i y and z belongs to capital d. If two curves C1 and C2 in a z plane intersect at z not, then the angle from curves f of c1 to f of c2 in a w plane intersect at f of z not is the same as the angle from c one and c 2. If the derivative of the function is nonzero, then there exists a conformal map for the analytic function. That is, the analytical function f of z is conformal at any point say z not where it has a non-zero derivative. A complex mapping of the form w equals f of z which again equals to az plus b where a and b are complex constants is called a linear mapping. Bilinear or Mobius transformation is given by w equals f of z and is equal to az plus b upon c z upon d, where a, b, c and d all belong to the set C and ad minus bc is not equal to zero. They are so called because both z and w in the above equation are linear. Cross-ratio is the term which remains invariant under bilinear transformation and is defined for w equals f of z which maps from distinct points z1, z2 and z3 onto the distinct points w1, w2 and w3 as follows. One last term of significant importance is Fixed points of Bilinear Transformation and are defined as the points that satisfy the equation f of z equals az plus b upon cz plus d equals to z. Hence, we first saw what conformal and linear mapping are and then went on to see what bilinear mapping is?
In the next episode of MechanicaLEi find out what standard transformations are?

Attributions: