Fractional Transformations, Cross Ratios and Conformal Mapping
Not what you're looking for?
1. a) Let z1,z2,z3,z4 lie on a circle. Show that z1,z3,z4 and z2,z3,z4 determine the same orientation iff (z1,z2,z,3,z4)>0
b) Let z1,z2,z3,z4 lie on a circle and be consecutive vertices of a quadrilateral. Prove that |z1-z3|*|z2-z4|=|z1-z2|*|z3-z4|+|z2-z3|*|z1-z4|
Purchase this Solution
Solution Summary
Fractional Transformations, Cross Ratios and Conformal Mapping are investigated. The solution is detailed and well presented.
Solution Preview
a) Let
a = Arg (z1,z3,z4) = Arg [(z4-z1)/(z3-z1)]
= Arg (z4-z1) - Arg (z3-z1)
= anticlockwise turn from >-Z1Z3-> to >-Z1Z4->
(negative means clockwise turn)
b = Arg (z2,z3,z4)
= anticlockwise turn from >-Z2Z3-> to >-Z2Z4->
From school geometry (and taking care of the clockwise /
anticlockwise senses) we get two cases:
Case 1
b = a (when Z1, Z2 are on same side of chord Z3Z4
i.e. when Z1,Z3,Z4 and Z2,Z3,Z4 have same orientation)
Case 2
b = -(pi - a) (when ...
Purchase this Solution
Free BrainMass Quizzes
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.