Introduction           Transformations           Lines             Unit Circle           More Problems
                   Geometry in the Complex Plane
                      Hongyi Chen          ⋊⋉    UNCAwards Banquet 2016
   Introduction           Transformations           Lines             Unit Circle           More Problems
   “All Geometry is Algebra”
         Many geometry problems can be solved using a purely algebraic
         approach - by placing the geometric diagram on a coordinate plane,
         assigning each point an x/y coordinate, writing out the equations
         of lines and circles, and solving these equations. This method of
         solving geo problems (often called coordinate bashing) can be quite
         powerful given the right conditions, but it has some problems.
         Issues with coordinate bash
               Equations for circles are ugly
               Two variables are necessary for each random point
               Rotations are extremely painful
               Attempting to solve the equations may result in massive 5th
               degree polynomials in 8 variables...
         Fortunately, these problems can be fixed by replacing the Cartesian
         plane with the complex plane...
   Introduction           Transformations           Lines             Unit Circle           More Problems
   Quick Introduction to Complex Numbers
               Acomplex number (in rectangular form) is a number of the
               form a +bi, where a and b are real and i2 = −1.
               Wedefine the real and imaginary parts of a complex
               z = a+bi as Re(z) = a and Im(z) = bi.
               Complex numbers can be plotted on the complex plane. The
               number a+bi is placed where the coordinate (a,b) is placed
               on the Cartesian plane. The horizontal axis is called the real
               axis and the vertical axis is called the imaginary axis.
               The conjugate of a complex number z, denoted by z¯, is its
               reflection about the real axis. For any z = a + bi we have
               z¯ = a − bi.
                           ¯                         ¯
               ab = a¯· b and a + b = a¯+ b.
               Re(z) = z +z¯ and Im(z) = z −z¯.
                               2                          2
               z is real if and only if Im(z) = 0, which occurs when z = z¯.
               Similarly a number z is pure imaginary iff z = −z¯.
   Introduction           Transformations           Lines             Unit Circle           More Problems
   Quick Introduction to Complex Numbers
               The magnitude of z = a +bi, denoted by |z|, is its distance
               from the origin in the complex plane. If z = a + bi then
                       √ 2         2
               |z| =      a +b .
                                                                     2
               Notice that for any complex z, zz¯ = |z| .
               |a − b| is the distance between a and b.
               Acomplex number z can also be expressed in polar form as
               r(cosθ +i sinθ) for a real r and angle θ, where r = |z| and θ
               is the angle formed by the positive real axis and the ray
               starting at the origin pointing towards z, measured
               counterclockwise.
               For simplicity we shall let cis θ = cosθ + i sinθ.
               The set of possible values of cis θ forms the unit circle on the
               complex plane - a circle centered at the origin with radius 1.
               For any angle θ we have cis θ =                 1    =cis (−θ)
                                                            cis θ