18.5: Argument and polar coordinates
As before, we assume that \(O\) and \(E\) are the points with complex coordinates \(0\) and \(1\) respectively.
Let \(Z\) be a point distinct form \(O\) . Set \(\rho=OZ\) and \(\theta=\measuredangle EOZ\) . The pair \((\rho,\theta)\) is called the polar coordinates of \(Z\) .
If \(z\) is the complex coordinate of \(Z\) , then \(\rho=|z|\) . The value \(\theta\) is called the argument of \(z\) (briefly, \(\theta=\arg z\) ). In this case,
\(z=\rho\cdot e^{i\cdot\theta}=\rho\cdot(\cos\theta+i\cdot\sin\theta).\)
Note that
\(\arg (z\cdot w) \equiv \arg z+\arg w\)
and
\(\arg \tfrac z w \equiv \arg z-\arg w\)
if \(z \ne 0\) and \(w \ne 0\) . In particular, if \(Z\) , \(V\) , \(W\) are points with complex coordinates \(z\) , \(v\) , and \(w\) respectively, then
\[\begin{aligned} \measuredangle VZW &=\arg\left(\frac{w-z}{v-z}\right)\equiv \\ &\equiv \arg(w-z)-\arg(v-z) \end{aligned}\]
if \(\measuredangle VZW\) is defined.
Use the formula 18.5.1 to show that
\(\measuredangle ZVW+\measuredangle VWZ+\measuredangle WZV\equiv \pi\)
for any \(\triangle ZVW\) in the Euclidean plane.
- Hint
-
Let \(z, v\), and \(w\) denote the complex coordinates of \(Z, V\), and \(W\) respectively. Then
\(\begin{array} {rcl} {\measuredangle ZVW + \measuredangle VWZ + \measuredangle WZV} & \equiv & {\arg \dfrac{w - v}{z- v} + \arg \dfrac{z-w}{v-w} + \arg \dfrac{v-z}{w-z} \equiv} \\ {} & \equiv & {\arg \dfrac{(w - v) \cdot (z - w) \cdot (v -z)}{(z - v) \cdot (v - w) \cdot (w - z)} \equiv} \\ {} & \equiv & {\arg (-1) \equiv \pi} \end{array}\)
Suppose that points \(O\) , \(E\) , \(V\) , \(W\) , and \(Z\) have complex coordinates \(0\) , \(1\) , \(v\) , \(w\) , and \(z=v\cdot w\) respectively. Show that
\(\triangle OEV\sim \triangle OWZ.\)
- Hint
-
Note and use that \(\measuredangle EOV = \measuredangle WOZ = \arg v\) and \(\dfrac{OW}{OZ} = \dfrac{OZ}{OW} = |v|\).
The following theorem is a reformulation of Corollary 9.3.2 which uses complex coordinates.
Let \(\square UVWZ\) be a quadrangle and \(u\) , \(v\) , \(w\) , and \(z\) be the complex coordinates of its vertices. Then \(\square UVWZ\) is inscribed if and only if the number
\(\dfrac{(v-u)\cdot(z-w)}{(v-w)\cdot(z-u)}\)
is real.
The value \(\dfrac{(v-u)\cdot(w-z)}{(v-w)\cdot(z-u)}\) is called the complex cross-ratio of \(u\) , \(w\) , \(v\) , and \(z\) ; it will be denoted by \((u,w;v,z)\) .
Observe that the complex number \(z\ne 0\) is real if and only if \(\arg z=0\) or \(\pi\) ; in other words, \(2\cdot\arg z\equiv 0\) .
Use this observation to show that Theorem \(\PageIndex{1}\) is indeed a reformulation of Corollary 9.3.2 .
- Hint
-
Note that
\(\arg \dfrac{(v-u) \cdot (z-w)}{(v -w) \cdot (z -u)} \equiv \arg \dfrac{v - u}{z - u} + \arg \dfrac{z- w}{v -w} = \measuredangle ZUV + \measuredangle VWZ.\)
The statement follows since the value \(\dfrac{(v - u) \cdot (x - w)}{(v - w) \cdot (z - u)}\) is real if and only if
\(2 \cdot \arg \dfrac{(v - u) \cdot (z - w)}{(v - w) \cdot (z - u)} \equiv 0.\)