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Mathematics LibreTexts

18.5: Argument and polar coordinates

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    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,

    \(z=\rho\cdot e^{i\cdot\theta}=\rho\cdot(\cos\theta+i\cdot\sin\theta).\)

    截屏2021-03-01 下午2.35.41.png

    Note that

    \(\arg (z\cdot w) \equiv \arg z+\arg w\)


    \(\arg \tfrac z w \equiv \arg z-\arg w\)

    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.

    Exercise \(\PageIndex{1}\)

    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.


    Let \(z, v\), and \(w\) denote the complex coordinates of \(Z, V\), and \(W\) respectively. Then

    \(\begin{array} {rcl} {\measuredangle ZVW + \measuredangle VWZ + \measuredangle WZV} & \equiv & {\arg \dfrac{w - v}{z- v} + \arg \dfrac{z-w}{v-w} + \arg \dfrac{v-z}{w-z} \equiv} \\ {} & \equiv & {\arg \dfrac{(w - v) \cdot (z - w) \cdot (v -z)}{(z - v) \cdot (v - w) \cdot (w - z)} \equiv} \\ {} & \equiv & {\arg (-1) \equiv \pi} \end{array}\)

    Exercise \(\PageIndex{2}\)

    Suppose that points \(O\), \(E\), \(V\), \(W\), and \(Z\) have complex coordinates \(0\), \(1\), \(v\), \(w\), and \(z=v\cdot w\) respectively. Show that

    \(\triangle OEV\sim \triangle OWZ.\)


    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.

    Theorem \(\PageIndex{1}\)

    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


    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 \((u,w;v,z)\).

    Exercise \(\PageIndex{1}\)

    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.


    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 

    \(2 \cdot \arg \dfrac{(v - u) \cdot (z - w)}{(v - w) \cdot (z - u)} \equiv 0.\)