Let \(C\) be a curve in the complex plane where \(C = \{a(t): 0 \leq t \leq 1\},\) where \(a(0) \neq a(1)\). The authors discuss a dyadic paramerization for \(C\) first introduced by the first two authors [Michigan Math. J. 41, 269-284 (1994; Zbl 0815.30010)]. Let \(z(0) = a(0)\) and \(z(1) = a(1)\) and let \(C_{1,1}\) denote the directed segment from \(z(0)\) to \(z(1)\). Let \(L_{1,1}\) be the perpendicular bisector of \(C_{1,1}\), and let \(z(1/2) = a(q)\) where \(q = \inf{\{t: a(t) \in L_{1,1}\}}\). Now, define \(z(t)\) for dyadic values of \(t\) inductively as follows. Suppose that for a \(k \geq 2\), and for \(j\) odd, \(1 \leq j < 2^{k}\), points \(z(j/2^{k})\) and \(z((j+1)/2^{k})\) have already been chosen, together with the directed segment \(C_{k,j}\) from \(z(j/2^{k})\) to \(z((j+1)/2^{k})\). Let \(L_{k,j}\) be the perpendicular bisector of \(C_{k,j}\), and let \(z((2j+1)/2^{n+1})\) be the first point of \(C \cap L_{k,j}\) along the arc of \(C\) going from \(z(j/2^{k})\) to \(z((j+1)/2^{k})\). This inductive process defines \(z(t)\) for each dyadic number \(t = j/2^{k}, 0 \leq j \leq 2^{k}, 1 \leq k < \infty\). For \(t\) not a dyadic number, for each \(k\) let \(j_{k}(t)\) be the integer such that \(j_{k}(t)/2^{k} < t < (j_{k}(t)+1)/2^{k}\), and define \[ z(t) = \lim_{k \to \infty} z\bigg(\frac {j_{k}(t)} {2^{k}}\bigg) \;. \] This gives the full dyadic parameterization for \(C\). For each \(j\) odd, \(1 \leq j \leq 2^{k}\), define \(\theta_{k,j}\) to be the signed angle at \(z(j/2^{k})\) going from \(C_{k,j}\) to \(C_{k+1,2j+1}\), that is, \[ \theta_{j,k} = \arg{z\bigg(\frac {2j+1} {2^{k+1}}\bigg)} - \arg{z\bigg(\frac{j+1} {2^{k}}\bigg)} \;, \] and define \(\theta_{k,j+1} = - \theta_{k, j}\). For each point \(z = z(t) \in C\) and for each positive integer \(k\), define \(\theta_{k}(z) = \theta_{k,j(k)}\). If \(z\) is a non-dyadic point of \(C\) we say that \(C\) has a \textit{dyadic tangent} at \(z\) if \(\sum_{k} \theta_{k}(z) < \infty\), and we say that \(z\) is a \textit{dyadic twist point} if both \[ \limsup_{n \to \infty} \sum_{k=1}^{n} \theta_{k}(z) = + \infty \;\;\;\text{and} \;\;\;\liminf_{n \to \infty} \sum_{k=1}^{n} \theta_{k}(z) = - \infty \;. \] The functions \(\theta_{k}(z(t)) = \theta_{k}(t)\) can be considered as random variables for \(t \in [0, 1]\). For any random variable \(\theta = \theta(t)\) on \([0, 1]\), we denote the expected value by \(E(\theta) = \int_{[0, 1]} \theta(t) dt\) and define \(\| \theta\| _{p} = (E(| \theta| ^{p}))^{1/p}\). For each positive integer \(k\), define \(\sigma_{k} = \| \theta_{k}\| _{2}\). Finally, we say that the Lyapunov Ratio Condition (L.R.C.) holds if \[ \sup_{k} \frac {\| \theta_{k}\| _{4}} {\| \theta_{k}\| _{2}} < \infty \;. \] (Here, if the denominator is zero then the ratio is considered to be zero.) The authors prove two results about the dyadic representation of curves. Theorem 1. If \(\sigma_{k} \to 0\) as \(k \to \infty\) and if the L. R. C. holds, then \[ \dim{\{ t \in [0, 1]: C \text{ has a dyadic tangent at } z(t) \}} = 1 \;. \] Theorem 2. If \(\sum_{k} \sigma_{k} = \infty\) and the L. R. C. holds, then \[ \dim{ \{ t: z(t) \text{ is a dyadic twist point} \}} = 1 \;. \] These theorems improve results due to the first two authors with \textit{M. E. Grifkin} [Q. J. Math 52, 403-413 (2001; Zbl 1013.53003)].