= Signed swept area
{title2=$A(t)-A(t_0)=\frac12\int_{t_0}^t(x\dot y-y\dot x)\,ds$}
Signed swept area counts the <area> traversed by a <position> <vector>, positively or negatively according to its direction of turning. For a continuously differentiable planar trajectory, its accumulated value relative to $t_0$ is the <line integral>
$$
A(t)-A(t_0)=\frac12\int_{t_0}^t\mathbf n\cdot(\mathbf r(s)\times\dot{\mathbf r}(s))\,ds.
$$
A short segment contributes the oriented <triangle> <area> $\tfrac12\mathbf n\cdot(\mathbf r\times d\mathbf r)$; taking its <time> <derivative> gives <areal velocity>. Repeated turns count repeated <area>; a reversal subtracts it. Thus this quantity need not equal the <area> of the union of all previously swept regions. The origin matters: the relevant origin for <Kepler's second law> is the attracting focus of the <ellipse>, rather than its geometric centre.
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