The main facts about Hausdorff and packing measures and dimensions of a Borel set$E$are revisited, using$determining$set functions $\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)$ , where $\mathcal{B}_E$ is the family of all balls centred on$E$and α is a real parameter. With mild assumptions on φ_{α}, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set$V$in ℝ^{$D$}, these notions are used to introduce the$interior$and$exterior$measures and dimensions of any Borel subset of ∂$V$. We stress that these dimensions depend on the choice of φ_{α}. Two determining functions are considered, φ_{α}($B$)=Vol_{$D$}($B$∩$V$)diam($B$)^{α-$D$}and φ_{α}($B$)=Vol_{$D$}($B$∩$V$)^{α/$D$}, where Vol_{$D$}denotes the$D$-dimensional volume.

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In plane kinematics, the knowledge of the location of the instantaneous centre — deriving from the eighteenth century — affords information which is partial and very incomplete. But it is the case that there always exists an enumerable set of$cardinal points$, as they may be named, having the following characteristic: all the properties of the path of any, and every, element or series of elements, fixed in the moving plane, are completely determined by the configuration of this set of cardinal points. In this manner is given a very simple and complete synthesis of the whole realm of plane kinematics.