remark. SDFs in a Euclidean space [ag-0012]

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If \(X\) is also a normed space, i.e. for a point \(\boldsymbol {p} \in X\), its norm can be defined, e.g. in a Euclidean space \(\mathbb R^n\) where \(\boldsymbol {p}\) can be expressed by basis vectors as \((p_1, \ldots , p_n)\), then its norm \(\lVert \boldsymbol {p}\rVert \) can be defined as the Euclidean norm \[ \lVert \boldsymbol {p}\rVert _2:=\sqrt {p_1^2+\cdots +p_n^2} \] which is essentially the distance from the origin \(\boldsymbol {o}=(0, \ldots , 0)\) to the point \(\boldsymbol {p}\) \[ \lVert \boldsymbol {p}\rVert =d(\boldsymbol {p}, \boldsymbol {o})=\lVert \boldsymbol {p} - \boldsymbol {o}\rVert \] where \(d\) is the Euclidean distance function.

In the following, we will be working with Euclidean space \(\mathbb R^n\), which is both a metric space and a normed space. Most of the time, we will focus on \(\mathbb R^3\) for visualization purposes.