Liebe Freundinnen und Freunde des Zählens,
In[94]:= (* correct *)
RegionMeasure[Point[{672, 125}]]
Out[94]= 1
In[95]:= (* correct: dimension is 0 - RegionDimension counts *)
RegionDimension[Point[{{672, 125}, {672, 125}}]]
Out[95]= 0
In[96]:= (* still correct: number of points *)
RegionMeasure[Point[{{672, 125}, {672, 125}}]]
Out[96]= 1
In[97]:= (* now dimension is 1 and R. E. Ference says: "RegionMeasure \
corresponds to curve length for one-dimensional regions:"*)
RegionDimension[Line[{{672, 125}, {672, 125}}]]
Out[97]= 1
In[98]:= (* wrong: counts points instead of giving the ArcLength in
dimension 1 *)
RegionMeasure[Line[{{672, 125}, {672, 125}}]]
Out[98]= 1
In[99]:= RegionMeasure[Region[Line[{{672, 125}, {672, 125}}]]]
Out[99]= 1
In[100]:= ArcLength[Line[{{672, 125}, {672, 125}}]]
Out[100]= 0
the latest and greatest
------------------------------------------------------------------------
In[101]:= $Version
Out[101]= "12.2.0 for Microsoft Windows (64-bit) (December 12, 2020)"
------------------------------------------------------------------------
has the curve length of a single point line equal to 1 and some more
Monty Python results can be obtained:
In[102]:= RegionMeasure[Line[{{Missing[]}, {Missing[]}}]]
Out[102]= 1
In[103]:= RegionMeasure[Line[{{Null}, {Null}}]]
Out[103]= 1
Missing[] and Null seemingly considered as identical points, but in
principle RegionMeasure knows what to do:
In[105]:= RegionMeasure[Line[{{"Hans"}, {"Oderhauer"}, {"weiss"},
{"nicht"}, {"was"}, {"ein"}, {"Punkt"}, {"ist!"}}]]
Out[105]= Sqrt[("Hans" - "Oderhauer")^2] + Sqrt[("ein" - "Punkt")^2]
+ Sqrt[(-"ist!" + "Punkt")^2] + Sqrt[("nicht" - "was")^2]
+ Sqrt[(-"ein" + "was")^2] + Sqrt[("Oderhauer" - "weiss")^2] +
Sqrt[(-"nicht" + "weiss")^2]
Mit den besten Grüssen
Udo.
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