File:FS QQ dia.png
![File:FS QQ dia.png](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a2/FS_QQ_dia.png/580px-FS_QQ_dia.png?20230412213342)
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Contents
Summary
[edit]DescriptionFS QQ dia.png |
English: Smallest square inscribed (touching all sides) in a square
Deutsch: Kleinstes in ein Quadrat eingeschriebenes Quadrat, das alle Seiten des äußeren Quadrats berührt. |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a2/FS_QQ_dia.png/220px-FS_QQ_dia.png)
Task
[edit]Smallest square inscribed (touching all sides) in a square of side length
The square as base element. Inscribed is the smallest square that touches all sides of the base element
General case
[edit]Segments in the general case
[edit]0) Side length of the base square
1) Side length of the inscribed square: , see Calculation 1
Perimeters in the general case
[edit]0) Perimeter of base square:
1) Perimeter of the inscribed square:
S) Sum of perimeters:
Areas in the general case
[edit]0) Area of the base square:
1) Area of the inscribed square:
Centroids in the general case
[edit]0) Centroid position of the base square:
1) Centroid position of the inscribed square:
W) Weighted centroid:
Normalised case
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/FS_QQ.png/220px-FS_QQ.png)
In the normalised case the area of the base square is set to 1.
So
Segments in the normalised case
[edit]0) Side length of the base square:
1) Side length of the inscribed square:
Perimeters in the normalised case
[edit]0) Perimeter of base square:
1) Perimeter of the inscribed square:
S) Sum of perimeters:
Areas in the normalised case
[edit]0) Area of the base square is by definition
1) Area of the inscribed square:
Centroids in the normalised case
[edit]0) Centroid position of the base square:
1) Centroid position of the inscribed square:
W) Weighted centroid:
Distances of centroids
[edit]The distance between the centroid of the base element and the centroid of the inscribed square is:
Identifying number
[edit]Apart of the base element there is one shape allocated. Therefore the integer part of the identifying number is 1.
The decimal part of the identifying number is the decimal part of the sum of the perimeters and the distances of the centroids in the normalised case.
So the identifying number is:
Calculations
[edit]Calculation 1
[edit](1)
(2) , since
is the inscribed square
(3) , applying equation (1)
(4) , applying the Pythagorean theorem
, applying equation (2)
, applying equation (3)
, adding the two terms
, squaring the bracket term
, multiplying
, applying the root
Licensing
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 21:33, 12 April 2023 | ![]() | 1,672 × 1,728 (49 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
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Horizontal resolution | 129.92 dpc |
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Vertical resolution | 129.92 dpc |