File:FS CV.1C dia.png

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FS_CV.1C_dia.png (569 × 573 pixels, file size: 33 KB, MIME type: image/png)

Captions

Captions

Largest circle in a circle that contains the largest quarter circle

Summary

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Description
English: Largest circle in a circle that contains the largest quarter circle
Deutsch: Größter Kreis in eime Kreis, der den größten Viertelkreis enthält
Date
Source Own work
Author Hans G. Oberlack

Shows the largest circle within a circle that contains the largest quarter circle.

Elements

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Base is the circle of given radius around point and the resulting quarter circle of radius around point . From the developing of FS_CV we know that .
Inscribed is the largest possible circle in the shape having radius around point . This is also the largest circle in a CV type sangaku.

In order to find radius of the circle, the following reasoning is used:
The line segment corresponds to the radius of the quarter circle around . So we have:
(1)

The line segments correspond to the radius of the circle around . This means that:
(2)

It also means that the points form a square with side length . So applying the Pythagorean theorem we get:
(3)

Using (2) and (3) on (1) gives:



General case

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Segments in the general case

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0) The radius of the base circle
1) Radius of the quarter circle
2) Radius of the additional circle

Perimeters in the general case

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0) Perimeter of base circle
1) Perimeter of the quarter circle
2) Perimeter of additional circle

Areas in the general case

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0) Area of the base circle
1) Area of the inscribed quarter circle
2) Area of the additional circle

Covered surface of the base shape:

Centroids in the general case

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1) Centroids as graphically displayed

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Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base square:
1) Centroid positions of the inscribed quarter circle:
2) Centroids of the additional circle: , see calculation (1)

2) Orientated centroids

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The centroid positions of the following shapes will be expressed orientated so that the first shape n with will be of type with . This means that the graphical representation will not correspond to the mathematical expression.
0) Orientated centroid position of the base circle:
1) Orientated centroid positions of the inscribed quarter circle:
2) Orientazed centroid of the additional circle:

Normalised case

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In the normalised case the area of the base is set to 1.

Segments in the normalised case

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0) Radius of the base circle
1) Radius of the inscribed quarter circle
2) Radius of the additional circle

Perimeters in the normalised case

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0) Perimeter of base square
1) Perimeter of the inscribed quarter circle
2) Perimeter of additional circle
S) Sum of perimeters

Areas in the normalised case

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0) Area of the base square
1) Area of the inscribed quarter circle
2) Area of the additional circle

Centroids in the normalised case

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1) Centroids as graphically displayed

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Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base square:
1) Centroid positions of the inscribed quarter circle:

2) Centroid of the additional circle:

2) Orientated centroids

[edit]

The centroid positions of the following shapes will be expressed orientated so that the first shape n with will be of type with . This means that the graphical representation will not correspond to the mathematical expression.
0) Orientated centroid position of the base circle:
1) Orientated centroid positions of the inscribed quarter circle:
2) Orientazed centroid of the additional circle:


Calculation 1

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Calculating













Licensing

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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current23:48, 6 January 2022Thumbnail for version as of 23:48, 6 January 2022569 × 573 (33 KB)Hans G. Oberlack (talk | contribs)Enhanced
22:34, 26 December 2021Thumbnail for version as of 22:34, 26 December 2021569 × 574 (32 KB)Hans G. Oberlack (talk | contribs)Missing letter added
22:21, 26 December 2021Thumbnail for version as of 22:21, 26 December 2021569 × 574 (33 KB)Hans G. Oberlack (talk | contribs)Uploaded own work with UploadWizard

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