File:FS RCC dia.png
Original file (681 × 622 pixels, file size: 24 KB, MIME type: image/png)
Captions
Contents
Summary
[edit]DescriptionFS RCC dia.png |
English: Largest circle in a RC sangaku
Deutsch: Größter Kreis in einem RC Sangaku |
Date | |
Source | Own work |
Author | Hans G. Oberlack |
Shows the largest circle within RC sangaku.
Elements
[edit]Base is the right isosceles triangle of side length and centroid
Inscribed is the largest possible circle with radius around point
Inscribed is the largest possible circle outside the first circle with radius around point
General case
[edit]Segments in the general case
[edit]0) The side length of the base right triangle
1) Radius of the first circle (See FS RC).
2) Radius of the second circle , (see Calculation 1)
Perimeters in the general case
[edit]0) Perimeter of base triangle
1) Perimeter of the first circle (See FS RC )
2) Perimeter of the second circle
Areas in the general case
[edit]0) Area of the base triangle
1) Area of the inscribed circle (See FS RC)
2) Area of the second circle
Centroids in the general case
[edit]Centroid positions are measured from the centroid point of the base shape
0) Centroid positions of the base triangle:
1) Centroid positions of the inscribed circle:
2) Centroid positions of the second circle:
, (see Calculation 2)
, (see Calculation 3)
Normalised case
[edit]In the normalised case the area of the base is set to 1.
Segments in the normalised case
[edit]0) Side length of the base triangle
1) Radius of the inscribed circle
2) Radius of the second inscribed circle
Perimeters in the normalised case
[edit]0) Perimeter of base triangle
1) Perimeter of the inscribed circle
2) Perimeter of the second circle
S) Sum of perimeters
Areas in the normalised case
[edit]0) Area of the base triangle
1) Area of the inscribed circle
2) Area of the second inscribed circle
Centroids in the normalised case
[edit]Centroid positions are measured from the centroid point of the base shape.
0) Centroid positions of the base triangle:
1) Centroid positions of the inscribed circle:
2) Centroid positions of the second circle:
Distances of centroids
[edit]01) The distance between the centroid of the base element and the centroid of the circle is:
02) The distance between the centroid of the base element and the centroid of the second circle is:
12) The distance between the centroids of the circles is:
The sum of the distances is:
Identifying number
[edit]Apart of the base element there are two shapes allocated. Therefore the integer part of the identifying number is 2.
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]The following equations hold:
(1)
(2)
(3)
(4)
(5)
The triangle is similar to triangle . So the angle in is the same as the angle in .
The line segments and are tangent to the circle around with radius and the circle around with radius . So the angle in in bisects the angle in . Since the angle in is 45° we get:
(6)
, by putting
Looking at , we get:
(7) , since is the opposite and is the hypothenuse of
, applying equations (2) and (3)
, applying equation (6)
, applying equation (6)
, applying the equation for r_1
Calculation 2
[edit]
Calculation 3
[edit]
Licensing
[edit]- 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/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 23:00, 1 February 2022 | 681 × 622 (24 KB) | Hans G. Oberlack (talk | contribs) | Uploaded own work with UploadWizard |
You cannot overwrite this file.
File usage on Commons
The following 2 pages use this file:
Metadata
This file contains additional information such as Exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. If the file has been modified from its original state, some details such as the timestamp may not fully reflect those of the original file. The timestamp is only as accurate as the clock in the camera, and it may be completely wrong.
Horizontal resolution | 59.06 dpc |
---|---|
Vertical resolution | 59.06 dpc |
Software used |