Number-Theoretic Construction of Digital Circles

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

This is a construction of a digital circle. The algorithm used for the construction is based on number theory. It relates the run length to the number of square numbers in a certain interval that is calculated using a formula given in the Details section.

Contributed by: Aniket Jha (BTech Student IIT KGP), Partha Bhowmick (IIT KGP), and B. B. Bhattacharya (ISI Kolkata) (January 2009)
After work by: Partha Bhowmick and B. B. Bhattacharya
Open content licensed under CC BY-NC-SA


Snapshots


Details

The Demonstration has been prepared based on the paper:

P. Bhowmick and B. B. Bhattacharya, "Number-Theoretic Interpretation and Construction of a Digital Circle," Discrete Applied Mathematics, 156(12), 2008, pp. 2381–2399.

The main idea:

Let be the radius of the circle and the run number from the top (starting from 0).

The squares of the abscissas of the grid points lying on the digital circle corresponding to radius and having ordinate lie in the interval , where and are defined by:

if if if if if .

So the algorithm goes on calculating the consecutive square numbers, adding a point to the current run for each square number. It changes the run when the square number moves to the next interval.

It may be noted that the square number can be obtained easily (without using any multiplication) from the previous square number , since , which is equivalent to adding a "gnomon".

For details visit : http://dx.doi.org/10.1016/j.dam.2007.10.022



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send