![2/17/20103ExampleX = [0, 1], Y = RFor tε [0, 1], define f(t) = 2πtSee the figure](https://image.slidesharecdn.com/ncurving-100217111627-phpapp02/85/n-Curving-A-Transformation-of-Curves-3-320.jpg)
![2/17/20104f(t) = 2πtf(1)f(0)2π010As t varies from 0 to 1, = 2πt varies from 0 to 2πAnd f([o, 1]) = [0, 2π]](https://image.slidesharecdn.com/ncurving-100217111627-phpapp02/85/n-Curving-A-Transformation-of-Curves-4-320.jpg)
n-Curving - A Transformation of Curves
- 1. 1n-CurvingA Method of Creating Beautiful Mathematical CurvesBased on Functional Theoretic AlgebrasReferences: Sebastian Vattamattam & R. Sivaramakrishnan, “A Note on Convolution Algebras”, Chapter 6, Recent Trends in Mathematical Analysis, Allied Publishers, 2003http://en.wikipedia.org/wiki/Functional-theoretic_algebra
- 2. 2/17/20102Basic DefinitionsFunctionIf X and Y are two sets, Then, f is a function from X to Y if every element in X is related to exactly one element in Y. If x is related to y we write y = f(x).
- 3. 2/17/20103ExampleX = [0, 1], Y = RFor tε [0, 1], define f(t) = 2πtSee the figure
- 4. 2/17/20104f(t) = 2πtf(1)f(0)2π010As t varies from 0 to 1, = 2πt varies from 0 to 2πAnd f([o, 1]) = [0, 2π]
- 6. 2/17/20106f(t) = 2πt, 0 ≤ t ≤ 1 as a curve
- 8. 2/17/20108Examples of Closed Curves(Loops)01
- 14. 2/17/20101401
- 15. 2/17/20101501
- 16. 2/17/201016Functional Theoretic AlgebraIn a nutshell
- 17. 2/17/201017Ref: Sebastian Vattamattam, “Non-Commutative Function Algebras”,Bulletin of Kerala Mathematics Association, Vol. 4, No. 2(2007 December)
- 18. 2/17/201018
- 19. 2/17/201019Examples of Open Curves
- 20. 2/17/2010200120
- 21. 2/17/20102101
- 22. 2/17/20102201
- 23. 2/17/20102301
- 24. 2/17/20102401
- 25. 2/17/201025n-Curving ?Ref: Sebastian Vattamattam, Transforming Curves by n-Curving, Bulletin of Kerala Mathematics Association, Vol. 5, No.1(2008 December)
- 26. 2/17/201026N-Curving the Line Segment l
- 27. 2/17/201027Unit Circle – Line Segment
- 28. 2/17/201028N-Curving the Line Segment l
- 29. 2/17/201029Cardioid – Line Segment
- 30. 2/17/201030N-Curving the Line Segment l
- 31. 2/17/201031Rhodonea-Cos2 – Line Segment
- 32. 2/17/201032N-Curving the Line Segment l
- 33. 2/17/201033Rhodonea-Cos3 – Line Segment n = 1n = 2n = 10n = 3
- 34. 2/17/201034N-Curving the Line Segment l
- 36. 2/17/201036N-Curving the Line Segment l
- 37. 2/17/201037DoubleFolium – LineSegmentn = 2n = 1n = 4n = 3
- 38. 2/17/201038N-Curving the Line Segment l
- 41. Example 1n-Curved Cosine with Rhodonea-cos22/17/201041
- 42. 2/17/201042Example 2n-Curved Cosine with Double Folium
- 43. 2/17/201043Example 3n-Curved Archimedean Spiral with Unit Circle
- 44. 2/17/201044Example 4n-Curved Archimedean Spiral with Cardioid