n-curving - a transformation of curves

1

n-CurvingA Method of Creating

Beautiful Mathematical Curves

References: 1. Sebastian Vattamattam & R. Sivaramakrishnan, “A Note on

Convolution Algebras”, Chapter 6, Recent Trends in Mathematical Analysis, Allied Publishers, 2003

2. http://en.wikipedia.org/wiki/Functional-theoretic_algebra

Based on Functional Theoretic Algebras

http://en.wikipedia.org/wiki/Functional-theoretic_algebra















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Basic Definitions

FunctionIf 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).

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ExampleX = [0, 1], Y = RFor t ε [0, 1], define f(t) = 2πt

t 0 1/4 1/2 3/4 1

0 π/2 π 3π/2 2π

sin2πt 0 1 0 -1 0

cos2πt 1 0 -1 0 1

See the figure

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0 1

0 2π

f(0) f(1)

As t varies from 0 to 1, = 2πt varies from 0 to 2π

And f([o, 1]) = [0, 2π]

f(t) = 2πt

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)1( )0( :

)1( )0(:

)1( and )0(

)(),(

);()()( f

]1,0[:function continuousA

int

CurveOpen

CurveClosed

tytx

tittI

C

sPoEnd

FormParametric

Curve

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figuretheSee

tytx

ttittiftt

iscurveThe

ttftfG

isfofGraphThe

2,

10,2)(

]1,0[:))(,()(

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Examples ofClosed Curves

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Examples of Closed Curves(Loops)

1

)1(1)0(

2sin)(

2cos)(

)2sin()2()(

1

atLoop

FormParametric

uu

tty

ttx

titCostu

eUnit Circl

0 1

2/17/20102/17/2010

099

11/4 1/2 3/4

1

2sin4cos)(

2cos4cos)(

2cos

Cos2-Rhodonea 2

atLoop

ttty

tttx

r

FormParametric

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1at Loop

t)t)sin(2cos(6y

t)t)cos(2cos(6x

3cos

Cos3-Rhodonea 3r

0

1

1/21/4

3/4

(-1/2, 1/2)

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1at Loop

cardioid theas Take1at loop a be will1-c

t)t))sin(2cos(2(1(t)

1-t)t))cos(2cos(2(1(t)

2at loop a is

(1)2(0)

t)t))sin(2cos(2i(1t)t))cos(2cos(2(1(t)

cos1: 4

c

c

c

rCardioid

0

1

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1at Loop

umDoubleFoli theas Take

1at loop a be will1

t)t)sin(2t)sin(4cos(24(t)

t)t)cos(2t)sin(4cos(241)(

0at loop a is

(1)0(0)

t)t)sin(2t)sin(4cos(24it)t)cos(2t)sin(4cos(24(t)

2sincos4: 5

t

rFoliumDouble

0

1

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1at Loop

Folium Double theas Take

1at loop a be will2

t)(2sin))t2cos(21((t)

2-t)cos(2))t2cos(21()(

3at loop a is

(1)3(0)

t)(2sin))t2cos(21(it)cos(2))t2cos(21((t)

cos21: 6

t

rPascalofLimacon

0

1

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1at loop a is

)sin()3

sin3

(cosi)cos()3

sin3

(cos(t)(1)1(0)

3sin3cos:Egg rooked 7 rC

0

1

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Nephroid theas Take

1at loop a be will22iat loop a is

(1)2(0)

t)](12cos)t4cos(i[3t)(12sin)t4sin(3(t)

)cos(6-)3cos(2y

),sin(6)-3sin(2x

:ephroid 8

i

i

N

0

1

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a called isit and , if

algebra, ecommutativ-non a becomes

thislike defined productsWith

)()()()(

,,

)1()1(

,

Algebra. Theoretic Functional21

2121

1

2

1

1

21

LL

V

eyLxLxyLyxLyx

defineVyxIf

LLe

sfunctionallineartwoLL

FfieldtheoverspacevectoraV

F

F

FF

F

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1

, ,

1

]1,0[,1)(

lg ]1,0[

],1,0[ ,

]1,0[

)1()0()1()0(

HIf

atloopsofsetH

ttebydefinedunitywith

ebraAecommutativnonaisCThen

CIf

CincurvescontinuousofsetC

Ref: Sebastian Vattamattam, “Non-Commutative Function Algebras”,Bulletin of Kerala Mathematics Association, Vol. 4, No. 2(2007 December)

Curves ofProduct Theoretic Functional

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1 ,

])[()( [0,1], tIf

1] [0,[x]-x

int ][,

int

1

n

atloopacurvenancalledisn

ntntt

xegergreatestthexRxIf

egerpositivean

atloopa

CurvenDefining

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Examples of

Open Curves

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i

titttSegmentLineA)1(,1)0(

10,1)( :

20

0 1

ii

ttittPA1)1(,1)0(

10,)12(12)( :2

arabola

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icic

ttittc

2)1(,)0(

10,2cos2)(

Curve Cosine

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4)1(,0)0(

4sin4)(

ss

titts

Sine Curve

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4)1(,0)0(

t)sin(4t4y

t)cos(4t4x

))in(4it)t(cos(44)(

40,:

tst

SpiralnArchimedia

01

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n-Curving ?

Ref: Sebastian Vattamattam, Transforming Curves by n-Curving, Bulletin of Kerala Mathematics Association, Vol. 5, No.1(2008 December)

).()( then sin and cos of functions are

of partsimaginary and real thesuch that 1,at loop a

. with curved-n called is ),1)](0()1([)(

curve,-nan is and curveopen an is

n

n

n

:n

ntttt

α(t)isIf

If

Theorem

Curvingn

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N-Curving the Line Segment

ntnttty

ntntttx

iyxIf

i

nu

titCostu

titttCircleUnittheWith

2sin2cos1)(

2sin2cos2)(

)(

1)0()1(

)2sin()2()(

10,1)( .1

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Unit Circle – Line Segment

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ntnttty

ttx

iyxIf

t

t

i

nc

cCardioid

titttCtheWith

2sint))cos(2(12cost))cos(2(12)(

t)t))sin(2cos(2(1t)t))cos(2cos(2(13)(

)(

t)t))sin(2cos(2(1)(

1-t)t))cos(2cos(2(1)(

1)0()1(

10,1)(ardioid .2


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Cardioid – Line Segment

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).sin(2pintcos(4pint)+).cos(2pintcos(4pint)+1-t=y

).sin(2pintcos(4pint)-).cos(2pintcos(4pint)+1- t=x

)(

sin(2pit)cos(4pit).=y

cos(2pit)cos(4pit).=x

1)0()1(

2

10,1)(Cos2-Rhodonea With 3.

iyxIf

i

nc

CosRhodonea

tittt


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Rhodonea-Cos2 – Line Segment

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)(



1)0()1(

3

10,1)(Cos3-Rhodonea With 4.

iyxIf

i

nc

CosRhodonea

tittt


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Rhodonea-Cos3 – Line Segment

n = 1

n = 2

n = 3n = 10

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)(



1)0()1(

2

10,1)(Sin2-Rhodonea With 5.

iyxIf

i

nc

CosRhodonea

tittt


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RhodoneaSin2 – LineSegment

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)(

t)t)sin(2t)sin(4cos(24(t)

t)t)cos(2t)sin(4cos(241)(

1)0()1(

10,1)(

.6

iyxIf

t

i

tittt

n

umDoubleFoliWith


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DoubleFolium – LineSegment

n = 1 n = 2

n = 3 n = 4

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nt)nt))sin(22cos(2(1-nt)nt))cos(22cos(2(13-ty

nt)nt))sin(22cos(2(1-nt)nt))cos(22cos(2(1-t-4 x

)(

t)t))sin(2cos(221((t)

2-t)t))cos(2cos(221()(

1)0()1(

10,1)(

Pascal ofimacon .7

iyxIf

t

PascalofLimacon

i

tittt

n

LWith

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Limacon– LineSegment

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Random Examples

Example 1n-Curved Cosine with Rhodonea-cos2

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nt)(nt)(t)(ty

nt)(nt)(t-tx

2sin4cos22cos)(

)2cos4cos1(2)(

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Example 2n-Curved Cosine with Double Folium

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Example 3n-Curved Archimedean Spiral with Unit Circle

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Example 4n-Curved Archimedean Spiral with Cardioid

n-curving - a transformation of curves

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