1 n-Curving A 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_algebr a Based on Functional Theoretic Algebras
A method of transforming plane curves, developed on the basis of Functional Theoretic Algebra. http://en.wikipedia.org/wiki/Functional-theoretic_algebra
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1
n-CurvingA Method of Creating
Beautiful Mathematical Curves
References: 1. Sebastian Vattamattam & R. Sivaramakrishnan, “A Note on
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
2/17/2010 210 1
icic
ttittc
2)1(,)0(
10,2cos2)(
Curve Cosine
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4)1(,0)0(
4sin4)(
ss
titts
Sine Curve
2/17/2010 230 1
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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