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
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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
N-Curving the Line Segment
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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
N-Curving the Line Segment
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Rhodonea-Cos2 – Line Segment
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).sin(2pintcos(6pint)+).cos(2pintcos(6pint)+1-t=y
).sin(2pintcos(6pint)-).cos(2pintcos(6pint)+1- t=x
)(
sin(2pit)cos(6pit).=y
cos(2pit)cos(6pit).=x
1)0()1(
3
10,1)(Cos3-Rhodonea With 4.
iyxIf
i
nc
CosRhodonea
tittt
N-Curving the Line Segment
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Rhodonea-Cos3 – Line Segment
n = 1
n = 2
n = 3n = 10
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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)(Sin2-Rhodonea With 5.
iyxIf
i
nc
CosRhodonea
tittt
N-Curving the Line Segment
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RhodoneaSin2 – LineSegment
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).sin(2pintcos(4pint)+).cos(2pintcos(4pint)+1-t=y
).sin(2pintcos(4pint)-).cos(2pintcos(4pint)+1- t=x
)(
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
N-Curving the Line Segment
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DoubleFolium – LineSegment
n = 1 n = 2
n = 3 n = 4
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N-Curving the Line Segment
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