transformation of curves
DESCRIPTION
Two methods of transforming a curve, developed on the basis of Functional Theoretic Algebra.TRANSCRIPT
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Two Methods of Creating Beautiful
Mathematical Curves
Ref: http://en.wikipedia.org/wiki/Functional-theoretic_algebra
Based on Functional Theoretic Algebras
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)1( )0( : )1( )0(:
)1( and )0(
)(),();()()( f
]1,0[:function continuousA
int
CurveOpenCurveClosed
tytxtittI
C
sPoEnd
FormParametric
Curve
Basic Definitions
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Closed CurvesExamples
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Examples of Closed Curves(Loops)
1
)1(1)0(2sin)(2cos)(
)2sin()2()( 1
atLoop
FormParametric
uuttyttx
titCostueUnit Circl
0 1
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055
11/4 1/2 3/4
1 2sin4cos)(2cos4cos)(
2cos
Cos2-Rhodonea 2
atLooptttytttx
rFormParametric
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1at Loopt)t)sin(2cos(6yt)t)cos(2cos(6x
3cosCos3-Rhodonea 3
r
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
ccc
rCardioid
01
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1at Loop
umDoubleFoli theas Take1at 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 Take1at 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()3sin3(cosi)cos()3sin3(cos(t)(1)1(0)
3sin3cos:Egg rooked 7
rC
0
1
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Nephroid theas Take1at loop a be will2
2iat 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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Method1
n-Star Transformation
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1, ,
1 ]1,0[,1)( lg ]1,0[
],1,0[ , ]1,0[
Pr
)1()0()1()0(
HIfatloopsofsetH
ttebydefinedunitywithebraAecommutativnonaisCThen
CIfCincurvescontinuousofsetC
CurvesofoductTheoreticFunctional
Ref: Sebastian Vattamattam, Non-Commutative Function Algebras, Bulletin of Kerala Mathematics Association, Vol. 4, No. 2(2007 December)
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, 1 ,
])[()( [0,1], tIf1] [0,[x]-x
int ][, int
1
n
oftiontransformaStarnthe
calledisnHIfatloopacurvenancalledis
ntntt
xegergreatestthexRxIfegerpositivean
atloopa
n
CurvenDefining
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EXAMPLES
Of
n-Star Transform
ation
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n-Star Transformation of Unit Circle
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n-Star Transformation of Double Folium
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n-Star Transformation of Rhodonea-Cos2
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n-Star Transformation of Crooked Egg
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n-Star Transformation of Nephroid
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Examples
of
Open Curves
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ititttSegmentLineA
)1(,1)0(10,1)( :
22
0 1
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iittittPA
1)1(,1)0(10,)12(12)( : 2arabola
0 1
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icicttittc
2)1(,)0(10,2cos2)(
Curve Cosine
0 1
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4)1(,0)0(4sin4)(
sstitts
Sine Curve
0 1
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4)1(,0)0(t)sin(4t4yt)cos(4t4x
))in(4it)t(cos(44)(40,:
tstSpiralnArchimedia
01
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Method2
n-Curving
Ref: Sebastian Vattamattam, Transforming Curves by n-Curving, Bulletin of Kerala Mathematics Association, Vol. 5, No.1(2008 December)
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Defining n-Curving
. with curved-n called is ),1)](0()1([)(curve,-nan is and curveopen an is
).()( then sin and cos of functions are of partsimaginary and real thesuch that 1,at loop a
n
n
n
n
:
If
nttttα(t)isIf
Curvingn
Theorem
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N-Curving the Line Segment l
ntntttyntntttx
iyxIf
i
nu
titCostu
titttCircleUnittheWith
2sin2cos1)(2sin2cos2)(
)(
1)0()1(
)2sin()2()(
10,1)(
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Unit Circle – Line Segment
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ntntttyttx
iyxIftt
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
N-Curving the Line Segment l
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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-RhodoneaWith
iyxIf
i
nc
CosRhodonea
tittt
N-Curving the Line Segment l
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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-RhodoneaWith
iyxIf
i
nc
CosRhodonea
tittt
N-Curving the Line Segment l
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Rhodonea-Cos3 – Line Segment
n = 1n = 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-RhodoneaWith
iyxIf
i
nc
CosRhodonea
tittt
N-Curving the Line Segment l
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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)(
iyxIf
ti
tittt
n
umDoubleFoliWith
N-Curving the Line Segment l
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DoubleFolium – LineSegment
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N-Curving the Line Segment l
nt)nt))sin(22cos(2(1-nt)nt))cos(22cos(2(13-tynt)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
iyxIf
tPascalofLimaconi
tittt
n
LWith
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Limacon– LineSegment
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Random
Examples
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Example 1n-Curved Cosine with Rhodonea-cos2
nt)(nt)(t)(tynt)(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
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