Fish Curve

Author

Eric W. Weisstein
June 27, 2013

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/FishCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FishCurve.html.

©2013 Wolfram Research, Inc. except for portions noted otherwise

Equation

eq=FullSimplify[With[{b=a/Sqrt[2]},NegativePedalCurve[{a Cos[t],b Sin[t]},{Sqrt[a^2-b^2],0},t]
],a>0]

curve=a{Cos[t]-(((Sin[t])^2)/(Sqrt[2])),Cos[t] Sin[t]};

Polar

soln=Solve[Tan[θ]==1/Divide@@curve,t]//Simplify

Special Points

Function[t,Evaluate[curve]]/@{0,Pi/2,Pi}

Plot of Negative Pedal Curve

With[{a=1,b=1/Sqrt[2]},
ParametricPlot[{{a Cos[t],b Sin[t]},{a Cos[t]-((a ((Sin[t])^2))/(Sqrt[2])),a Cos[t] Sin[t]}},{t,0,2Pi},
PlotStyle->{Black,{Thickness[.01],Red}},AspectRatio->Automatic
]
]

Equations

Polar

expr=(Sqrt[#.#&[curve]])

FullSimplify[expr,a>0&&0<t<2π]

Refine[expr,a>0&&0<t<2π]

FullSimplify[(Sqrt[Expand[expr^2]]),a>0&&0<t<2π]

Plot[(( (Sqrt[-2 (Sqrt[2]) Cos[t] ((Sin[t])^2)+((Sin[t])^4)+2 ((Cos[t])^2) (1+((Sin[t])^2))]))/(Sqrt[2])),{t,0,2π},AxesOrigin->{0,0}]

Simplify[1/Divide@@curve]

Solve[Tan[θ]==((Sin[2 t])/(2 Cos[t]-(Sqrt[2]) ((Sin[t])^2))),t]//Simplify

PolarPlot[(( (Sqrt[-2 (Sqrt[2]) Cos[t] ((Sin[t])^2)+((Sin[t])^4)+2 ((Cos[t])^2) (1+((Sin[t])^2))]))/(Sqrt[2])),{t,0,2π}]

Parametric

ParametricPlot[curve/.a->1,{t,0,2π}]

Cartesian

Eliminate

Eliminate[{x==a Cos[t]-((a ((Sin[t])^2))/(Sqrt[2])),y==a Cos[t] Sin[t],Cos[t]^2+Sin[t]^2==1},{Cos[t],Sin[t]}]

FullSimplify[Subtract@@%]

GroebnerBasis

Factor/@GroebnerBasis[Append[Thread[{x,y}==curve],Sin[t]^2+Cos[t]^2==1],{x,y},{Cos[t],Sin[t]},MonomialOrder->EliminationOrder]

FullSimplify[%]

fish=-2 (a^4)-4 (Sqrt[2]) (a^3) x-2 (a^2) ((x^2)-5 (y^2))+(((2 (x^2)+(y^2)))^2)+2 (Sqrt[2]) a x (2 (x^2)+5 (y^2));

fish2=(2x^2+y^2)^2-2Sqrt[2]a x(2x^2-3y^2)+2a^2(y^2-x^2);

Factor[FullSimplify[fish-fish2]]

Area

Plot

V5.2

ImplicitPlot[fish2==0/.a->1,{x,-1,2}]

ImplicitPlot[fish==0/.a->1,{x,-2,1}]

FullSimplify[Subtract@@{fish/.x->(x-a/Sqrt[2]),fish2}]

<<Graphics`InequalityGraphics`

InequalityPlot[fish<0/.a->1,{x},{y}]

V6

RegionPlot[fish<0/.a->1,{x,-1.2,1},{y,-.5,.5},AspectRatio->Automatic,MaxRecursion->5,PlotPoints->50,PlotStyle->Red,Axes->True]

RegionPlot[fish2<0/.a->1,{x,-.5,2},{y,-.5,.5},AspectRatio->Automatic,MaxRecursion->5,PlotPoints->50,PlotStyle->Red,Axes->True]

Integrals

Simplify[Abs[Area[curve,{t,Sequence@@#}]&/@{{π/2,3π/2},{-π/2,π/2}}],a>0]

Total[%]//Simplify

V5 (?)

<<Calculus`Integration`

Integrate[Boole[fish<0/.a->1],{x},{y}]

N[%]

Integrate[Boole[fish<0/.a->1],{y},{x}]

N[%]

area=FullSimplify[x[t]y'[t]-y[t]x'[t]]/a^2

FindRoot[(1/4) (3 Cos[t]+Cos[3 t]+2 (Sqrt[2]) ((Sin[t])^2))==0,{t,2,2.5}]

FindRoot[(1/4) (3 Cos[t]+Cos[3 t]+2 (Sqrt[2]) ((Sin[t])^2))==0,{t,3.9,4}]

{3,5}Pi/4.

Plot[Evaluate[area],{t,-2Pi,2Pi}]

Tail

Plots

Plot[Evaluate[area],{t,Pi/2,3Pi/2},PlotStyle->Red]

With[{a=1},ParametricPlot[{a Cos[t]-((a ((Sin[t])^2))/(Sqrt[2])),a Cos[t] Sin[t]},{t,Pi/2,Pi},AspectRatio->Automatic,PlotStyle->Red,
PlotRange->{{-1.2,.1},{-.5,.5}},AxesOrigin->{0,0},
Ticks->None,
Epilog->Line[{{-1,0},{10,0}}]]
]

Function[t,{a Cos[t]-((a ((Sin[t])^2))/(Sqrt[2])),a Cos[t] Sin[t]}/a]/@{Pi/2,2.35,Pi}//N

2(1-.707)(.5)/2

tail=With[{a=1},ParametricPlot[{a Cos[t]-((a ((Sin[t])^2))/(Sqrt[2])),a Cos[t] Sin[t]},{t,Pi/2,3Pi/2},AspectRatio->Automatic,PlotStyle->Red,
PlotRange->{{-1.2,.1},{-.5,.5}},AxesOrigin->{0,0},
Ticks->None,
Epilog->Line[{{-1,0},{10,0}}]]
]

tailarea=Abs[FullSimplify[Integrate[area,{t,Pi/2,Pi}]]]

N[%]

Area of Head

Plot[Evaluate[area],{t,-Pi/2,Pi/2},PlotStyle->Red]

head=With[{a=1},ParametricPlot[{a Cos[t]-((a ((Sin[t])^2))/(Sqrt[2])),a Cos[t] Sin[t]},{t,-Pi/2,Pi/2},AspectRatio->Automatic,PlotStyle->Red,
PlotRange->{{-.8,1},{-.5,.5}},
Ticks->None]
]

headarea=Simplify[Integrate[area,{t,-Pi/2,Pi/2}]/2]

Whole Fish

headarea+tailarea

fish=With[{a=1},ParametricPlot[{a Cos[t]-((a ((Sin[t])^2))/(Sqrt[2])),a Cos[t] Sin[t]},{t,0,2π},AspectRatio->Automatic,PlotStyle->Red,
PlotRange->{{-1.2,1},{-.5,.5}},
Ticks->None]
]

Show[GraphicsArray[{
tail,
head,
fish
}],ImageSize->400]

Inequality

Assuming[a>0,Area[-2 (a^4)-4 (Sqrt[2]) (a^3) x-2 (a^2) ((x^2)-5 (y^2))+(((2 (x^2)+(y^2)))^2)+2 (Sqrt[2]) a x (2 (x^2)+5 (y^2))<0,{x,y}]]

Assuming[a>0,Area[-2 (a^4)-4 (Sqrt[2]) (a^3) x-2 (a^2) ((x^2)-5 (y^2))+(((2 (x^2)+(y^2)))^2)+2 (Sqrt[2]) a x (2 (x^2)+5 (y^2))<0,{y,x}]]

fish2

Assuming[a>0,Area[(2x^2+y^2)^2-2Sqrt[2]a x(2x^2-3y^2)+2a^2(y^2-x^2)<0,{x,y}]]//Timing

Assuming[a>0,Area[(2x^2+y^2)^2-2Sqrt[2]a x(2x^2-3y^2)+2a^2(y^2-x^2)<0,{y,x}]]//Timing

Parametric

fish=a{Cos[t]-Sin[t]^2/Sqrt[2],Cos[t] Sin[t]};

orientedfish=
a{Cos[t]-Sin[t]^2/Sqrt[2],If[π/2<t<3π/2,-1,1]Cos[t] Sin[t]};

PiecewiseExpand[If[π/2<t<3π/2,-1,1],Assumptions->0<t<2π]

Manipulate[ParametricPlot[orientedfish/a,{t,0,tmax}],{{tmax,2π},0,2π}]

Assuming[a>0,Area[orientedfish,{t,0,2π}]]

Assuming[a>0,Area[fish,{t,0,2π}]]

Head

p1=Assuming[a>0,Area[fish,{t,-π/2,π/2}]]

Tail

p2=Assuming[a>0,Area[fish,{t,π/2,3π/2}]]

p1-p2//Simplify

p1+p2//Simplify

Arc Length Function

Sqrt[#.#&[D[curve,t]]]

FullSimplify[%,a>0&&0<t<2π]

Integrate[a (Sqrt[((Cos[t])^4)+Sin[t] (Sin[t]+((Sin[t])^3)+(Sqrt[2]) Sin[2 t])]),{t,0,t},Assumptions->a>0&&0<t<2π]

ArcLength[curve,t,Assumptions->a>0&&0<t<2π]

Reduce[5+2 (Sqrt[2]) Cos[t]+Cos[4 t]≥2 (Cos[2 t]+(Sqrt[2]) Cos[3 t])&&((Cos[t])^4)+Sin[t] (Sin[t]+((Sin[t])^3)+(Sqrt[2]) Sin[2 t])≥0,t,Reals]

Plot[Boole[5+2 (Sqrt[2]) Cos[t]+Cos[4 t]≥2 (Cos[2 t]+(Sqrt[2]) Cos[3 t])&&((Cos[t])^4)+Sin[t] (Sin[t]+((Sin[t])^3)+(Sqrt[2]) Sin[2 t])≥0],{t,0,2π}]

g[t_]:=((Sqrt[5+2 (Sqrt[2]) Cos[t]-2 Cos[2 t]-2 (Sqrt[2]) Cos[3 t]+Cos[4 t]]) (4 Sin[t]+2 (Sqrt[2]) (t-Cos[t] Sin[t])))/(8 (Cos[t]+(Sqrt[2]) ((Sin[t])^2)))

FullSimplify[g'[t]-Sqrt[Cos[t]^4+Sin[t] (Sin[t]+Sin[t]^3+Sqrt[2] Sin[2 t])]]

g[2π]

Plot[g[t],{t,0,2π}]

Plot[g[t],{t,2.2,2.4}]

h[x_?NumericQ]:=NIntegrate[ (Sqrt[((Cos[t])^4)+Sin[t] (Sin[t]+((Sin[t])^3)+(Sqrt[2]) Sin[2 t])]),{t,0,x}]

Plot[h[t],{t,0,2π}]

Remove Discontinuities

dcad[t_]=Sqrt[2] ((Sqrt[2]-Cos[t]) Sin[t]+ArcTan[Tan[t-Pi/4]]) Sqrt[(Sqrt[2] Cos[t]+1)^2 (Cos[t]-Sqrt[2])^2]/((Sqrt[2] Cos[t]+1) (-Cos[t]+Sqrt[2]));

Plot[dcad[t]-g[t],{t,0,2π}]

D[dcad[t],t]-2 Sqrt[(Sqrt[2] Cos[t]+1)^2 (Cos[t]-Sqrt[2])^2]//FullSimplify

In[292]:=
Differences[ Table[Limit[dcad[t], t -> 3 Pi/4 + 2 Pi n, Direction -> d, Assumptions -> n ∈ Integers], {d, {-1, 1}}]] // Expand

Out[292]= {3 Sqrt[2]}

In[293]:=
Differences@ Table[Limit[dcad[t], t -> (7 Pi)/4 + 2 Pi n, Direction -> d, Assumptions -> n ∈ Integers], {d, {-1, 1}}] // Expand

Out[293]= {Sqrt[2] π}

In[294]:=
Differences@ Table[Limit[dcad[t], t -> 5 Pi/4 + 2 Pi n, Direction -> d, Assumptions -> n ∈ Integers], {d, {-1, 1}}] // Expand

Out[294]= {3 Sqrt[2]}

Plot[dcad[t]+Sqrt[2] Sum[3 UnitStep[t-3/4 Pi-2 Pi n]+3 UnitStep[t-5/4 Pi-2 Pi n]+Pi UnitStep[t-7/4 Pi-2 Pi n],{n,0,5}]//Evaluate,{t,0,2 Pi}]

dcad[t]+Sqrt[2] (6+Pi+Pi Floor[(-7 Pi+4 t)/(8 Pi)]+3 Floor[(-5 Pi+4 t)/(8 Pi)]+3 Floor[(-3 Pi+4 t)/(8 Pi)])/.t->0

i[t_]:=(dcad[t]+Sqrt[2] (6+Pi+Pi Floor[(-7 Pi+4 t)/(8 Pi)]+3 Floor[(-5 Pi+4 t)/(8 Pi)]+3 Floor[(-3 Pi+4 t)/(8 Pi)])+(π/(2 (Sqrt[2]))))/2

{i[0],i[2π]}

FullSimplify[i[t],0<t<2π]

This needs to be simplified a bit more

Plot[i[t],{t,0,2π},AxesOrigin->{0,0}]

Plot[i[t],{t,0,24},AxesOrigin->{0,0},Exclusions->None]

Plot[Sign[(Sqrt[2])+Cos[t]-(Sqrt[2]) ((Cos[t])^2)],{t,0,6π}]

Reduce[(Sqrt[2])+Cos[t]-(Sqrt[2]) ((Cos[t])^2)==0,t,Reals]

N[%]

PiecewiseExpand[Cos[

?PiecewiseExpand

PiecewiseExpand[2ArcTan[Cot[π t/2]]/π,0<t<20,Reals]//InputForm

PiecewiseExpand[1-Mod[t,2],0<t<20,Reals]//InputForm

{Plot[2ArcTan[Cot[π t/2]]/π,{t,0,10}],Plot[1-Mod[t,2],{t,0,10},Exclusions->None]}

Plot[{ArcTan[Cot[(π/4)+t]],1-ArcTan[t]},{t,0,2π}]

FunctionExpand[ArcCot[π/4+t]]

Int

Continuous version of i[t] obtained using

Floor[z_]:>z-1/Pi ArcTan[Tan[Pi z-Pi/2]]+(1/2) ((-1)^(Floor[-z]+Floor[z]))

cad[t_]:=(t/Sqrt[2]+(3*Sqrt[2]*t)/Pi+Sqrt[2]*ArcTan[Cot[Pi/8+t/2]]+(3*Sqrt[2]*ArcTan[Tan[Pi/8-t/2]])/Pi-(3*Sqrt[2]*ArcTan[Tan[Pi/8+t/2]])/Pi+(Sqrt[2]*Sqrt[(-Sqrt[2]+Cos[t])^2*(1+Sqrt[2]*Cos[t])^2]*(-ArcTan[Tan[Pi/4-t]]+(Sqrt[2]-Cos[t])*Sin[t]))/((Sqrt[2]-Cos[t])*(1+Sqrt[2]*Cos[t]))-(π/(4 (Sqrt[2]))))/2

Plot[cad[t],{t,0,2π}]

D[cad[t],t]//FullSimplify[#,t∈Reals]&//FullSimplify[#-2 Sqrt[(Sqrt[2] Cos[t]+1)^2 (Cos[t]-Sqrt[2])^2],t∈Reals]&

With[{ϵ=1`50*10^-30},Table[{cad[x-ϵ]-cad[x+ϵ]},{x,Pi Range[0,16]/8}]]//Chop

i[.2]

Plot[i[t]-cad[t],{t,0,2π},PlotStyle->Red]

FullSimplify[i[t],t∈Reals]

FullSimplify[cad[t],t∈Reals]

FunctionExpand[Sign[z]]

FullSimplify[(z/(Abs[z]))]

Plot[Sign[(Sqrt[2])+Cos[t]-(Sqrt[2]) ((Cos[t])^2)],{t,0,4π}]

Arc Length

fish={x[t],y[t]}/.a->1

Integral Kernel

FullSimplify[Sqrt[x'[t]^2+y'[t]^2],t>0&&a>0]

Plot[(Sqrt[((Cos[t])^4)+(1+2 (Sqrt[2]) Cos[t]) ((Sin[t])^2)+((Sin[t])^4)]),{t,-2Pi,2Pi},
PlotStyle->Red]

FullSimplify[Sqrt[x'[t]^2+y'[t]^2]/.Sin[t]:>Sqrt[1-Cos[t]^2],t>0&&a>0]

Correct

Lockwood's result

Sqrt[2](Pi/2+3)//N

Together[Sqrt[2](Pi/2+3)]

Numerical

NIntegrate[Sqrt[Cos[t]^4+(1+2*Sqrt[2]*Cos[t])*Sin[t]^2+Sin[t]^4],{t,0,2*Pi}]

((6+π)/(Sqrt[2]))//N

Symbolic

V5

ArcLength[fish,{t,0,2π}]//Timing

N[Last[%]]

Integrate[Sqrt[Cos[t]^4+(1+2*Sqrt[2]*Cos[t])*Sin[t]^2+Sin[t]^4],{t,0,2*Pi}]//Timing

Integrate[Sqrt[Cos[t]^4+(1+2*Sqrt[2]*Cos[t])*Sin[t]^2+Sin[t]^4],{t,0,3π/4,5π/4,2π}]//Timing

Simplify[Last[%]]

N[%]

V6.0

ArcLength[curve/.a->1,{t,0,2π}]//FullSimplify

V6.1

FullSimplify[ArcLength[curve,{t,0,2π}],a>0]

Special Factorization

poly=Factor[Together[TrigToExp[Cos[t]^4+(1+2*Sqrt[2]*Cos[t])*Sin[t]^2+Sin[t]^4]/.Power[E,Times[Complex[0,n_],t]]:>x^n],Extension->Sqrt[2]]

FullSimplify[ExpToTrig[PowerExpand[Sqrt[poly]]/.x->Exp[I t]]]

s=FullSimplify[Integrate[Abs[Cos[t]+(Sqrt[2]) ((Sin[t])^2)],{t,0,2Pi}]]

2N[s-3Pi/Sqrt[2]]

NIntegrate[Sqrt[Cos[t]^4 + (1 + 2*Sqrt[2]*Cos[t])*Sin[t]^2 + 
    Sin[t]^4], {t, 0, 2*Pi}]

Integrate[Sqrt[Cos[t]^4+(1+2*Sqrt[2]*Cos[t])*Sin[t]^2+Sin[t]^4],{t,0,2*Pi}]

Properties

fish=a{Cos[t]-Sin[t]^2/Sqrt[2],Cos[t] Sin[t]};

orientedfish=
a{Cos[t]-Sin[t]^2/Sqrt[2],If[π/2<t<3π/2,-1,1]Cos[t] Sin[t]};

AreaInertiaTensor

Parametric

Assuming[a>0,AreaInertiaTensor[orientedfish,{t,0,2π}]]

Inequalities

Assuming[a>0,AreaInertiaTensor[-2 (a^4)-4 (Sqrt[2]) (a^3) x-2 (a^2) ((x^2)-5 (y^2))+(((2 (x^2)+(y^2)))^2)+2 (Sqrt[2]) a x (2 (x^2)+5 (y^2))<0,{x,y}]]//Timing

Assuming[a>0,AreaInertiaTensor[-2 (a^4)-4 (Sqrt[2]) (a^3) x-2 (a^2) ((x^2)-5 (y^2))+(((2 (x^2)+(y^2)))^2)+2 (Sqrt[2]) a x (2 (x^2)+5 (y^2))<0,{y,x}]]//Timing

Centroid

Parametric

Assuming[a>0,Centroid[orientedfish,{t,0,2π}]]

Inequalities

Assuming[a>0,Centroid[-2 (a^4)-4 (Sqrt[2]) (a^3) x-2 (a^2) ((x^2)-5 (y^2))+(((2 (x^2)+(y^2)))^2)+2 (Sqrt[2]) a x (2 (x^2)+5 (y^2))<0,{x,y}]]//Timing

Assuming[a>0,Centroid[-2 (a^4)-4 (Sqrt[2]) (a^3) x-2 (a^2) ((x^2)-5 (y^2))+(((2 (x^2)+(y^2)))^2)+2 (Sqrt[2]) a x (2 (x^2)+5 (y^2))<0,{y,x}]]//Timing

Curvature

Factor[FullSimplify[Curvature[curve,t],a>0]]

Plot[((2 (Sqrt[2])+3 Cos[t]-Cos[3 t])/(2(((((Cos[t])^4)+((Sin[t])^2)+((Sin[t])^4)+(Sqrt[2]) Sin[t] Sin[2 t]))^(3/2)))),{t,0,2π},PlotRange->{0,10}]

SortBy[FullSimplify[Curvature[fish,{x,y}],a>0&&(x|y)∈Reals],ByteCount]

ByteCount/@%

Inflection Points

InflectionPoints[fish,{x,y}]

Singular Points

SingularPoints[fish,{x,y}]

Union[%]

Slope

Slope[a{Cos[t]-Sin[t]^2/Sqrt[2],Cos[t] Sin[t]},t]//FullSimplify

Slope[-2 a^4-4 Sqrt[2] a^3 x-2 a^2 (x^2-5 y^2)+(2 x^2+y^2)^2+2 Sqrt[2] a x (2 x^2+5 y^2)==0,{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[fish,t],a>0&&0<t<2π]

TangentialAngle

TangentialAngle[curve,t]

FullSimplify[((2 (Sqrt[2])+3 Cos[t]-Cos[3 t])/(2 a (((((Cos[t])^4)+((Sin[t])^2)+((Sin[t])^4)+(Sqrt[2]) Sin[t] Sin[2 t]))^(3/2))))Sqrt[#.#&[D[curve,t]]],a>0]

FullSimplify[((2 (Sqrt[2])+3 Cos[t]-Cos[3 t])/(2 (((Cos[t])^4)+Sin[t] (Sin[t]+((Sin[t])^3)+(Sqrt[2]) Sin[2 t])))),0<t<2π]

Integrate[((2 (Sqrt[2])+3 Cos[t]-Cos[3 t])/(2 (((Cos[t])^4)+Sin[t] (Sin[t]+((Sin[t])^3)+(Sqrt[2]) Sin[2 t])))),{t,0,t}]

f[x_?NumericQ]:=NIntegrate[((2 (Sqrt[2])+3 Cos[t]-Cos[3 t])/(2 (((Cos[t])^4)+Sin[t] (Sin[t]+((Sin[t])^3)+(Sqrt[2]) Sin[2 t])))),{t,0,x}]

Plot[f[x],{x,0,2π},PlotPoints->10]

TrigToExp[((2 (Sqrt[2])+3 Cos[t]-Cos[3 t])/(2 (((Cos[t])^4)+Sin[t] (Sin[t]+((Sin[t])^3)+(Sqrt[2]) Sin[2 t]))))]

Simplify[Together[%]]

Integrate[-(2 (E^(I t)) (1-3 (E^(2 I t))-4 (Sqrt[2]) (E^(3 I t))-3 (E^(4 I t))+(E^(6 I t))))/(1-2 (Sqrt[2]) (E^(I t))-2 (E^(2 I t))+2 (Sqrt[2]) (E^(3 I t))+10 (E^(4 I t))+2 (Sqrt[2]) (E^(5 I t))-2 (E^(6 I t))-2 (Sqrt[2]) (E^(7 I t))+(E^(8 I t))),{t,0,t}]

ExpToTrig[%]//FullSimplify

f[t_]:=π-(1/2) I Log[2]-I Log[-1+(Sqrt[2])-(E^(I t))]+I Log[(Sqrt[2])+(-2+(Sqrt[2])) (E^(I t))]

N[f[0],10]

Limit[f[t],t->0]//FullSimplify

Plot[f[t],{t,0,2π}]

fn[t_?NumericQ]:=π-(1/2) I Log[2]-I Log[-1+(Sqrt[2])-(E^(I t))]+I Log[(Sqrt[2])+(-2+(Sqrt[2])) (E^(I t))]

Plot[fn[t],{t,0,2π},PlotPoints->10]

f[t]

Plot[π+Log[(((((Sqrt[2])+(-2+(Sqrt[2])) (E^(I t))))^I)/((2^(I/2))(((-1+(Sqrt[2])-(E^(I t))))^I)))],{t,0,2π}]

Plot[π+I Log[((((Sqrt[2])+(-2+(Sqrt[2])) (E^(I t))))/((2^(1/2))(-1+(Sqrt[2])-(E^(I t)))))],{t,0,2π}]

FullSimplify[((((Sqrt[2])+(-2+(Sqrt[2])) (E^(I t))))/((2^(1/2))(-1+(Sqrt[2])-(E^(I t)))))]

FullSimplify[ComplexExpand[Im[Log[-1+(Sqrt[2])-(2/(-1+(1+(Sqrt[2])) (E^(I t))))]]],0<t<2π]

Plot[π-Arg[-1+(Sqrt[2])-(2/(-1+(1+(Sqrt[2])) (E^(I t))))],{t,0,2π}]

Limit[π-Arg[-1+(Sqrt[2])-(2/(-1+(1+(Sqrt[2])) (E^(I t))))],t->#1,Direction->#2]&@@@{{0,-1},{2π,1}}

Vector Length

FullSimplify[VectorLength[a{Cos[t]-Sin[t]^2/Sqrt[2],Cos[t] Sin[t]},t],a>0&&0<t<2π]

ExpandAll@FullSimplify[VectorLength[a{Cos[t]-Sin[t]^2/Sqrt[2],Cos[t] Sin[t]},t],a>0&&0<t<2π]//FullSimplify

Vector Properties

parm=a{Cos[t]-Sin[t]^2/Sqrt[2],Cos[t] Sin[t]};
assum=a>0&&0<t<2π;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};

VectorLength

Assuming[assum,FullSimplify[Norm[parm]/.xform]]

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]/.xform]]

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]/.xform]]