Parabola

Author

Eric W. Weisstein
June 6, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Code

ParabolaMovie[a_,n_]:=Module[{g=ParametricPlot[{a t^2,t},{t,0,2},DisplayFunction->Identity][[1]],pts,p,t},
pts={PointSize[.02],Point[{a,0}]};
Table[Show[Graphics[
{g,
{RGBColor[1,0,0],Line[{{-f,0},{Cos[t],b Sin[t]},{f,0}}]},
{PointSize[.02],pts,Point[{Cos[t],b Sin[t]}]}
},AspectRatio->Automatic]],{t,0,2Pi-2Pi/n,2Pi/n}]
]

Equations

parabola=a{2t,t^2};

Cartesian

Factor/@GroebnerBasis[Thread[{x,y}==a{2t,t^2}],{x,y},t]

Solve[First[%]==0,y]

Polar Coordinates centered on focus

r =- ((2a)/(1+cos θ))

Together/@Cos[ArcTan[y/(x-a)]]

FullSimplify[PowerExpand[%]]

Simplify[Sqrt[(x-a)^2 +y^2](1+((x-a)/(Sqrt[(((x-a))^2)+(y^2)])))+2a]

Solve[%==0,x]

eps=.7;
Show[GraphicsArray[{
PolarPlot[-2*.7/(1+Cos[t]),{t,-2.5,2.5},Ticks->None,PlotRange->{{3,-1},{-3,3}},Epilog->{PointSize[.06],Point[{0,0}]},DisplayFunction->Identity],
PolarPlot[Evaluate[Table[-2/(1+Cos[t])i,{i,.1,1,.1}]],{t,0,2Pi},Ticks->None,PlotRange->{{3,-1},{-3,3}},
Epilog->{PointSize[.06],Point[{0,0}]},DisplayFunction->Identity]
}]]

Polar coordinates centered on vertex

Sqrt[#.#]&[{2t,t^2}]//Simplify

Solve[θ==ArcTan[((t^2)/(2t))],t]

FullSimplify[(Sqrt[(t^2) (4+(t^2))])/.t->2Tan[q],0<q<2π]

PolarPlot[4 Sec[q]Tan[q],{q,-1,1}]

Parabola

Plot[x^2,{x,-1,1},Ticks->None,PlotStyle->Red,AspectRatio->Automatic]

With[{p=1/30},
Plot[{2Sqrt[p x],-2Sqrt[p x]},{x,0,1/2},
Ticks->None,AspectRatio-> Automatic,PlotStyle->Red]
]

Parabola through 3 Points

Parabola1[m_List,{x_,y_}]:=Det[Prepend[{#1^2,#1,#2,1}&@@@m,{x^2,x,y,1}]]/;Dimensions[m]=={3,2}
Parabola2[m_List,{x_,y_}]:=Det[Prepend[{#2^2,#1,#2,1}&@@@m,{y^2,x,y,1}]]/;Dimensions[m]=={3,2}

pts=Table[Random[],{3},{2}]

<<Graphics`ImplicitPlot`
<<Graphics`Colors`

ImplicitPlot[{Parabola1[pts,{x,y}]==0,Parabola2[pts,{x,y}]==0},{x,-1,2},Epilog->{Red,PointSize[.03],Point/@pts},PlotRange->{{-.5,2},{-.1,2}}]

Envelopes

ParabolaStrings[a_,n_]:=Module[{i},
		Graphics[{
				Line[{{0,0},{1,0}}],Line[{{0,0},{Cos[a],Sin[a]}}],
				Table[Line[{{i/n,0},(1-i/n){Cos[a],Sin[a]}}],{i,n-1}]
				}]
]

ParabolaStringsPoints[a_,n_]:=Module[{i},
		Graphics[{
				ParabolaStrings[a,n][[1]],
{PointSize[.03],
				Point/@Table[{i/n,0},{i,n-1}],
				Point/@Table[(1-i/n){Cos[a],Sin[a]},{i,n-1}]
					}
			}]
	]

Show[GraphicsArray[{
Show[ParabolaStringsPoints[60 Degree,5],
				AspectRatio->Automatic,DisplayFunction->Identity],
Show[ParabolaStrings[60 Degree,20],
				AspectRatio->Automatic,DisplayFunction->Identity]
		}]]

Focus

Ray[y_,p_]:=Module[{r=1.5},
		Line[{{r,y},{y^2/(4p),y},
			{p r+(1-r)y^2/(4p),y(1-r)}}]
	]

p=1/4;
r=10;
Plot[{2Sqrt[p x],-2Sqrt[p x]},{x,0,1/2},
AspectRatio->Automatic,Ticks->None,
Prolog->{{GrayLevel[.5],Table[Ray[i,p],{i,0,.6,.1}]},
			{PointSize[.05],Point[{p,0}]}}
		]

Tangents

p=1/4;
P={.6,.9};
Plot[{2Sqrt[p x],-2Sqrt[p x]},{x,0,2},
AspectRatio->Automatic,Ticks->None,PlotRange->{{-.1,2.5},{-1.5,1.5}},
Epilog->{Circle[({p,0}+P)/2,Sqrt[({p,0}-P).({p,0}-P)]/2],
		{Circle[P,.03],Circle[{p,0},.03]}}
		]

Directrix

Parabola[a_]:=Show[Graphics[{
ParametricPlot[{t^2/4/a,t},{t,-2,2},Ticks->None,DisplayFunction->Identity][[1]],
PointSize[.05],Point[{a,0}],Line[{{-a,-2},{-a,2}}],
{Dashing[{.05,.05}],Line[{{-a-.2,0},{2^2,0}}]}
},AspectRatio->Automatic]
]

Parabola[a_,n_]:=Module[{dy=.1},Show[Graphics[{{RGBColor[1,0,0],Table[Line[{{-a,t},{t^2/4/a,t},{a,0}}],{t,0,2,2/n}]},
ParametricPlot[{t^2/4/a, t},{t,-2-dy,2+dy},Ticks->None,DisplayFunction->Identity][[1]],
PointSize[.05],Point[{a,0}],
Line[{{-a,-2-dy},{-a,2+dy}}],
{Dashing[{.05,.05}],Line[{{-a-.2,0},{2^2,0}}]},
{RGBColor[1,0,0],Table[Point[{t^2/4/a,t}],{t,0,2,2/n}]}
},AspectRatio->Automatic]
]
]

Show[GraphicsArray[{
Parabola[.5],Parabola[.5,5]
}]]

Properties

parabola=a{2t,t^2};

ParametricPlot[{2t,t^2},{t,-1,1}]

Arc Length

FullSimplify[ArcLength[parabola,t],a>0]

Curvature

Parametric

FullSimplify[Curvature[parabola,t],a>0]

Polar

FullSimplify[Curvature[4 a Sec[q] Tan[q],q],a>0&&q∈Reals]

Implicit

FullSimplify[Curvature[x^2-4 a y,{x,y}],a>0]

Differential Equation

xy=(y[x]-y0)^2-4a(x-x0);
eqs=Table[D[xy,{x,i}],{i,0,3}];

gb=GroebnerBasis[eqs,{x,y[x],y'[x],y''[x],y'''[x]},{x0,y0,a},MonomialOrder->EliminationOrder]

TraditionalForm[gb[[1]]==0]

Solve[xy==0,y[x]]

expr=2 (Sqrt[a x-a x0])+y0;

Simplify[expr/.{x->0,x0->0,y0->0},a>0]

Simplify[Limit[D[expr,x],x->x0],a>0]

Slope

Slope[{2a t,a t^2},t]

Slope[x^2-4 a y==0,{x,y}]

Slope[4 a Sec[q] Tan[q],q]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[parabola,t],a>0]

Tangential Angle

FullSimplify[TangentialAngle[parabola,t],t>0]

VectorLength

FullSimplify[VectorLength[{2a t,a t^2},t],a>0&&t>0]

FullSimplify[VectorLength[x^2-4 a y==0,{x,y}],a>0&&t>0&&(x|y)∈Reals]

ByteCount/@%

ByteCount[((Abs[x])/(4a))(Sqrt[16(a^2)+(x^2)])]

FullSimplify[((Abs[x])/(4a))(Sqrt[16(a^2)+(x^2)])-(1/4) (Sqrt[16 (x^2)+((x^4)/(a^2))]),a>0&&x∈Reals]

VectorLength[4 a Sec[q] Tan[q],q]

Tangent and Normal Vectors

FullSimplify[TangentVector[parabola]]

NormalTangentPlot[{2t,t^2},{t,-4,4,.2},20]

Vector Properties

parm={2a t,a t^2};
assum=t∈Reals;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n,Sqrt[x_^(n_/;Mod[n,2]==0)]:>Abs[x]^(n/2)};

VectorLength

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

NormalVector

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

TangentVector

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