Parabolic Segment

Author

Eric W. Weisstein
June 28, 2013

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/ParabolicSegment.nb.

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

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

Full Segment

curve={t,h(1-t^2/a^2)};

Plot

<<Graphics`InequalityGraphics`

InequalityPlot[ y<=1-x^2, {x,-1,1},{y,0,1},Ticks->None]

RegionPlot[ y<=1-x^2, {x,-1,1},{y,0,1},Frame->False,Axes->True,Ticks->None,PlotStyle->Hue[.5],AspectRatio->Automatic]

Arc Length

D[h(1-x^2/a^2),x]

s=2Integrate[Sqrt[1+(-((2 h x)/(a^2)))^2],{x,0,a},Assumptions->a>0&&h>0] //Simplify

Simplify[Limit[(Sqrt[(a^2)+4 (h^2)])+(((a^2) ArcSinh[((2 h)/a)])/(2 h)),h->0],Assumptions->a>0]

expr=TrigToExp[2((2 h (Sqrt[(a^2)+4 (h^2)])+(a^2) Log[((2 h+(Sqrt[(a^2)+4 (h^2)]))/a)])/(4 h))-s]//Factor

FullSimplify[expr,a>0&&h>0]

Area

h (1-((x^2)/(a^2)))

Integrate[h(1-x^2/a^2),{x,-a,a}]

Assuming[a>0&&h>0,Area[h x^2+a^2 (-h+y)<0&&y>0,{x,y}]]

Centroid

Integrate[1,{x,-a,a},{y,0,h(1-x^2/a^2)}]

Integrate[y,{x,-a,a},{y,0,h(1-x^2/a^2)}]

Integrate[y,{x,-a,a},{y,0,h(1-x^2/a^2)}]/Integrate[1,{x,-a,a},{y,0,h(1-x^2/a^2)}]

With[{h=2,a=1},ParametricPlot[{t,h(1-t^2/a^2)},{t,-a,a}]]

With[{h=2,a=1},RegionPlot[h x^2+a^2 (-h+y)<0&&y>0,{x,-1,1},{y,-1,2}]]

Assuming[a>0&&h>0,Centroid[h x^2+a^2 (-h+y)<0&&y>0,{x,y}]]

AreaInertiaTensor

Assuming[a>0&&h>0,AreaInertiaTensor[h x^2+a^2 (-h+y)<0&&y>0,{x,y}]]//Timing

Curvature

FullSimplify[Curvature[{t,h(1-t^2/a^2)},t],a>0&&-a<t<a&& h>0]

ImplicitCurvature

Curvature[h x^2+a^2 (-h+y),{x,y}]

Slope

Slope[h(1-t^2/a^2),t]//FullSimplify

Slope[h x^2+a^2 (-h+y)==0,{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&h>0&&-a<t<a]

TangentialAngle

FullSimplify[Assuming[a>0&&t<a&& h>0,TangentialAngle[{t,h(1-t^2/a^2)},t]],a>0&&t<a&& h>0]

Parabolic Segment cut off by line

Plot

Show[{
InequalityPlot[y>x^2∧y<2/5+1/3x, {x,-1,1},{y,0,1},Ticks->None,DisplayFunction->Identity],
Plot[{1,x^2},{x,-1.1,1.1},DisplayFunction->Identity]
},DisplayFunction->$DisplayFunction]

Area

FullSimplify[Integrate[(a x+b)-x^2,{x,(a-Sqrt[a^2+4b])/2,(a+Sqrt[a^2+4b])/2}]]

Maximum Area

(SuperMinus[x])=(a-Sqrt[a^2+4b])/2;
(SuperPlus[x])=(a+Sqrt[a^2+4b])/2;
(SuperMinus[y])=(SuperMinus[x])^2;
(SuperPlus[y])=(SuperPlus[x])^2;
(SuperStar[y])=(SuperStar[x])^2;

A=FullSimplify[Det[{{(SuperPlus[x]),(SuperPlus[y]),1},{(SuperMinus[x]),(SuperMinus[y]),1},{(SuperStar[x]),(SuperStar[y]),1}}]]/2

D[A,(SuperStar[x])]

Solve[D[A,(SuperStar[x])]==0,(SuperStar[x])]//Flatten

FullSimplify[A /.%]

Vector Properties

parm={t,h(1-t^2/a^2)};
assum=a>0&&h>0&&-a<t<a;
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]]/.xform

NormalVector

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

TangentVector

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