Hügelschäffer Egg

A geometric construction of an egg-shaped curve and its surface of revolution

Summary

This notebook presents the Hügelschäffer egg, a classical egg-shaped curve obtained from two circles by a simple geometric construction. Starting from the planar curve, we derive its parametric and Cartesian equations and then generate the corresponding three-dimensional egg by rotation about the x-axis.

1. Geometric idea

Consider two circles:
◼
  • a major circle centered at the origin with radius
    R
    • ◼
  • a minor circle centered at
    (d,0)
    with radius r
    • with
    R > r
    .
    A line is drawn from the center
    (d,0)
    of the minor circle, forming an angle θ with the positive x-axis. This line intersects:
    ◼
  • the major circle at
    P
    1
    • ◼
  • the minor circle at
    P
    2
    • The point
    P
    E
    = (x,y)
    on the egg curve is obtained by combining:
    ◼
  • the x-coordinate of
    P
    1
    • ◼
  • the y-coordinate of
    P
    2
    • This produces an asymmetric closed curve with the characteristic shape of an egg.

    Reference

    ◼
  • MathCurve
    • ◼
  • AI-assisted development and personal notes

    • 2. Planar setup

    We begin with the two circles in parametric form.
    ζ[t_]:={RCos[t],RSin[t]}​​
    ζ
    min
    [t_]:={rCos[t]+d,rSin[t]}
    A line through
    (d,0)
    and the point
    ζ[t]
    on the major circle can be written as:
    {d,0}+w(ζ[t]-{d,0})
    which expands to:
    {d+w(-d+RCos[t]),RwSin[t]}

    3. Intersection with the minor circle

    To locate the intersection with the minor circle, substitute the line into
    2
    (x-d)
    +
    2
    y
    
    2
    r
    and solve for w.
    Solve[
    2
    (d+w(-d+RCos[t])-d)
    +
    2
    (RwSin[t])
    ==
    2
    r
    ,w]
    w-
    r
    2
    d
    -2dRdCos[t]+
    2
    R
    2
    Cos[t]
    +
    2
    R
    2
    Sin[t]
    ,w
    r
    2
    d
    -2dRdCos[t]+
    2
    R
    2
    Cos[t]
    +
    2
    R
    2
    Sin[t]
    
    Using the positive branch gives the y-coordinate inherited from the minor-circle intersection. Combining it with the x-coordinate from the major circle yields the egg curve.

    4. Parametric equation of the Hügelschäffer egg

    P
    E
    [t_]:=RCos[t],R
    r
    2
    d
    -2dRCos[t]+
    2
    R
    2
    Cos[t]
    +
    2
    R
    2
    Sin[t]
    Sin[t]
    This parametrization makes the construction especially transparent:
    ◼
  • x(t) = R Cos[t]
    comes from the major circle
    • ◼
  • y(t)
    comes from the minor-circle intersection along the same ray

    • 5. Cartesian equation

    Starting from
    Eliminatex==RCos[t],y==R
    r
    2
    d
    -2dRCos[t]+
    2
    R
    2
    Cos[t]
    +
    2
    R
    2
    Sin[t]
    Sin[t],t//FullSimplify
    we obtain the Cartesian equation
    2
    r
    (R-x)(R+x)(
    2
    d
    +
    2
    R
    -2dx)
    2
    y
    or equivalently
    2
    r
    (
    2
    R
    -
    2
    x
    )(
    2
    d
    +
    2
    R
    -2dx)
    2
    y
    This form clearly shows the algebraic relation satisfied by the egg curve.

    6. Interactive planar construction

    In[]:=
    ManipulateR=6;r=4;P2[t_]:=d+
    r
    2
    d
    -2dRCos[t]+
    2
    R
    2
    Cos[t]
    +
    2
    R
    2
    Sin[t]
    (-d+RCos[t]),R
    r
    2
    d
    -2dRCos[t]+
    2
    R
    2
    Cos[t]
    +
    2
    R
    2
    Sin[t]
    Sin[t];​​
    P
    E
    [t_]:=RCos[t],R
    r
    2
    d
    -2dRCos[t]+
    2
    R
    2
    Cos[t]
    +
    2
    R
    2
    Sin[t]
    Sin[t];​​ζ[t_]:={RCos[t],RSin[t]};
    ζ
    min
    [t_]:={rCos[t]+d,rSin[t]};​​O1={0,0};O2={d,0};​​​​Show[{​​ParametricPlot[ζ[θ],{θ,0,2π},PlotStyle->{Blue,Thick}],​​ParametricPlot[
    ζ
    min
    [θ],{θ,0,2π},PlotStyle->{Black,Thick}],​​ParametricPlot[
    P
    E
    [θ],{θ,0,t},ColorFunction->"BeachColors"],​​Graphics[{Black,PointSize[0.01],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[O2],Text[Style["O'",14,Bold],O2,{0,-1.2}],​​Point[ζ[t]],Text[Style["
    P
    1
    ",14,Bold],ζ[t],{0,-1.2}],​​Point[P2[t]],Text[Style["
    P
    2
    ",14,Bold],P2[t],{0,-1.2}],​​Point[
    P
    E
    [t]],Text[Style["
    P
    E
    ",14,Bold],
    P
    E
    [t],{0,-1.2}],​​Thick,{DarkYellow,Line[{O2,ζ[t]}],Line[{
    P
    E
    [t],P2[t]}],Line[{
    P
    E
    [t],ζ[t]}]}​​}]},AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-6.5,6.5},{-6,6}},ImageSize700],Style["Hügelschäffer Egg",Bold,Large],{{t,0.00002,"Value (t)"},0.000001,2π,0.00001},Delimiter,{{d,-1.60421,"Value (d)"},-2,2,0.00001},ControlPlacementTop
    Out[]=
    ​
    Hügelschäffer Egg
    Value (t)
    Value (d)

    What the animation shows

    7. From curve to solid: a surface of revolution

    Rotating the planar egg around the x-axis produces a three-dimensional egg-shaped surface.

    Interpretation

    8. Cartesian equation of the surface of revolution

    Starting from
    we obtain
    This is the Cartesian equation of the three-dimensional egg-shaped surface.

    Verification by implicit plotting

    9. Parametric equation of the surface of revolution

    Verification by parametric plotting

    10. Remarks

    The Hügelschäffer egg is a nice example of how a simple geometric prescription can produce:
    ◼
  • a visually natural asymmetric curve
    • ◼
  • an exact parametric representation
    • ◼
  • an implicit algebraic equation
    • ◼
  • a convincing three-dimensional egg by revolution
    • The construction is especially appealing because the geometry, algebra, and visualization reinforce each other.

    11. Possible extensions

    Some natural directions for further exploration are:
    ◼
  • study how the shape changes as d varies
    • ◼
  • compare the curve with other classical egg curves
    • ◼
  • examine curvature along the planar profile
    • ◼
  • compute volume and surface area of the solid of revolution