Modeling the Shape of an Egg

I-Heng McComb, Jun 23, 2018

Exploration of five models for the oval shape of an egg....

Table of Contents

  1. Not Quite an Ellipse
    1. Meet Our Egg
    2. Ellipse Model: Implicit Cartesian Equation (Transformed Circle)
    3. Ellipse: Locus Relative to Foci
    4. Ellipse: Polar Parametric Equations
  2. Descartes Oval
    1. Descartes Oval: Geometric Construction
    2. Descartes Oval: Locus Relative to Foci
  3. Cassini Oval
    1. Cassini Oval: Locus Relative to Foci
    2. Cassini Oval: Implicit Cartesian Equation
    3. Cassini Oval: Polar Parametric Equations
  4. Tweaked x-Coordinates
    1. Adjusted Ellipse: Polar Parametric Equations
  5. Pieces of Circles?
    1. Curvature Modeling

1.

Not Quite an Ellipse

  • 1. Meet Our Egg
  • 2. Ellipse Model: Implicit Cartesian Equation (Transformed Circle)
  • 3. Ellipse: Locus Relative to Foci
  • 4. Ellipse: Polar Parametric Equations

1.1.

Meet Our Egg

With the egg centered at the origin, "blunt" end at the bottom and "pointy" end at the top, note that with this scaling, the height of the egg is (approximately) 6 units and the width at midline is (approximately) 4.5 units. Also note that the width at midline is slightly less than the maximum width of the egg...
– I-Heng McComb

1.2.

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1.3.

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1.4.

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2.

Descartes Oval

  • 1. Descartes Oval: Geometric Construction
  • 2. Descartes Oval: Locus Relative to Foci

2.1.

Descartes Oval: Geometric Construction

Move the green point along the top circle and watch Point [i]P[/i]....
– I-Heng McComb

2.2.

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3.

Cassini Oval

  • 1. Cassini Oval: Locus Relative to Foci
  • 2. Cassini Oval: Implicit Cartesian Equation
  • 3. Cassini Oval: Polar Parametric Equations

3.1.

Cassini Oval: Locus Relative to Foci

[size=150]For an ellipse, the [b]sum[/b] of the distances from the foci is constant. For a Cassini oval, the [b]product[/b] of the distances from the foci is constant.[/size]
– I-Heng McComb

3.2.

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3.3.

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4.

Tweaked x-Coordinates

  • 1. Adjusted Ellipse: Polar Parametric Equations

4.1.

Adjusted Ellipse: Polar Parametric Equations

Here we start with the parametric equations of the ellipse model, [math]x\left(\theta\right)=2.25\cdot cos\left(\theta\right)[/math] and [br][math]y\left(\theta\right)=3\cdot sin\left(\theta\right)[/math] , but adjust the [i]x[/i] coordinate with the factor [math]1+a\ast sin\left(\theta\right)+b\ast sin^2\left(\theta\right)[/math].
– I-Heng McComb

5.

Pieces of Circles?

  • 1. Curvature Modeling

5.1.

Curvature Modeling

This one uses the "tweaked" ellipse idea, but models the egg shape around 0 degrees, 90 degrees, 180 degrees, and 270 degrees as arcs of circles matching the curvature at those angles.
– I-Heng McComb

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