Astroids

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Animation of the astroids
Animation astroids as an envelope
Astroids as the envelope of a family of ellipses, in which a + b = const.

The astroide (also known as the star curve ) is a flat curve that is determined by the parametric equations with a parameter

or by the implicit equation

To which equivalent   is

can be described, where is a fixed positive, real number . It is the curve that describes a point on a circle with a radius that rolls inside on a circle with a radius . So it is a special hypocycloid .

The following applies to their area

.

The length of the entire curve is . Within a curve quarter applies to the arc length

and for the radius of curvature

.

The astroids are also similar to the diamonds on ordinary playing cards.

main emphasis

Focus of the astroids
interval
Flat curve piece 0 ≤ t ≤
0 ≤ t ≤ 0
Level figure 0 ≤ t ≤
0 ≤ t ≤ 0
Rotating body * 0 ≤ t ≤ 0

* When rotating around the X axis

Leaning astroids

A generalization is the skewed astroids, which are represented by the parametric equations

or by the implicit equation

can be described. The evolution of an ellipse is also a crooked astroid.

See also

Web links

Commons : Astroid  - collection of pictures, videos and audio files
Wiktionary: Astroids  - explanations of meanings, word origins, synonyms, translations