Astroids
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
- Eric W. Weisstein : Astroid . In: MathWorld (English).
- Astroide at 2dcurves.com (English)
- Stoner-Wohlfarth Astroids Applet (physics). (English)