Astroid
 

Annotation 2019 04 13 110248
The curve called the Astroid is shown to the left with its algebraic equation.  Astroid (also sometimes referred to as the tetracuspid), of course, means star-shaped.  More details about this curve can be found in my book Playing with Dynamic Geometry, Chapter 11.



There are many different geometric constructions for the Astroid (and over time, I will use a lot of them).  The animations below are presented in pairs.  The first of the pairs is the basic geometric construction that either generates the Astroid or demonstrates one of its properties.  The second of the pairs is my “artistic” rendering based on that basic construction.

 

The Astroid by Parametric Equation

The Astroid by Parametric Equation Art

The Astroid by Compass only

The Astroid by Compass only art

A Pedal Curve of the Astroid

A Pedal Curve of the Astroid Art

The Astroid as an Envelope of Lines

The Astroid as an Envelope of Lines Art

A Revolving Astroid and Equilateral Triangle

A Revolving Astroid and Equilateral Triangle Art

An Astroid, a Deltoid, and a Common Tangent

An Astroid, a Deltoid, and a Common Tangent art

Three Deltoids Inside an Asrtoid

Three Deltoids Inside an Asrtoid art

The Astroid as a Hypocycloid Revisited

The Astroid as a Hypocycloid  Revisited Art

The Astroid as an Envelope of Ellipses

The Astroid as an Envelope of Ellipses art

The Astroid as a Hypocycloid

The Astroid as a Hypocycloid Art

Two Astroids for the price of one

Two Astroids for the price of one Art

Astroid 4 cusped Epicycloid

Astroid 4 cusped Epicycloid Art

Astroid Trammel of Archimedes

Astroid Trammel of Archimedes art1

Astroid's Osculating Circle

Astroid's Osculating Circle Art

Astroid Compass Only

Astroid Compass Only Art

Concurrent Tangents of the Astroid 

Concurrent Tangents of the Astroid Art

Astroid by Envelope of Line Segments

Astroid by of Envelope Line Segments Art