Astroid -- from Wolfram MathWorld
WEBAug 22, 2024 · Download Wolfram Notebook. A 4-cusped hypocycloid which is sometimes also called a tetracuspid, cubocycloid, or paracycle. The parametric equations of the astroid can be obtained by plugging in or into the equations for a general hypocycloid, giving parametric equations.
Astroid - Encyclopedia of Mathematics
WEBMar 26, 2023 · Astroid. A plane algebraic curve of order six, described by a point $M$ on a circle of radius $r$ rolling on the inside of a circle of radius $R=4r$; a hypocycloid with module $m=4$. Its equation in orthogonal Cartesian coordinates is. $$x^ {2/3}+y^ {2/3}=R^ {2/3};$$ and a parametric representation is. $$x=R\cos^3\frac t4,\quad y=R\sin^3\frac t4.$$
calculus - Length of an Astroid - Mathematics Stack Exchange
WEBA potentially easier way to do this is to parametrize the astroid by taking advantage of the trig identity cos2(θ) + sin2(θ) = 1. Take x = cos3(t) and y = sin3(t) where 0 ≤ t ≤ 2π. By symmetry, we can simply find the arclength of 1 / 4 th the astroid and multiply by 4 …
Astroid with two Constructions - Virtual Math Museum
WEBThe astroid is defined by these equations: c(t) = RR * [ cube(cos(t)), cube(sin(t)) ] Of course it is a special case of the Hypocycloids with frequency = -3 and stick = 1.
WEBAn Astroid is a curve traced out by a point on the circum-ference of one circle (of radius r) as that circle rolls without slipping on the inside of a second circle having four times or four-thirds times the radius of the first. The latter is known as double generation.
1.11: Optional — The Astroid - Mathematics LibreTexts
WEBJan 11, 2022 · Imagine a ball of radius \(a/4\) rolling around the inside of a circle of radius \(a\text{.}\) The curve traced by a point \(P\) painted on the inner circle (that's the blue curve in the figures below) is called an astroid 1. We shall find its equation. Define the angles \(\theta\) and \(\phi\) as in the figure in the left below. That is
How Can I Calculate Area of Astroid Represented by Parameter?
WEBLet $x=2\cos^3\theta$ and $y=2\sin^3\theta$ known as the astroid. In this case, radius $r=2$. and gray part's $x$ range is $1/\sqrt{2}\leq x\leq 2$. this deal with $0\leq\theta\leq \pi/4$. Question. How can I calculate area of gray part in this picture?
Astroid - Vocab, Definition, and Must Know Facts | Fiveable
WEBAn astroid is a parametric curve that resembles a four-lobed rose shape. It has four cusps, or sharp points, where the curvature is infinite, and four points of inflection where the curvature changes sign. The astroid is defined by the parametric equations $x (t) = a\cos^3 (t)$ and $y (t) = a\sin^3 (t)$, where $a$ is the radius of the astroid.
ASTROID - MATHCURVE.COM
WEBThe astroid is a hypocycloid with four cusps (a circle having a/4 (or 3a/4) for radius running in a circle (C) having a for radius).
Astroid - Michigan State University
WEBMay 25, 1999 · Astroid. A 4-cusped Hypocycloid which is sometimes also called a Tetracuspid, Cubocycloid, or Paracycle. The parametric equations of the astroid can be obtained by plugging in or into the equations for a general Hypocycloid, giving. In Cartesian Coordinates, (3) In Pedal Coordinates with the Pedal Point at the center, the equation is. (4)