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Astroid

An astroid is a plane algebraic curve consisting of a four-cusped hypocycloid, resembling a diamond shape with pointed vertices, and is mathematically defined as the locus of a point on the circumference of a circle rolling inside a fixed circle of four times the radius.^[1]^[2] Also known as a tetracuspid, cubocycloid, or paracycle, it features four symmetric cusps and is a special case of a hypocycloid where the rolling circle has a radius one-fourth that of the fixed circle.^[1] The astroid can be generated parametrically by the equations x = a \cos^3 \theta and y = a \sin^3 \theta, where a > 0 is a scaling parameter and \theta ranges from 0 to $2\pi, tracing the full curve.^[1]^[2] In Cartesian coordinates, it satisfies the implicit equation x^{2/3} + y^{2/3} = a^{2/3}, which highlights its superellipse-like form and algebraic degree of six.^[1] Geometrically, the curve also arises as the envelope of line segments of fixed length sliding with endpoints on two perpendicular axes, or as the envelope of a family of ellipses with constant sum of semi-axes.^[1]^[3] Key properties include a perimeter of $6a and an enclosed area of \frac{3}{8} \pi a^2, both derived from integrating along the parametric form.^[1]^[4] The astroid's evolute is another scaled hypocycloid, and its tangents from any cusp form a constant length in certain quadrants, making it useful in studies of envelopes and roulettes.^[1]^[5] While primarily of theoretical interest in algebraic geometry and calculus, it illustrates concepts like parametric tracing and cusp formation without direct widespread practical applications beyond mathematical modeling of sliding mechanisms, such as the path of a ladder corner against walls.^[6] Historically, the astroid was first investigated by Johann Bernoulli in 1691–1692, appearing in Gottfried Wilhelm Leibniz's correspondence by 1715.^[4] It was later named "astroid" (from Greek aster meaning star) in 1836, distinguishing it from earlier terms like cubocycloid.^[4]

Definition and Equations

Parametric Representation

The astroid is commonly represented in parametric form using the equations

x(\theta) = a \cos^3 \theta, \quad y(\theta) = a \sin^3 \theta,

where a > 0 is a scaling parameter that determines the size of the curve, and \theta is the parameter ranging from $0 to $2\pi.^[1] This parameterization allows for straightforward plotting and generation of the curve, with \theta tracing the entire astroid exactly once over its full interval without repetition.^[1] These equations arise from the astroid's interpretation as a hypocycloid, specifically the path traced by a point on the circumference of a circle of radius a/4 rolling inside a fixed circle of radius a.^[7] As \theta varies, the resulting curve forms a diamond-like shape with four cusps located at the points (a, 0), (-a, 0), (0, a), and (0, -a).^[1]

Implicit Equation

The implicit equation of the astroid in Cartesian coordinates is given by

x^{2/3} + y^{2/3} = a^{2/3},

where a > 0 is a scaling parameter that determines the size of the curve.^[1] This form represents the astroid as a special case of a superellipse, also known as a Lamé curve, where the general superellipse equation \left| \frac{x}{a} \right|^n + \left| \frac{y}{a} \right|^n = 1 has exponent n = 2/3.^[1]^[8] To verify this implicit equation, substitute the parametric equations x = a \cos^3 \theta and y = a \sin^3 \theta (where $0 \leq \theta < 2\pi) into the left side: (a \cos^3 \theta)^{2/3} + (a \sin^3 \theta)^{2/3} = a^{2/3} (\cos^2 \theta + \sin^2 \theta) = a^{2/3}, confirming it satisfies the equation.^[1] Clearing the fractional exponents in the implicit equation yields an equivalent polynomial form of degree 6 (a sextic), such as (x^2 + y^2 - a^2)^3 + 27 a^2 x^2 y^2 = 0.^[9]

Derivation

From Hypocycloid Construction

A hypocycloid is the curve traced by a fixed point on the circumference of a small circle of radius r that rolls without slipping inside the fixed circle of radius R > r.^[7] The astroid arises as a special case of this construction when R = 4r, or equivalently, when the rolling circle has radius r = a/4 inside a fixed circle of radius R = a, producing a four-cusped curve.^[1] To derive the parametric equations geometrically, consider the position of the tracing point as the vector sum of the position of the center of the rolling circle and the position relative to that center. The center moves along a circle of radius R - r = 3r with angle \phi from a reference axis. The rolling circle rotates by an additional angle -\frac{R - r}{r} \phi = -3\phi relative to the center (the negative sign accounts for the internal rolling direction). Thus, the coordinates are

\begin{align*} x &= 3r \cos \phi + r \cos(3\phi), \\ y &= 3r \sin \phi - r \sin(3\phi). \end{align*}

This follows from the standard hypocycloid parametrization with the angle \phi as the rolling parameter.^[7]^[1] These equations match the standard parametric form of the astroid x = a \cos^3 \theta, y = a \sin^3 \theta upon applying the triple-angle identities \cos 3\phi = 4 \cos^3 \phi - 3 \cos \phi and \sin 3\phi = 3 \sin \phi - 4 \sin^3 \phi, with the reparametrization \theta = \phi and a = 4r.^[1] The astroid was first discussed as a hypocycloid by Johann Bernoulli in 1691–1692.^[4]

Algebraic Derivation

The astroid is given in parametric form by the equations x = a \cos^3 \theta and y = a \sin^3 \theta, where a > 0 is a scaling parameter and \theta ranges from 0 to $2\pi.^[1] To derive the implicit equation algebraically, eliminate the parameter \theta through trigonometric identities. First, solve for the trigonometric functions: \cos \theta = \left( \frac{x}{a} \right)^{1/3} and \sin \theta = \left( \frac{y}{a} \right)^{1/3}, where the cube roots are the principal real roots (negative for negative arguments).^[1] Square these expressions to obtain \cos^2 \theta = \left( \frac{x}{a} \right)^{2/3} and \sin^2 \theta = \left( \frac{y}{a} \right)^{2/3}.^[1] Apply the Pythagorean identity \cos^2 \theta + \sin^2 \theta = 1:

\left( \frac{x}{a} \right)^{2/3} + \left( \frac{y}{a} \right)^{2/3} = 1.

Multiplying through by a^{2/3} yields the implicit equation

x^{2/3} + y^{2/3} = a^{2/3}.

^[1] The fractional exponents involve cube roots, which are multi-valued in the complex plane but restricted to principal real branches for the astroid in the real plane; this ensures the equation traces the four-cusped curve without extraneous branches, as the parametric form covers all points uniquely over \theta \in [0, 2\pi).^[10] To obtain a polynomial form, start with the implicit equation and cube both sides of x^{2/3} + y^{2/3} = a^{2/3}:

(x^{2/3} + y^{2/3})^3 = a^2.

Expanding the left side gives

x^2 + y^2 + 3x^{4/3} y^{2/3} + 3x^{2/3} y^{4/3} = a^2.

Rearranging yields

a^2 - x^2 - y^2 = 3 x^{2/3} y^{2/3} (x^{2/3} + y^{2/3}).

Since x^{2/3} + y^{2/3} = a^{2/3}, substitute to get

a^2 - x^2 - y^2 = 3 x^{2/3} y^{2/3} a^{2/3}.

Cubing both sides now produces

(a^2 - x^2 - y^2)^3 = 27 x^2 y^2 a^2.

Adjusting signs via (a^2 - x^2 - y^2)^3 = - (x^2 + y^2 - a^2)^3 leads to the standard polynomial form

(x^2 + y^2 - a^2)^3 + 27 a^2 x^2 y^2 = 0,

or equivalently,

(x^2 + y^2 - a^2)^3 = -27 a^2 x^2 y^2.

^[11] This sextic equation fully describes the astroid algebraically without fractional powers.^[11]

Geometric Properties

Shape and Symmetry

The astroid is a closed plane curve that resembles a squished square or diamond, with its sides curving inward in a concave manner, giving it a star-like appearance without any self-intersections.^[1] This distinctive shape arises as a specific type of hypocycloid, forming a compact, bounded figure that fits snugly within a square region.^[12] The curve is oriented such that it lies entirely within the bounding box defined by the interval [-a, a] \times [-a, a] for a positive parameter a, touching the boundary of this square at four points: the intercepts on the coordinate axes at (\pm a, 0) and (0, \pm a).^[1] These intercepts mark the extremal points where the astroid aligns with the axes, emphasizing its compact and centered geometry around the origin.^[4] The astroid exhibits rich symmetry, including reflectional symmetry across the lines x=0, y=0, y=x, and y=-x, which together form the principal axes of the coordinate system and its diagonals.^[4]^[12]^[13] Additionally, it possesses four-fold rotational symmetry, remaining invariant under rotations by multiples of $90^\circ (or \pi/2 radians) about the origin, which underscores its balanced and equitable form.^[1] This combination of symmetries contributes to the astroid's elegant, diamond-like profile, featuring four cusps that enhance its star-shaped aesthetic.^[4]

Cusps and Singularities

The astroid, parametrized by x = a \cos^3 \theta and y = a \sin^3 \theta for $0 \leq \theta < 2\pi, exhibits four cusps located at the parameter values \theta = 0, \pi/2, \pi, 3\pi/2, corresponding to the points (a, 0), (0, a), ([-a](/page/List_of_New_Zealand_actors), 0), and (0, [-a](/page/List_of_female_bass_guitarists)), respectively.^[1] These points align with the coordinate axes, reflecting the curve's fourfold rotational symmetry.^[1] At each cusp, the partial derivatives with respect to the parameter vanish: dx/d\theta = -3a \cos^2 \theta \sin \theta = 0 and dy/d\theta = 3a \sin^2 \theta \cos \theta = 0, confirming these as singular points where the curve is not differentiable.^[1] The parametric speed, given by \sqrt{(dx/d\theta)^2 + (dy/d\theta)^2}, also equals zero at these locations, resulting in sharp corners rather than smooth transitions.^[1] In contrast to smooth curves, where the tangent vector remains nonzero and the derivative exists everywhere, the astroid's cusps introduce points of nondifferentiability, altering local geometric behavior.^[1]^[14] These cusps are ordinary cusps, characterized by the tangent vector turning by $3\pi (or 540 degrees) as the parameter passes through the singular point, a property shared with the standard cusp form y^2 = x^3.^[15] This turning reflects the curve's reversal along the common tangent line from both approaching branches, distinguishing ordinary cusps from other singularities like nodes or higher-order types.^[14] For the astroid, the second and third derivatives at the cusps are linearly independent and nonzero, satisfying the criteria for ordinary cusps.^[16]

Metric Properties

Arc Length

The arc length of the astroid is computed using the standard formula for a parametric curve (x(\theta), y(\theta)), $0 \leq \theta \leq 2\pi:

L = \int_0^{2\pi} \sqrt{ \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 } \, d\theta.

Using the parametric representation x(\theta) = a \cos^3 \theta, y(\theta) = a \sin^3 \theta, the derivatives are \frac{dx}{d\theta} = -3a \cos^2 \theta \sin \theta and \frac{dy}{d\theta} = 3a \sin^2 \theta \cos \theta.^[17] The squared terms simplify as follows:

\left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 = 9a^2 \cos^4 \theta \sin^2 \theta + 9a^2 \sin^4 \theta \cos^2 \theta = 9a^2 \cos^2 \theta \sin^2 \theta (\cos^2 \theta + \sin^2 \theta) = 9a^2 \cos^2 \theta \sin^2 \theta.

Thus, the integrand is \sqrt{ \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 } = 3a |\cos \theta \sin \theta| = \frac{3a}{2} |\sin 2\theta|.^[17] The astroid's fourfold rotational symmetry allows the total length to be four times the arc length over one quadrant, from \theta = 0 to \pi/2, where \sin 2\theta \geq 0 and the integrand is \frac{3a}{2} \sin 2\theta. The integral evaluates to

\int_0^{\pi/2} \frac{3a}{2} \sin 2\theta \, d\theta = \frac{3a}{2} \left[ -\frac{1}{2} \cos 2\theta \right]_0^{\pi/2} = \frac{3a}{2} \cdot \frac{1}{2} (1 - (-1)) = \frac{3a}{2}.

Therefore, L = 4 \times \frac{3a}{2} = 6a.^[17] The cusps occur at \theta = 0, \pi/2, \pi, 3\pi/2, where the integrand vanishes since \cos \theta = 0 or \sin \theta = 0; however, the integrand remains continuous over the interval, ensuring the improper integral converges.^[17]

Enclosed Area

The astroid, parametrized by x(\theta) = a \cos^3 \theta and y(\theta) = a \sin^3 \theta for \theta \in [0, 2\pi], encloses a single bounded region without self-intersections, as the curve traces a diamond-like shape with four cusps, returning to the starting point after one full period.^[1] To compute the enclosed area A, apply Green's theorem in the form for a positively oriented, simple closed curve:

A = \frac{1}{2} \int_0^{2\pi} \left( x \frac{dy}{d\theta} - y \frac{dx}{d\theta} \right) d\theta.

The derivatives are \frac{dx}{d\theta} = -3a \cos^2 \theta \sin \theta and \frac{dy}{d\theta} = 3a \sin^2 \theta \cos \theta. Substituting yields

x \frac{dy}{d\theta} - y \frac{dx}{d\theta} = 3a^2 \cos^4 \theta \sin^2 \theta + 3a^2 \sin^4 \theta \cos^2 \theta = 3a^2 \cos^2 \theta \sin^2 \theta = \frac{3}{4} a^2 \sin^2 (2\theta).

Thus,

A = \frac{1}{2} \int_0^{2\pi} \frac{3}{4} a^2 \sin^2 (2\theta) \, d\theta = \frac{3}{8} a^2 \int_0^{2\pi} \sin^2 (2\theta) \, d\theta.

Using the identity \sin^2 \phi = \frac{1 - \cos 2\phi}{2} with \phi = 2\theta, the integral simplifies to \int_0^{2\pi} \frac{1 - \cos (4\theta)}{2} \, d\theta = \pi, so A = \frac{3\pi a^2}{8}.^[1]^[18] This area represents \frac{3}{8} of the area of a circle with radius a, highlighting the astroid's compact enclosure relative to the circumscribing circle of that radius.^[1] An alternative computation via the implicit equation x^{2/3} + y^{2/3} = a^{2/3} uses symmetry to integrate over one quadrant and multiply by four, yielding the same result through substitution y = a (1 - (x/a)^{2/3})^{3/2}, though the parametric approach is more straightforward due to the curve's natural representation.^[19]

Analytic Properties

Evolute and Radius of Curvature

The radius of curvature \rho for a parametric plane curve (x(\theta), y(\theta)) is given by the formula

\rho(\theta) = \frac{\left[ \left( \frac{dx}{d\theta} \right)^2 + \left( \frac{dy}{d\theta} \right)^2 \right]^{3/2}}{\left| \frac{dx}{d\theta} \frac{d^2 y}{d\theta^2} - \frac{dy}{d\theta} \frac{d^2 x}{d\theta^2} \right|}.

For the astroid parametrized as x(\theta) = a \cos^3 \theta and y(\theta) = a \sin^3 \theta, the first derivatives are \frac{dx}{d\theta} = -3a \cos^2 \theta \sin \theta and \frac{dy}{d\theta} = 3a \sin^2 \theta \cos \theta, while the second derivatives are \frac{d^2 x}{d\theta^2} = 3a \cos \theta (2 \sin^2 \theta - \cos^2 \theta) and \frac{d^2 y}{d\theta^2} = 3a \sin \theta (2 \cos^2 \theta - \sin^2 \theta). Substituting these into the formula yields \rho(\theta) = 3a |\sin \theta \cos \theta|. At the cusps of the astroid, which occur where \theta = k \pi / 2 for integer k, \sin \theta or \cos \theta vanishes, making \rho = 0 and thus the curvature \kappa = 1/\rho infinite (undefined). Away from these cusps, the radius of curvature is finite and varies continuously along each smooth arc, reaching a maximum value of $3a/2 (curvature minimum of $2/(3a)) at the midpoints of the arcs (e.g., \theta = \pi/4). The evolute of the astroid, which is the locus of its centers of curvature, has parametric equations

x_e(\theta) = a \cos \theta (1 + 2 \sin^2 \theta), \quad y_e(\theta) = a \sin \theta (1 + 2 \cos^2 \theta),

or equivalently,

x_e(\theta) = \frac{a}{2} [3 \cos \theta - \cos 3\theta], \quad y_e(\theta) = \frac{a}{2} [3 \sin \theta + \sin 3\theta].

These equations describe another astroid similar to the original, scaled by a linear factor of 2 and rotated by $45^\circ. The cusps of this evolute correspond to the points of minimum curvature on the original astroid's smooth arcs. Due to this self-similarity under the evolute transformation (up to scaling and rotation), the astroid is also its own involute in the sense that the involute of the astroid is another similar astroid, scaled by a factor of $1/2 and rotated by $45^\circ.^[20]

Relation to Other Curves

The astroid is a special case of the Lamé curve, also known as a superellipse, corresponding to the exponent n = \frac{2}{3} in the general equation \left| \frac{x}{a} \right|^n + \left| \frac{y}{b} \right|^n = 1 with a = b.^[1] This places it within the broader family of Lamé curves, which generalize ellipses and include both convex and star-shaped forms depending on the value of n.^[21] The astroid can also be constructed as the envelope of line segments of fixed length whose endpoints slide along a pair of perpendicular axes.^[1] In this glissette formation, the curve emerges as the boundary traced by the positions of the moving segment, highlighting its relation to envelope geometries in classical curve theory.^[4] The pedal curve of the astroid with respect to its center (the origin) is a quadrifolium, a four-petaled rose curve given parametrically by x = a \cos^3 \theta \cos 2\theta, y = a \cos^3 \theta \sin 2\theta.^[22] This connection underscores the astroid's ties to polar and rhodonea curves, where the feet of the perpendiculars from the origin to the astroid's tangents form the quadrifolium's lobes.^[12] An involute of the astroid, obtained by unwinding a taut string from the curve, traces another astroid similar to the original but scaled by a factor of \frac{1}{2} and rotated by \frac{1}{8} turn.^[23] Conversely, the evolute of the astroid is a similar astroid enlarged by a factor of 2, linking it to the family of hypocycloids through these dual transformations.^[12] As a singular plane algebraic curve of degree 6, the astroid has arithmetic genus 10 for a smooth sextic but geometric genus 0 due to its four cusps, which resolve the singularities and confirm its rationality via parametric equations with rational functions.^[24] This genus-zero property aligns it with rational curves like conics, despite its higher degree and non-smooth nature.^[25] In modern applications, the astroid's parametric form finds utility in computer graphics for generating astroid-like fractal designs and animations, such as in zipper fractal Bézier curves that produce decorative patterns resembling four-cusped stars for visual effects and curve-based modeling.^[26] These uses leverage the curve's symmetry and parametric simplicity to create smooth transitions in educational simulations and artistic renderings.^[27]

References

[1]
Astroid -- from Wolfram MathWorld
A 4-cusped hypocycloid which is sometimes also called a tetracuspid, cubocycloid, or paracycle. The parametric equations of the astroid can be obtained by ...Missing: definition | Show results with:definition
[2]
https://math.libretexts.org/Bookshelves/Calculus/CLP-4_Vector_Calculus_(Feldman_Rechnitzer_and_Yeager)/01%3A_Curves/1.11%3A_Optional__The_Astroid
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The Astroid as envelope of segments and ellipses
An astroid is a curve with four cups formed by the envelope of moving segments on perpendicular lines, and it is also the envelope of ellipses with constant ...<|control11|><|separator|>
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Astroid
### Summary of Astroid Curve
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What are the practical applications of the Astroid curve?
Nov 19, 2018 · The astroid curve has few real-world applications, but is the envelope of rhombuses and the exterior of points swept by a sliding ladder.Proof of Astroid? - Math Stack ExchangeHow to make a sharp 5-pointed astroid in parametric coordinates?More results from math.stackexchange.com
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Hypocycloid -- from Wolfram MathWorld
The curve produced by fixed point P on the circumference of a small circle of radius b rolling around the inside of a large circle of radius a>b.
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MacTutor History of Mathematics Archive. "Lamé Curves." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Lame.html.
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astroid - two dimensional curves
The astroid is the hypocycloid for which the rolled circle is four times as large as the rolling circle. The curve can be written in a Whewell equation as s ...
[10]
Optional — The Astroid
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.
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[PDF] (cos (t),sin (t)). Implicit equation: x + y = aa . Description An Astroid is a
(t)). Implicit equation: x. 2/3. + y. 2/3. = aa. 2/3 . Description. An Astroid is a curve traced out by a point on the circum- ference of one circle (of radius ...
[12]
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It can be easily defined as the envelopes of the straight lines crossing Ox and Oy in P and Q while the sum or difference of OP² and OQ² stays constant.
[13]
[PDF] Solids of Revolution and the Astroid - Solutions Math 125 1 If the ...
The volumes are the same because the astroid curve is symmetric about the line y = x. Notice that shells, here, are more difficult than disks. 3. Use any method ...Missing: rotational | Show results with:rotational
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None
### Summary of Sections on Astroid, Cusps, Singularities, Derivatives Vanishing, and Cusp Nature/Tangent Behavior
[15]
[PDF] ON THE FUNDAMENTAL GROUP OF PLANE CURVE ...
distinct roots of the ordinary cusp (y2 − x3 = 0) applied to our loop b(t) gives a loop that traverses 3π rather than just 2π, which gives the extra permutation ...
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[PDF] Exercises for Elementary Differential Geometry
2 has an ordinary cusp at the origin;. (iii) if γγ has an ordinary cusp at a point pp, so does any reparametrization of γγ. ... For the astroid in (iv) ...
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As in Example 3, the perimeter of the astroid is 4 times the length of the curve in the first quadrant. dx dt. = −3 cos. 2. (t) sin(t) dy dt. = 3 sin. 2. (t) ...
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[PDF] Math 214 — Solutions to Assignment #5 11.1 10. Let x = t 2, y = t (a ...
Find the area of the region enclosed by the astroid x = a cos3 θ, y = a sin3 θ. Solution. The graph of the astroid is a. −a. 0 x y. −a a. 4. Page 5. Using ...<|control11|><|separator|>
[19]
Area inside the astroid - Math Stack Exchange
Sep 4, 2016 · ... the part where x>0 and y>0, so: y=(22/3−x2/3)3/2. Now we get that ∫20(22/3−x2/3)3/2=3π8. So the area of the whole astroid is 3π2.Calculating area of astroid $x^{2/3}+y^{2/3}=a^{2/3}$ for $a>0 ...Find a volume of a figure given by an astroid rotating around an axisMore results from math.stackexchange.com
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[PDF] Differential Geometry - UCSD Math
... equations. * = fi(t), y. = /2W>. * = /s(0 are called the equations of the curve y in the parametric form. A curve is defined uniquely by its equations in the ...
[21]
[PDF] Iterating evolutes and involutes - The University of Texas at Dallas
Oct 28, 2015 · Typically, an evolute has cusp singularities, generically ... hypocycloid, generically an astroid (Theorem 2). We also provide ...
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Lame Curves - MacTutor History of Mathematics
Lame Curves ... View the interactive version of this curve. Description. In 1818 Lamé discussed the curves with equation given above. He considered more general ...
[23]
Astroid Pedal Curve -- from Wolfram MathWorld
The pedal curve of an astroid with pedal point at the center is the quadrifolium. See also Astroid, Pedal Curve, Quadrifolium.
[24]
Astroid Involute -- from Wolfram MathWorld
The involute of the astroid is a hypocycloid involute for n=4. Surprisingly, it is another astroid scaled by a factor (n-2)/n=2/4=1/2 and rotated ...
[25]
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Using the zipper fractal Bézier curves, we create attractive deltoid-like, astroid-like, exoid-like, and six-leaf flower designs. Our findings have applications ...
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Nov 28, 2019 · This paper introduces a new teaching–learning technique that utilizes STEAM-based methods to explore cycloidal curves for Computer Science ...