Fish curve

Last updated November 25, 2022

A fish curve is an ellipse negative pedal curve that is shaped like a fish. In a fish curve, the pedal point is at the focus for the special case of the squared eccentricity $e^{2}={\tfrac {1}{2}}$ .^[1] The parametric equations for a fish curve correspond to those of the associated ellipse.

Equations

For an ellipse with the parametric equations

\textstyle {x=a\cos(t),\qquad y={\frac {a\sin(t)}{\sqrt {2}}}},

the corresponding fish curve has parametric equations

\textstyle {x=a\cos(t)-{\frac {a\sin ^{2}(t)}{\sqrt {2}}},\qquad y=a\cos(t)\sin(t)}.

When the origin is translated to the node (the crossing point), the Cartesian equation can be written as:^[2]^[3]

\left(2x^{2}+y^{2}\right)^{2}-2{\sqrt {2}}ax\left(2x^{2}-3y^{2}\right)+2a^{2}\left(y^{2}-x^{2}\right)=0.

Area

The area of a fish curve is given by:

A={\frac {1}{2}}\left|\int {\left(xy'-yx'\right)dt}\right|

={\frac {1}{8}}a^{2}\left|\int {\left[3\cos(t)+\cos(3t)+2{\sqrt {2}}\sin ^{2}(t)\right]dt}\right|

,

so the area of the tail and head are given by:

A_{\mathrm {Tail} }=\left({\frac {2}{3}}-{\frac {\pi }{4{\sqrt {2}}}}\right)a^{2}

A_{\mathrm {Head} }=\left({\frac {2}{3}}+{\frac {\pi }{4{\sqrt {2}}}}\right)a^{2}

giving the overall area for the fish as:

A={\frac {4}{3}}a^{2}

.^[2]

Curvature, arc length, and tangential angle

The arc length of the curve is given by $a{\sqrt {2}}\left({\frac {1}{2}}\pi +3\right)$ .

The curvature of a fish curve is given by:

K(t)={\frac {2{\sqrt {2}}+3\cos(t)-\cos(3t)}{2a\left[\cos ^{4}t+\sin ^{2}t+\sin ^{4}t+{\sqrt {2}}\sin(t)\sin(2t)\right]^{\frac {3}{2}}}}

,

and the tangential angle is given by:

\phi (t)=\pi -\arg \left({\sqrt {2}}-1-{\frac {2}{\left(1+{\sqrt {2}}\right)e^{it}-1}}\right)

where $\arg(z)$ is the complex argument.

Related Research Articles

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions $y (x)$ of Bessel's differential equation

<span class="mw-page-title-main">Ellipse</span> Plane curve: conic section

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity $, a number ranging from to .$

<span class="mw-page-title-main">Superellipse</span> Family of closed mathematical curves

A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.

<span class="mw-page-title-main">Spheroid</span> Surface formed by rotating an ellipse

A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry.

<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

<span class="mw-page-title-main">Tautochrone curve</span> Concept in geometry

A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point on the curve. The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity. The tautochrone curve is related to the brachistochrone curve, which is also a cycloid.

<span class="mw-page-title-main">Fermat's spiral</span> Spiral that surrounds equal area per turn

A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance from the spiral center, contrasting with the Archimedean spiral and the logarithmic spiral. Fermat spirals are named after Pierre de Fermat.

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form

In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points $F 1$ and $F 2$ , known as foci, at distance $2 c$ from each other as the locus of points $P$ so that $PF 1 \cdot PF 2 = c 2$ . The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscatus, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

In the physical sciences, the Airy function $Ai(x)$ is a special function named after the British astronomer George Biddell Airy (1801–1892). The function $Ai(x)$ and the related function $Bi(x)$ , are linearly independent solutions to the differential equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization of the object.

<span class="mw-page-title-main">Limaçon</span> Type of roulette curve

In geometry, a limaçon or limacon, also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp.

<span class="mw-page-title-main">Cardioid</span> Type of plane curve

A cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion. A cardioid can also be defined as the set of points of reflections of a fixed point on a circle through all tangents to the circle.

<span class="mw-page-title-main">Viviani's curve</span> Figure-eight shaped curve on a sphere

In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is the intersection of a sphere with a cylinder that is tangent to the sphere and passes through two poles of the sphere. Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval.

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle $, the sine and cosine functions are denoted simply as and .$

In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse whose center is the triangle's centroid. Named after Jakob Steiner, it is an example of a circumconic. By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral.

For a plane curve C and a given fixed point O, the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal $even though it is not an independent quantity and it relates to as .$

In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of $into an ordinary rational function of by setting . This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general transformation formula is:$

<span class="mw-page-title-main">Orthoptic (geometry)</span>

In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle.

References

↑ Lockwood, E. H. (1957). "Negative Pedal Curve of the Ellipse with Respect to a Focus". Math. Gaz. 41: 254–257. doi:10.1017/S0025557200037293. S2CID 125623811.
1 2 Weisstein, Eric W. "Fish Curve". MathWorld. Retrieved May 23, 2010.
↑ Lockwood, E. H. (1967). A Book of Curves. Cambridge, England: Cambridge University Press. p. 157.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Lockwood, E. H. (1957). "Negative Pedal Curve of the Ellipse with Respect to a Focus". Math. Gaz. 41: 254–257. doi:10.1017/S0025557200037293. S2CID 125623811.

[fish-2] 1 2 Weisstein, Eric W. "Fish Curve". MathWorld. Retrieved May 23, 2010.

[book-3] Lockwood, E. H. (1967). A Book of Curves. Cambridge, England: Cambridge University Press. p. 157.

[1]

[2]

[3]