Tschirnhausen cubic

Last updated October 16, 2024

In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation

History

The curve was studied by von Tschirnhaus, de L'Hôpital, and Catalan. It was given the name Tschirnhausen cubic in a 1900 paper by Raymond Clare Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan.

Other equations

Put $t=\tan(\theta /3)$ . Then applying triple-angle formulas gives

x=a\cos \theta \sec ^{3}{\frac {\theta }{3}}=a\left(\cos ^{3}{\frac {\theta }{3}}-3\cos {\frac {\theta }{3}}\sin ^{2}{\frac {\theta }{3}}\right)\sec ^{3}{\frac {\theta }{3}}=a\left(1-3\tan ^{2}{\frac {\theta }{3}}\right)

=a(1-3t^{2})

y=a\sin \theta \sec ^{3}{\frac {\theta }{3}}=a\left(3\cos ^{2}{\frac {\theta }{3}}\sin {\frac {\theta }{3}}-\sin ^{3}{\frac {\theta }{3}}\right)\sec ^{3}{\frac {\theta }{3}}=a\left(3\tan {\frac {\theta }{3}}-\tan ^{3}{\frac {\theta }{3}}\right)

=at(3-t^{2})

giving a parametric form for the curve. The parameter t can be eliminated easily giving the Cartesian equation

27ay^{2}=(a-x)(8a+x)^{2}

.

If the curve is translated horizontally by 8a and the signs of the variables are changed, the equations of the resulting right-opening curve are

x=3a(3-t^{2})

y=at(3-t^{2})

and in Cartesian coordinates

x^{3}=9a\left(x^{2}-3y^{2}\right)

.

This gives the alternative polar form

r=9a\left(\sec \theta -3\sec \theta \tan ^{2}\theta \right)

.

Generalization

The Tschirnhausen cubic is a Sinusoidal spiral with n = −1/3.

Related Research Articles

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions, under suitably restricted domains. 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 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, called a parametric curve and parametric surface, respectively. In such cases, the equations are collectively called a parametric representation, or parametric system, 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

In geometry, 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.

In geometry, the cissoid of Diocles is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the curve family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix.

<span class="mw-page-title-main">Projectile motion</span> Motion of launched objects due to gravity

Projectile motion is a form of motion experienced by an object or particle that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only. In the particular case of projectile motion on Earth, most calculations assume the effects of air resistance are passive.

<span class="mw-page-title-main">Folium of Descartes</span> Algebraic curve of the form x³ + y³ – 3axy = 0

In geometry, the folium of Descartes is an algebraic curve defined by the implicit equation

In geometry, a strophoid is a curve generated from a given curve $C$ and points $A$ and $O$ as follows: Let $L$ be a variable line passing through $O$ and intersecting $C$ at $K$ . Now let $P 1$ and $P 2$ be the two points on $L$ whose distance from $K$ is the same as the distance from $A$ to $K$ . The locus of such points $P 1$ and $P 2$ is then the strophoid of $C$ with respect to the pole $O$ and fixed point $A$ . Note that $AP 1$ and $AP 2$ are at right angles in this construction.

In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742.

<span class="mw-page-title-main">Rifleman's rule</span>

Rifleman's rule is a "rule of thumb" that allows a rifleman to accurately fire a rifle that has been calibrated for horizontal targets at uphill or downhill targets. The rule says that only the horizontal range should be considered when adjusting a sight or performing hold-over in order to account for bullet drop. Typically, the range of an elevated target is considered in terms of the slant range, incorporating both the horizontal distance and the elevation distance, as when a rangefinder is used to determine the distance to target. The slant range is not compatible with standard ballistics tables for estimating bullet drop.

There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary definitions are based on the geometry of right triangles and the ratio between their sides. The proofs given in this article use these definitions, and thus apply to non-negative angles not greater than a right angle. For greater and negative angles, see Trigonometric functions.

In inversive geometry, an inverse curve of a given curve $C$ is the result of applying an inverse operation to $C$ . Specifically, with respect to a fixed circle with center $O$ and radius $k$ the inverse of a point $Q$ is the point $P$ for which $P$ lies on the ray $OQ$ and $OP \cdot OQ = k 2$ . The inverse of the curve $C$ is then the locus of $P$ as $Q$ runs over $C$ . The point $O$ in this construction is called the center of inversion, the circle the circle of inversion, and $k$ the radius of inversion.

In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

<span class="mw-page-title-main">Limaçon trisectrix</span> Quartic plane curve

In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose, conchoid or epitrochoid. The curve is one among a number of plane curve trisectrixes that includes the Conchoid of Nicomedes, the Cycloid of Ceva, Quadratrix of Hippias, Trisectrix of Maclaurin, and Tschirnhausen cubic. The limaçon trisectrix a special case of a sectrix of Maclaurin.

<span class="mw-page-title-main">Lituus (mathematics)</span> Spiral

The lituus spiral is a spiral in which the angle $θ$ is inversely proportional to the square of the radius $r$ .

In Euclidean geometry, 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 $p c$ (the contrapedal coordinate) even though it is not an independent quantity and it relates to $(r, p)$ as

References

J. D. Lawrence, A Catalog of Special Plane Curves. New York: Dover, 1972, pp. 87-90.

External links

Weisstein, Eric W. "Tschirnhausen Cubic". MathWorld .
"Tschirnhaus' Cubic" at MacTutor History of Mathematics archive
Tschirnhausen cubic at mathcurve.com

This algebraic geometry–related article is a stub. You can help Wikipedia by expanding it.

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.