Tschirnhausen cubic

In mathematics, the Tschirnhausen cubic is a cubic plane curve defined by the polar equation $${\displaystyle r=a\sec ^{3}\left({\frac {\theta }{3}}\right)}$$ or the equivalent algebraic equation $${\displaystyle 27ay^{2}=(a-x)(8a+x)^{2}.}$$

Tschirnhausen cubic, case of a = 1

It is a nodal cubic, meaning that it crosses itself at one point, its node. Because it has this singularity, it can be given a parametric equation, and any arc of it can be drawn as a cubic Bézier curve. It is a special case of a sinusoidal spiral, of a pursuit curve, and of a Pythagorean hodograph curve.

The original study of this curve, by Ehrenfried von Tschirnhaus, found it as a caustic of light reflected in a parabolic mirror. It was used by Eugène Catalan in an angle trisection, and it appears among the geodesics of the Enneper surface.

History

The curve was studied by Ehrenfried von Tschirnhaus, Guillaume de l'Hôpital,^[1] Pierre Bouguer,^[2] and Eugène Catalan.^[1] It was given the name Tschirnhausen cubic in a 1900 paper by Raymond Clare Archibald,^[3] though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan.^[1][4]

Tschirnhaus wrote about this curve in a 1690 publication in Acta Eruditorum, concerned with geometrical optics. He showed that it arises as the caustic of parallel light rays (light rays from a source infinitely far away) reflected from a parabolic mirror. L'Hôpital included it in an 1696 textbook on calculus, Analyse des infiniment petits, pour l’intelligence des lignes courbes.^[1]

Bouguer included this curve in a 1732 study of pursuit curves. He observed that for a pursuer moving twice as fast as their linearly-moving target and always moving directly towards the target rather than anticipating their movement (intuitively, like a dog chasing its master on a walk), the curve resulting from this motion is cubic.^[2][5]

Much later, Catalan used this curve, and a parabola associated with it, as part of an angle trisection.^[6][7] Catalan published this work in a letter in 1885, but writes that he originally derived it in 1832, in a paper that by 1885 was "yellowed by age".^[7] As Catalan observed, and used in this construction, the Tschirnhausen cubic is the negative pedal curve of a parabola, with respect to the focus of the parabola.^[7]

Equations and construction

The Tschirnhausen cubic has the equation in polar coordinates $${\displaystyle r=a\sec ^{3}\left({\frac {\theta }{3}}\right),}$$ where ${\displaystyle \sec }$ is the secant function.^[4]

Define the parameter ${\displaystyle t}$ by ${\displaystyle t=\tan(\theta /3)}$. Then converting the curve from polar coordinates to Cartesian coordinates and applying De Moivre's triple-angle formulas leads to a parametric equation for the curve:^[4] $${\displaystyle {\begin{aligned}x&=a(1-3t^{2})\\y&=at(3-t^{2}).\end{aligned}}}$$ Eliminating the parameter ${\displaystyle t}$ gives an expression as an algebraic curve, a cubic plane curve:^[4] $${\displaystyle 27ay^{2}=(a-x)(8a+x)^{2}.}$$

Another way of parameterizing the same curve uses complex numbers, each representing a point in the complex plane. For a parameter ${\displaystyle \tau }$ ranging over the unit circle in the complex plane, the Tschirnhausen cubic can be described by the parametric equation^[8] $${\displaystyle z={\frac {2}{(1-\tau )^{3}}}.}$$

The Tschirnhausen cubic can also be constructed synthetically, as the negative pedal curve of the parabola with respect to its focus. For any curve ${\displaystyle C}$ and fixed point ${\displaystyle P}$, the negative pedal curve is the envelope of a family of lines, through points ${\displaystyle Q}$ on curve ${\displaystyle C}$, perpendicular to the lines ${\displaystyle PQ}$. For a curve to be the envelope of a given family of lines, the curve must have the lines in the given family as its tangent lines. The reverse construction, the pedal curve, finds the locus of points closest to ${\displaystyle P}$ on each tangent line; the pedal curve of the Tschirnhausen cubic, for ${\displaystyle P}$ at the origin of the polar coordinate system, is a parabola. In the same way, the parabola is the negative pedal curve of a line, with respect to a point off the line (the focus of the parabola), and the pedal curve of a parabola with respect to its focus is a line. A line can in turn be seen as the negative pedal curve of a single point, and the pedal curve of a line degenerates to a point. Therefore, the Tschirnhausen cubic is the three-times-iterated negative pedal curve of a point.^[3]

Applications

Caustic

Similar copies of the Tschirnhausen cubic created as caustics for varying directions of parallel light rays, reflected from a parabola

The reflected light rays for one direction of parallel light rays

Geometrical optics models light by rays (rather than, as in other forms of optics, particles or waves), emitted by a source of light and then transformed through reflections and refractions. When modeled in this way, caustics, arcs or sheets of intensified light created by reflection or refraction through curved surfaces, can be explained mathematically as the envelopes of the reflected or refracted light rays.^[9]

Light from an infinitely-far source (or from a source so far as to be indistinguishable from infinite, such as the stars) forms a parallel family of rays. When these rays are reflected by a parabolic mirror, whose symmetry axis is parallel to the rays, they converge at the focus point of the parabola, which may be thought of as a degenerate caustic. For any other direction of parallel rays, their reflections from the same mirror do not focus at a point, but instead form a non-degenerate caustic. This caustic (in its two-dimensional cross-section) takes the shape of a Tschirnhausen cubic, of varying scale and direction depending on the direction of the incoming rays. Its scale is largest for rays perpendicular to the symmetry axis of the parabolic mirror, and smallest for rays whose direction approaches that of the symmetry axis.^[10]

The Tschirnhausen cubic can also be found in a second way as a caustic, for reflection from a semicubical parabola of parallel light rays that are perpendicular to the shared symmetry axis of the semicubical parabola and the Tschirnhausen cubic.^[10]

Pursuit curve

The Tschirnhausen cubic is a radiodrome, a type of pursuit curve, obtained for a pursuer ("a dog trying to catch its master") whose target is moving linearly at half the speed of the pursuer. The curve constructed in this way is tangent to the line of the target's motion, at the vertex of the curve, its point of maximum curvature.^[2][10][11]

The same curve can be generated in a different way as a pursuit curve, again for a target moving linearly, pursued by a "crab" that always faces the target and moves perpendicularly to that direction, at the same speed as the target.^[10]

Trisectrix

Catalan's angle trisection, redrawn from Catalan 1885 with the same labeling. Angle BFD is one third of angle AFD.

As Catalan observed, the Tschirnhausen cubic can be used as a trisectrix, a curve that can be used to solve the trisection of the angle. Trisection cannot be solved with straightedge and compass, but can be solved when a copy of this curve and its associated parabola are available.^[6]

To trisect a given angle, given a copy of the Tschirnhausen cubic and its associated parabola, place the angle at the focus ${\displaystyle F}$ of the parabola, with one side of the angle along the ray ${\displaystyle FA}$ from the focus towards the vertex ${\displaystyle A}$ of the parabola, the point of maximum curvature where it is tangent to the Tschirnhausen cubic. Extend the other side of the angle to cross the Tschirnhausen cubic at point ${\displaystyle D}$, so that the angle to be trisected is ${\displaystyle AFD}$. Find a tangent line to the Tschirnhausen cubic, and extend this line to cross the parabola at point ${\displaystyle B}$. Then angle ${\displaystyle BFD}$ is one third of angle ${\displaystyle AFD}$.^[7]

To prove that angle ${\displaystyle BFD}$ is one third of angle ${\displaystyle AFD}$, Catalan uses several auxiliary points, as labeled in the figure. By the negative pedal property of the Tschirnhausen cubic, angle ${\displaystyle DBF}$ is a right angle, forming two sides of a rectangle ${\displaystyle DBFC}$ with center ${\displaystyle G}$, whose diagonal ${\displaystyle BC}$ is perpendicular to the parabola. One more point ${\displaystyle E}$ is the crossing point of diagonal ${\displaystyle BC}$ with the symmetry axis ${\displaystyle AF}$ of the two curves. Catalan observes that triangles ${\displaystyle BFE}$ and ${\displaystyle BGF}$ are two similar isosceles triangles. He lets ${\displaystyle \theta }$ denote the apex angle of these triangles, ${\displaystyle \theta =\angle BGF=\angle BFE}$, and he lets ${\displaystyle \omega }$ denote the angle ${\displaystyle \omega =\angle GFE}$ that is supplementary to the angle to be trisected, ${\displaystyle \pi -\omega }$. Then the base angle of the isosceles triangles is ${\displaystyle {\tfrac {1}{2}}(\pi -\theta )}$. Segment ${\displaystyle FG}$ divides the apex of isosceles triangle ${\displaystyle BFE}$ into two angles, one equal to this base angle and the other equal to ${\displaystyle \omega }$, from which it follows that ${\displaystyle \theta ={\tfrac {1}{2}}(\pi -\theta )+\omega }$ and ${\displaystyle \omega ={\tfrac {3}{2}}\theta -{\tfrac {1}{2}}\pi }$. Thus, the angle to be trisected, ${\displaystyle \angle AFD=\pi -\omega }$, is ${\displaystyle {\tfrac {3}{2}}(\pi -\theta )}$, three times the base angle ${\displaystyle \angle BFD={\tfrac {1}{2}}(\pi -\theta )}$.^[7]

Spline

An arc of a Tschirnhausen cubic, constructed as a cubic Bézier spline (red), such that the two consecutive triples of points along its control path (dotted black lines) form two congruent isosceles triangles

The cubic plane curves that have a singularity, such as the crossing point of the Tschirnhausen cubic, also have cubic parametric equations and representations as cubic Bézier splines.^[12] Any finite arc of this curve can be generated as a cubic Bézier spline for which the two consecutive triples of the four control points form two similar triangles (with the same orientation as each other), and each spline of this special type produces an arc of a scaled Tschirnhausen cubic.^[13]

For a version of Hermite interpolation that asks for a curve through two given points with given slopes at the two points, arcs of the Tschirnhausen cubic can provide a unique solution. In contrast, arbitrary cubic Bézier curves have infinitely many solutions, requiring arbitrary choices for how to fit them to this data, and quadratic Bézier curves do not provide a solution for inputs with parallel slopes. Fitting an arbitrary smooth function by splines of this special form gives an approximation whose error is quartic in the length of the spline segments.^[14][15] However, because the Tschirnhausen cubic has no finite inflection points, approximating a curve with an inflection point in this way requires a subdivision at the inflection point, to avoid loops or cusps in the approximating curve. The inability of these curves to inflect has been described as making them "inadequate ... for most practical design applications".^[16]

The parameterizations of the Tschirnhausen cubic have a special form: this curve is an example (up to similarity the only cubic example) of a Pythagorean hodograph curve, a parametric curve whose speed, arc length, curvature, and offset curves all have rational parameterizations.^[17] Specifically, for the parametric form with ${\displaystyle x=a(1-3t^{2})}$ and ${\displaystyle y=at(3-t^{2})}$, the speed is^[18] $${\displaystyle v(t)={\sqrt {x'^{2}+t'^{2}}}=3a+3at^{2},}$$ the arc length between parameters ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$ is the definite integral of speed,^[19] $${\displaystyle \int _{t_{0}}^{t_{1}}v(t)\,dt=(3at_{1}+at_{1}^{3})-(3at_{0}+at_{0}^{3}),}$$ and the curvature is^[19] $${\displaystyle \kappa (t)={\frac {2}{3a(1+t^{2})^{2}}}.}$$ As with any Pythagorean hodograph curve, the unit tangent vector can be given a rational parameterization by dividing the velocity vector ${\displaystyle (x',y')}$ by the speed.^[18] The unit normal vector is the same vector rotated by 90°,^[20] and each offset curve is obtained by adding a scalar multiple of the unit normal vector to the given curve.^[21] The involutes of the Tschirnhausen cubic are also rational curves, of degree 4, while its offset curves are rational of degree 5.^[22] Because its arc length has a rational parameterization, a space curve obtained from the Tschirnhausen cubic by using the arc length from a fixed reference point as its ${\displaystyle z}$-coordinate provides an example of an algebraic curve with constant slope: it maintains a fixed angle with respect to the ${\displaystyle xy}$-plane.^[16]

Other properties

Two copies of the Tschirnhausen cubic as geodesics of the Enneper surface

The Tschirnhausen cubic is a special case of a sinusoidal spiral ${\displaystyle r^{n}=a^{n}\cos n\theta }$ with ${\displaystyle n=-{\tfrac {1}{3}}}$.^[23] As with sinusoidal spirals more generally, its pedal equation has a particularly simple form. This equation relates the distance from the origin ${\displaystyle r}$ of any point on the curve with the distance from the origin ${\displaystyle p}$ of the tangent line through that point. For the Tschirnhausen cubic it is ${\displaystyle ar^{2}=p^{3}}$.^[4]

It has a single self-crossing (a node or crunode),^[24] so that an arc of the curve starting and ending at the node forms a bounded loop. The area of this loop, in the forms given above with parameter ${\displaystyle a}$, is:^[4] $${\displaystyle {\frac {72{\sqrt {3}}}{5}}a^{2}\approx 24.94a^{2}.}$$ The length of the loop, with the same parameter, is^[25] $${\displaystyle 12a{\sqrt {3}}\approx 20.78a.}$$

The Tschirnhausen cubic has no finite inflection points.^[26] However, an algebraic curve in the projective plane, the Tschirnhausen cubic is tangent to the line at infinity at the point where this line is crossed by the ${\displaystyle y}$-axis. It follows from the form of its algebraic equation that it has an inflection point at this point of tangency, which is also a vertex, a local minimum of curvature.^[27]

In its polar form, for angle parameter ${\displaystyle \theta }$, the angle of the (outward) normal vector perpendicular to the curve is ${\displaystyle 2\theta /3}$. Thus, for instance, at its crossing point, the parameters are ${\displaystyle \theta =\pm \pi }$ and the normal angles are ${\displaystyle \pm 2\pi /3}$, forming a crossing with acute angles of ${\displaystyle \pi /3}$ (60°) inside and opposite the loop, and obtuse angles of ${\displaystyle 2\pi /3}$ (120°) on the two sides of the loop. Every non-vertical line through the origin (the focus of the associated parabola) cuts the curve in three points whose tangent lines form an equilateral triangle. For this statement, one should count the symmetry axis as cutting the curve twice at its crossing point, with two tangent lines there forming an equilateral triangle with the tangent line at the vertex.^[3]

Each point of the Tschirnhausen cubic has a parabola tangent to it at that point with a focus at the origin. In the complex number parameterization where the Tschirnhausen cubic has the equation ${\displaystyle z=2/(1-\tau )^{3}}$, the parabola tangent to a point ${\displaystyle \tau _{0}}$ has the equation $${\displaystyle z={\frac {2}{(1-\tau _{0})(1-\tau )^{2}}}.}$$ The vertex of each parabola is at the point ${\displaystyle z=1/(2-2\tau _{0})}$, on the vertical line tangent to the vertex of the Tschirnhausen cubic. For any two of these tangent parabolas, the associated tangent lines to the Tschirnhausen cubic (at their points of tangency) form an isosceles triangle whose base is a bitangent of the two parabolas.^[8]

For the Enneper surface, with its standard representation as a parametric surface, fixing either the ${\displaystyle u}$ or ${\displaystyle v}$ parameter to zero results in a geodesic on the surface in the form of the Tschirnhausen cubic. These two geodesics meet at the point where both parameters are zero.^[10][28]