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I didn't yet encounter to a case that need to be solved by cubic equations (degree three) ! May you give me some information about the branches of science or criterion deal with such nature ?

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"I didn't yet encounter to a case that need to be solved by cubic equations": this is not really a surprise; third (and higher) degree equations are much harder to handle than the first and second degree cases. So by some kind of "natural selection", the are much less frequently met because theoricians and practitioners tend to avoid them. – Yves Daoust Jun 8 '14 at 10:36
My teacher uses them to make me bored :P – Awal Garg Jun 8 '14 at 17:41
cubic splines perhaps? – user13267 Jun 9 '14 at 2:07
Kepler's third law of planetary motion involves squares and cubes of physical quantities. – KCd Jun 9 '14 at 7:22
Finding eigenvalues of a $3\times 3$ matrix in general requires solving a cubic equation. This kind of problem is very common in teaching, but mysteriously one seems to only encounter examples where the eigenvalues can (also) be found without solving a cubic equation, or at least without using the general formula for doing so. – Marc van Leeuwen Jun 9 '14 at 10:07

14 Answers 14

up vote 51 down vote accepted

One example ...

The curves used in Postscript (including Postscript fonts), and in most drawing and graphics programs (like Adobe Illustrator, Powerpoint, etc.), are cubic Bezier curves.

The outlines of the characters you are reading right now might be strings of cubic Bezier curves. The prefered font for this page is "Georgia", which uses (TrueType) quadratic curves. But, if you don't have the Georgia font installed, Times Roman will be displayed instead, and that uses cubic curves. The seemingly circular dot above the "i" is actually four cubic curves.

Similarly, whenever you read a PDF document, or almost any printed document, you are (often) looking at strings of cubic curves.

Graphics packages like OpenGL and Direct3D also use cubic curves heavily. So the curves you see in games and other 3D programs are often cubics.

Whenever you do any calculations with cubic Bezier curves, you are using cubic equations. For example, if you calculate points on the curves so that you can draw them on a computer screen or a printer.

If you want to intersect one of these curves with a straight line, you will have to solve a cubic equation. This happens (for example) when you "clip" the curve to some rectangular boundary.

In short, cubic Bezier curves are everywhere. You can start reading about all of this here.

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+1, I believe these are also called splines? – DanielV Jun 8 '14 at 10:23
@DanielV -- Not really. A spline is (usually) a sequence of $n$ polynomial curve segments that are joined together smoothly. If the polynomisl pieces are cubics, then you have a cubic spline. A Bezier curve is a single polynomial segment, so it's the special case $n=1$. But cubic splines are another good example of the usage of cubic functions. – bubba Jun 8 '14 at 10:30
Ah thanks, I wasn't sure of the terminology but I know cubic splines are used a lot in rendering curves because once you match up endpoints, derivative, and second derivative, you have a cubic that is a very nice piecewise approximation. – DanielV Jun 8 '14 at 10:34
@bubba your answer startled me ! – Fereydoon Shekofte Jun 8 '14 at 10:43
@bubba: As for terminology, I think the character boundaries are defined by Bézier splines, which by the way aren't actually splines (because those are numeric functions), thought their two coordinates (as parametric functions) are. The notion of Bézier curve does not allow for multiple pieces to be joined together; your comment admits this, but doing so contradicts your answer. – Marc van Leeuwen Jun 9 '14 at 11:13

Eigenvalues and Eigenvectors are an essential tool in the theory of matrix computation. The case of 3x3 matrices is particularly important as it relates to geometric transforms of our good old 3D world.

The Eigenvalues of a 3x3 matrix are found from this appetizing equation:


As a "real world" example, consider fitting a plane to a point cloud in 3D space (for instance to model the facets of an object digitized with a 3D scanner). This leads to an Eigen problem of this kind (or similarly Singular Value Decomposition).

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can't the last term be simplified with $-\det(A)$? – Ant Jun 9 '14 at 13:22
You are quite right. I blindly copied this from a site and didn't check. I also question the $x$ term. – Yves Daoust Jun 9 '14 at 13:35
Eigen-stuff is not only important in matrix theory. The study of moments of inertia (principal axes, specifically) requires finding eigenvalues of $3\times3$ matrices. – bubba Jun 9 '14 at 22:55
@bubba: quite right. And I wouldn't be surprised that the best-fit plane is perpendicular to one of the axis of inertia of the point cloud. – Yves Daoust Jun 10 '14 at 8:04

The drag of airplanes is essentially the coefficient of drag of the airframe (complete) times velocity cubed.

So for a $10\%$ increase in speed, it requires $1.1^3$ more horse power ($33.1\%$ more horsepower) providing that:

  1. the additional horsepower adds no additional weight (not likely)
  2. the additional horsepower does not change the airframe's outline or shape in any way (except perhaps for a propeller’s blade profile, planform, or blade count) (also, not likely)

That is why recent advances in airplane design have focused on the two primary sources of drag and a third secondary (Which are LINEAR):

  1. Airframe and wetted area drag. (parasitic drag)
  2. Coefficient of drag for the lift produced by the wing. (Induced Drag)
  3. Drag produced lifting more or less weight. (eventually becomes Induced Drag)

The joke is that a aerodynamicist designing a plane would sell his/her grandmother for a few drag points, or counts at the third decimal place of the coefficient of drag, i.e. $C(f)= 0.26x$.

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Thank you for the edit, MattAllegro – Dennis Jun 10 '14 at 22:20
For similar reasons the maximum power that can be generated by a wind turbine is proportional to the cube of the wind speed. – Adam Bailey Jun 11 '14 at 8:08
@Adam Bailey - Very true Adam. Also the reason why there are such MASSIVE structures holding up signs, especially in the Gulf Coast area. A 50 mph wind might 800 lbs of horizontal force on a billboard. A 100 mph wind would put 6400 lbs and a 125 would put 12,202 lbs of horizontal force. – Dennis Jun 12 '14 at 15:56

In thermodynamics, the most commonly used equations of state in industry (for reseach and development in particular in oil and gas industry) are called cubic, even if they do not look to ba at first glance.

The first of them was introduced by Van der Waals in 1873. It write $$P =\frac{R T}{V-b}-\frac{a}{V^2}$$ (where $P$ is pressure, $V$ is volume and $T$ is temperature). For a lot of different reasons, it is expressed as a function of the compressibility factor $Z=\frac{P V}{R T}$ and this equation of state then write $$Z^3-(1-B)Z^2+A Z-A B=0$$ where $$A=\frac{a P}{(R T)^2}$$ $$B=\frac{b P}{R T}$$

I shall add that, in any steady-state or dynamic simulation of chemical or petrochemical plants, these cubic equations must be solved zillions of times and even more !

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Elliptic curves can be (are) considered a type of cubic.

$$ y^2=x^3+ax+b $$

These are used in many cryptographic algorithms.

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Well I can't comment because I don't have 50 rep. But I will expand on @DrkVenom's answer.

Cubic equations are used widely in Elliptic Curve Cryptography. So much so that you have probably used it quite a few times today on your phone or PC without realizing it. It is a pretty indepth/complex idea but is used widely in network security.

For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible — this is the "elliptic curve discrete logarithm problem" or ECDLP. The entire security of ECC depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points. The size of the elliptic curve determines the difficulty of the problem.

Another usage is in the popular Bitcoin crypto-currency and many other crypto-currencies based off Bitcoin (see here for others). The algorithm used to sign bitcoin transactions is called ECDSA. It is used to sign digital messages for transmission over insecure lines to assure that your message cannot be altered, and to ensure messages you receive have not been altered. Fun fact, there was recently a small vulnerability/concern found in Bitcoins usage of this algorithm where an attacker (by applying his/her knowledge of cubic algorithms) could modify the signature of a transaction, causing all sorts of havoc if changed and distributed correctly (but nothing major!). There was also something like this used to attack the playstation network back in 2010.

In all, I said the same things in school. Now I use a lot of what I would "never use" everyday plus much more I couldn't imagine.

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Now, you can comment. – Jun 10 '14 at 7:31

There exist cubic tweening functions; that is, for animations on computers, people sometimes use a cubic polynomial to ease (or smooth the animation). I personally find the cubic easing functions to feel the smoothest and thus use them a bunch.

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Your mass is $m$.
Your initial location is $x_0$.
Your initial velocity is $v_0$.
Your initial propulsion force is $F_0$.
Your force increases at a constant rate of $\Delta_F$.


Your force is $F(t) = F_0 + \Delta_F\,t$.

Your acceleration is $a(t) = \frac{F(t)}{m} = \frac{F_0 + \Delta_F\,t}{m} = \frac{F_0}{m} + \frac{\Delta_F}{m}t$

Your velocity is $v(t) = v_0 + \frac{F_0}{m}t + \frac{\Delta_F}{2m}t^2$

Your position is $x(t) = x_0 + v_0 \,t+ \frac{F_0}{2m}t^2 + \frac{\Delta_F}{6m}t^3$

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But you haven't given a real-world example of this happening, as was requested. – Blazemonger Jun 9 '14 at 13:52
For a real-world motivation, consider the inverse problem of having a path you'd like to follow, then deciding when to apply forces (and what magnitudes to use) to achieve the path. Motion controllers perform this kind of calculation quite often. In particular, industrial robots perform this kind of calculation and were likely involved in the manufacturing of a good proportion of the things you own. – Dan Bryant Jun 9 '14 at 18:17
@Blazemonger I believe that the fact that he is talking about classical mechanics is sufficient enough information that it's a real-world example. – Gary Drocella Jun 13 '14 at 16:48

This example arises in a bio-economic model of fishing (it’s a variant of the Gordon-Schaefer model). From simple and plausible assumptions it follows that the level of fishing effort for maximum sustainable profit is given by a cubic equation.

Natural growth $F$ of the fish stock $X$ in the absence of fishing is assumed to be given by the following logistic function, where $k$ is the carrying capacity (maximum fish population the fishery can support) and $r$ is a growth parameter (throughout I use capital letters for variables and small letters for parameters):

$$F = rX(1 – X/k).......(1)$$

Note that the above is the simplest plausible assumption: to assume that $F$ is a linear function of $X$, however large $X$ may be, would not be credible due to overcrowding etc.

The fish harvest $H$ from fishing effort $E$ is assumed given by:

$$H = gEX.......(2)$$

Costs $C$ are assumed to increase linearly with effort:

$$C = cE.......(3)$$

The unit price $P$ at which the harvested fish can be sold is often assumed to be exogenous, but let's assume (more realistically in some circumstances) a downward-sloping linear relation with harvest:

$$P = a – bH.......(4)$$

For sustainability the harvest must equal the natural growth of the fish population. From (1) and (2) this implies:

$$rX(1 – X/k) = gEX$$

$$1 – X/k = gE/r$$

$$X = k(1 – gE/r)$$

Substituting for $X$ in (2), the relation between sustainable harvest and effort is:

$$H = gEk(1 – gE/r)$$

Substituting for $H$ in (4), the relation between unit price and effort, given sustainability, is:

$$P = a – bgEk(1 – gE/r)$$

Sustainable profit $\Pi$ equals revenue (harvest times unit price) less cost, given by:

$$\Pi = HP-C = [gEk(1 – gE/r)][a – bgEk(1 – gE/r)] – cE$$

$$ = [gEk – g^2E^2k/r][a – bgEk + bg^2E^2k/r] – cE$$

$$ = agEk – bg^2E^2k^2 + bg^3E^3k^2/r – ag^2E^2k/r + bg^3E^3k^2/r – bg^4E^4k^2/r^2 – cE$$

$$ = - (bg^4k^2/r^2)E^4 + (2bg^3k^2/r)E^3 – (a/r + bk)(g^2k)E^2 + (agk – c)E$$

Hence the first-order condition for maximum profit is a cubic equation in $E$:

$$d\Pi/dE = - 4(bg^4k^2/r^2)E^3 + (6bg^3k^2/r)E^2 – 2(a/r + bk)(g^2k)E + (agk – c) = 0$$

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very amazing ! and thanks for demonstration of formulas . – Fereydoon Shekofte Jun 11 '14 at 10:02

Generally speaking, algebraic equations (roots of polynomials of some degree) are frequently used in relation with the dynamic behavior of linear systems.

[Linear systems are such that the effect of the sum of causes is the same as the sum of the effects of the individual causes. They are an important aspect of engineering science.]

When you study the behavior of such systems as times goes (like a mass tied to a spring), you see natural "modes" appear, such as natural vibration frequencies. These are found as roots of an algebraic equation corresponding to the ordinary differential equation that governs the dynamics of the system.

The degree of the algebraic equation is directly related to the number of linear components in the system. For instance, studying the behavior of some electric circuits with three capacitors can lead to a third degree equation.

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Catmull-Rom splines are a type of cubic Hermite spline curve that are widely used in modern video games (DirectX even has functionality for them built-in: They are well suited for use in real-time systems, as they are fairly quick to calculate:

Games use them to smooth out camera and object motion paths.

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To expand on @DrkVenom's post, elliptic curve cryptography (ECC) is a great example of the deep application of cubics. ECC is currently at the forefront of research in public key cryptography. It has the important benefit of requiring a smaller key size while maintaining strong security. Here is an accessible and well-illustrated introduction to the subject:

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Voltage-current relationships of Thyristors ,Unijunction transistors and Tunnel Diodes are cubic algebric expressions.......................... V = a(0) +a(1)*I +a(2)*I^2 +a(3)*I^ a general Voltage (V) and Current (I) relationship expression of these electronic devices.............Usually one of the roots is real and the other two are complex .

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We need to solve cubic and quartic equations on a regular basis for 4- and 5-precision synthesis of planar mechanisms.

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thanks a lot for your contribution ! you mentioned to an amazing subject that i am not familiar with it , at all . may you link a reference for this topic on web or just show us an example ? – Fereydoon Shekofte Dec 10 '15 at 8:02

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