# Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$?

Here is a plot from WA.

This curve is an example of a subset of $\mathbb R^3$ which has cardinality $\mathbb c$ and intersects every plane in at most three points. (See Problems and Theorems in Classical Set Theory by Péter Komjáth, Vilmos Totik, p.300. They mention Wacław Sierpiński's book Cardinal and Ordinal Numbers as a reference.)

The above fact is not very difficult to prove. (If a plane $ax+by+cz+d=0$ intersects the curve in four differen points, then the equation $d+at+bt^2+ct^3=0$ is fulfilled for 4 different values of parameter $t$. This give a system of linear equations with unknowns $a$, $b$, $c$, $d$. The matrix of this linear system is Vandermonde matrix and it is invertible. So there is only trivial solution for this linear system.)

I was wondering whether this curve might have some other interesting properties. Knowing the name (if it has one) would be useful for finding some more information about it.

EDIT: As the answers say, it is called twisted cubic. For other user's convenience, I will add Wikipedia link. Other pictures of this curve and some related objects can be found by Google.

• As the answers say, this curve (or its projective version) is usually called the twisted cubic. A fancier name is the rational normal curve of degree 3. It has lots of interesting properties: just look in a book on algebraic geometry (e.g. Harris' First Course) to find them! – user64687 Jan 12 '15 at 16:21
• See Wiki. – user64494 Jan 12 '15 at 16:24
• I think that Matousek calls it the power curve. It is widely used to prove theorems about the embeddability of a projective variety in $\mathbb{R}^n$. – Jack D'Aurizio Jan 12 '15 at 16:33

My book on ideals and varieties calls this the 'twisted cubic'.

More precisely I'm talking about the book Ideals, Varieties, and Algorithms where on page 20 it is mentioned that in the context of the variety $\textbf{V}(y-x^2,z-x^3)$ (which has previously been introduced as being the twisted cubic):

"... Note that setting $x=t$ in $y-x^2=z-x^3=0$ gives us a parametrization $$x=t$$ $$y=t^2$$ $$z=t^3$$ of the twisted cubic."

• When you say "your book", what do you mean by that? – Asaf Karagila Jan 14 '15 at 19:31
• I mean a book I have laying on my desk here, not a book written by me, if that's what you're wondering – user2520938 Jan 14 '15 at 19:43
• I was wondering that, yes. Which book is that? What page does this information appear on? Some citation that can be used to corroborate your claim that it appears in a book? – Asaf Karagila Jan 14 '15 at 19:48
• Ok added some info, is this what you mean? – user2520938 Jan 14 '15 at 19:58
• That is much better! – Asaf Karagila Jan 14 '15 at 20:41

The curve $(t,t^2,t^3\dots t^n)$ can also be called the moment curve of degree $n$. It has the property that every hyperplane intersects the moment curve in a finite set of at most $n$ points, which is why it is used alongside the ham sandwich theorem sometimes.

• I liked that this answer specifically show the generalization of the particular aspect which was used in that 3-dimensional case I stumbled upon. – Martin Sleziak Jan 15 '15 at 21:39

Isn't it called he twisted cubic?

It is the simplest example of third degree curve in 3D space. It ought to be perhaps called a "bent and twisted" cubic as $\kappa$ and $\tau$ scalars are non-zero. Curvature and Torsion are equally important for space curves to describe the way they bend and twist in 3D space.