On Maps Taking Lines to Plane Curves

Let $\mathbb{P}^{n}$ be the projective space of dimension $n$ over $\mathbb{R}$ or $\mathbb{C}$. In [Timorin2012], a planarization was defined as a mapping $\Phi:U\to\mathbb{P}^{n}$, where $U\subset\mathbb{P}^{2}$ is an open subset, such that $\Phi(\lambda\cap U)$ is a subset of a hyperplane, for every line $\lambda\subset\mathbb{P}^{2}$. Studying planarizations is closely related to studying maps taking lines to curves of certain linear systems; a classical result of this type is the Möbius–von Staudt theorem ([Möbius1885]; [von Staudt1847]) about maps taking lines to lines, sometimes called the Fundamental Theorem of Projective Geometry. We will always assume that the planarizations are sufficiently smooth, i.e., sufficiently many times differentiable. If the ground field is $\mathbb{C}$, then we assume analyticity. The main result of this paper, stated below, is a complete description of all planarizations in case $n=3$.

Before stating the main result, we recall the motivation for studying planarizations ([Timorin2012]). An interesting class of problems consists of finding all transformations of a planar domain into a (in general) different planar domain that take line segments to arcs of curves in a given linear system $\mathcal{L}$. Say, if $\mathcal{L}$ is the linear system of all circles, then we are finding transformations taking lines to circles (including degenerations, i.e., straight lines). This particular problem was initially motivated by Nomography ([Khovanskii1976]), and it was solved in [Khovanskii1980]. Maps taking lines to circles have also been investigated in higher dimensions ([Izadi2005]; [Timorin2003]; [Timorin2004]). Large classes of such maps are provided by generalized Hopf fibrations ([Timorin2006]) and, more generally, by quadratic rational parameterizations of spheres ([Timorin2005]). Another motivation comes from freeform architecture, for which surfaces formed by simple planar curves are of special interest, see [Deng et al.2011].

We now state the problem more precisely. Let $\mathcal{L}$ be a linear system of algebraic curves in $\mathbb{P}^{2}$, and let $U\subset\mathbb{P}^{2}$ be an open set. We say that a map $f:U\to\mathbb{P}^{2}$ takes all lines to $\mathcal{L}$-curves if $f(U\cap L)$ is a subset of a curve from $\mathcal{L}$, for every line $L\subset\mathbb{P}^{2}$. For various linear systems $\mathcal{L}$, we would like to know all sufficiently smooth maps that take lines to $\mathcal{L}$-curves. If a sufficiently smooth map $f:U\to\mathbb{P}^{2}$ takes lines to $\mathcal{L}$-curves, then it gives rise to a planarization $F=\Phi_{\mathcal{L}}\circ f:U\dashrightarrow\mathbb{P}^{n}$, where $\Phi_{\mathcal{L}}:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{n}$ is the rational map, whose homogeneous components are generating equations for $\mathcal{L}$. This was the original motivation for studying planarizations. For example, to describe maps taking line segments to conics, we need to consider planarizations with values in $\mathbb{P}^{5}$.

We need to explain the terminology. A planarization $\Phi:V\to\mathbb{P}^{3}$ is said to be trivial if $\Phi(V)$ is a subset of a plane. A planarization $\Phi:V\to\mathbb{P}^{3}$ is said to be co-trivial if there exists a point $b\in\mathbb{P}^{3}$ such that, for every line $\lambda\subset\mathbb{P}^{2}$, the set $\Phi(\lambda\cap V)$ lies in a plane containing $b$. Of course, logically, co-trivial planarizations include trivial planarizations. On the other hand, there are “more” trivial planarizations than co-trivial planarizations that are not trivial. This is one of the reasons for distinguishing trivial planarizations as a separate class; the second reason being a partial duality between trivial and co-trivial planarizations. Trivial planarizations can be described in terms of an arbitrary map from $\mathbb{P}^{2}$ to $\mathbb{P}^{2}$, and co-trivial planarizations can be described in terms of an arbitrary function on $\mathbb{P}^{2}$ (see Theorem 6.1).

A map $\Phi:V\to\mathbb{P}^{3}$ is said to be a quadratic rational map if in some (hence any) system of homogeneous coordinates it is given by quadratic homogeneous polynomials. In other words, there are quadratic homogeneous polynomials $Q_{0}$, $Q_{1}$, $Q_{2}$ and $Q_{3}$ in $x_{0}$, $x_{1}$, $x_{2}$ such that $\Phi$ maps a point with homogeneous coordinates $[x_{0}:x_{1}:x_{2}]$ to the point with homogeneous coordinates $[y_{0}:y_{1}:y_{2}:y_{3}]$, where

It is easy to see that every quadratic rational map takes lines to conics, in particular, every quadratic rational map is a planarization. Some quadratic planarizations are neither trivial nor co-trivial.

Another class of examples is provided by duality. Let $\Phi:V\to\mathbb{P}^{3}$ be a planarization. Recall that the dual projective plane $\mathbb{P}^{2*}$ consists of lines in $\mathbb{P}^{2}$, and the dual projective space $\mathbb{P}^{3*}$ consists of planes in $\mathbb{P}^{3}$. Let $V^{*}$ be the subset of $\mathbb{P}^{2*}$ consisting of all lines $\lambda\subset\mathbb{P}^{2}$ with the following property: the set $\Phi(\lambda\cap V)$ lies in a unique plane $P_{\lambda}$. Note that the set $V^{*}$ is open, possibly empty. The dual planarization $\Phi^{*}:V^{*}\to\mathbb{P}^{3*}$ is by definition the map taking $\lambda$ to $P_{\lambda}$. Given a coordinate representation of $\Phi$, it is easy to write explicit formulas for $\Phi^{*}$. It turns out that a planarization dual to a quadratic rational map is a special kind of cubic rational map. Such planarization is called dual quadratic. It is rather obvious that the duality is symmetric: if $\Phi:V\to\mathbb{P}^{3}$ is a planarization, $\Phi^{*}:V^{*}\to\mathbb{P}^{3*}$ is the dual planarization with a nonempty domain $V^{*}$, and $V^{**}\cap V\neq\varnothing$, then $\Phi=\Phi^{**}$ on $V^{**}\cap V$.

It is proved in [Timorin2012] that a planarization $\Phi:U\to\mathbb{P}^{3}$ that is neither trivial nor co-trivial must be a rational map of degree two or three, at least on some nonempty open subset of $U$. Thus, to prove the Main Theorem, it suffices to describe all cubic planarizations. A cubic planarization is defined globally as a rational map from $\mathbb{P}^{2}$ to $\mathbb{P}^{3}$ (it may have some points of indeterminacy). Moreover, it suffices to assume that the ground field is $\mathbb{C}$.

It is natural to consider the following equivalence relation on the set of all planarizations. Given two planarizations $\Phi:V\to\mathbb{P}^{3}$, $\Phi^{\prime}:V^{\prime}\to\mathbb{P}^{3}$ we say that they are equivalent if they coincide on some nonempty open set after suitable projective coordinate changes in $\mathbb{P}^{2}$ and in $\mathbb{P}^{3}$. In other terms, there are projective automorphisms $\eta\in\mathrm{PGL}_{2}$, $\mu\in\mathrm{PGL}_{3}$, and a nonempty open subset $W\subset V\cap\eta^{-1}(V^{\prime})$ such that $\Phi=\mu\circ\Phi^{\prime}\circ\eta$ on $W$. In Sect. 6, we describe all equivalence classes of planarizations over real numbers by specifying a representative in each class. These representatives will also be called normal forms of planarizations, so that every planarization is projectively equivalent to some normal form. Of course, there are infinitely many classes of trivial and co-trivial planarizations. These classes can be described by means of function parameters, i.e., they depend on some arbitrary functions. Other than that, there are 16 classes. Our classification is based on the classification of equivalence classes of quadratic rational maps ([Coffman et al.1996]).

Studying planarizations for $n>3$ is also important, e.g., the case $n=5$ relates to studying maps that take lines to conics. As in the case $n=3$, we expect that, apart from (co-)trivial cases, planarizations must be rational maps of low degree. An explicit upper bound for the degree is easy to conjecture: $\frac{n(n-1)}{2}$. On the other hand, we do not know even a conjectural description of all planarizations for $n>3$.

Let us introduce some notation and conventions that will be used throughout the paper. First, all spaces and maps are defined over complex numbers. Let $\Phi:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{3}$ be a cubic rational map sending a point of $\mathbb{P}^{2}$ with homogeneous coordinates $[x_{0}:x_{1}:x_{2}]$ to a point of $\mathbb{P}^{3}$ with homogeneous coordinates $[y_{0}:y_{1}:y_{2}:y_{3}]$, where

and $\varphi_{\alpha}$ is a homogeneous polynomial in three variables of degree 3. Recall that an indeterminacy point of $\Phi$ is a point $x$ in $\mathbb{P}^{2}$ such that $\varphi_{\alpha}(x)=0$ for all $\alpha=0$, 1, 2, 3. This is precisely a point that does not have an image under $\Phi$.

Recall that $\Phi$ defines a linear system $\mathcal{L}_{\Phi}$ of plane cubics of dimension 3 (a three-dimensional linear system is called a linear web). By definition, $\mathcal{L}_{\Phi}$ is generated by the cubics $\varphi_{\alpha}=0$, i.e., the equation of any cubic in $\mathcal{L}_{\Phi}$ has the form

where the coefficients $c_{0}$, $c_{1}$, $c_{2}$, $c_{3}$ are complex numbers not vanishing simultaneously, thus $[c_{0}:c_{1}:c_{2}:c_{3}]$ can be thought of as a point in $\mathbb{P}^{3}$, or, in more invariant terms, as a point in the dual projective space $\mathbb{P}^{3*}$ defining a plane $P$ in $\mathbb{P}^{3}$. Indeterminacy points of $\Phi$ are also called the base points of $\mathcal{L}_{\Phi}$. We will write $B_{\Phi}$ for the set of all base points of $\mathcal{L}_{\Phi}$. Every cubic from $\mathcal{L}_{\Phi}$ contains the set $B_{\Phi}$. To each plane $P$ in $\mathbb{P}^{3}$, we may associate the set $\varkappa_{P}=B_{\Phi}\cup\Phi^{-1}(P)$ in $\mathbb{P}^{2}$; clearly, if $\varphi_{0},\ldots,\varphi_{3}$ are linearly independent, then $\varkappa_{P}$ is a plane cubic curve, possibly degenerate.

For this reason, we will mostly assume that $B_{\Phi}$ is a finite set of points. There is a natural way of assigning multiplicity to every point $b\in B_{\Phi}$. Namely, the multiplicity $m(b)$ is equal to the minimum intersection index of two cubics in $\mathcal{L}_{\Phi}$ at $b$. Since all cubics in $\mathcal{L}_{\Phi}$ pass through $b$, we have $m(b)\geq 1$. We will write $|B_{\Phi}|$ for the number of points in $B_{\Phi}$ counting multiplicities. In other terms, we have by definition

We will write $S_{\Phi}=\Phi(\mathbb{P}^{2}{\setminus}B_{\Phi})$ for the image of $\Phi$ and $\overline{S}_{\Phi}$ for its projective closure in $\mathbb{P}^{3}$. Note that if $\Phi$ is a planarization and dimension of $S_{\Phi}$ is one, then $S_{\Phi}$ is a planar curve, hence $\Phi$ is trivial. Thus, the case when $S_{\Phi}$ is a curve is not interesting.

In what follows, we recall how to define the topological degree of the map $\Phi$. Suppose that $B_{\Phi}$ is finite and that $S_{\Phi}$ has dimension two. By [Beauville1996], Theorem II.7 (elimination of indeterminacy), there exists a smooth compact projective surface $X$, a regular morphism $\pi:X\to\mathbb{P}^{2}$ that is a finite composition of blow-ups, and a regular morphism $\Psi:X\to\mathbb{P}^{3}$ with the property $\Psi=\Phi\circ\pi$, which holds where the right-hand side is defined. By the Specialization Principle, cf. [Mumford1976], Theorem (3.25), and since the dimension of $\Psi(X)\supset S_{\Phi}$ is 2, a generic point in $\Psi(X)$ has exactly $k$ preimages in $X$, where $k$ is the degree of the field extension $\mathbb{C}(X)/\Psi^{*}\mathbb{C}(\Psi(X))$. The difference $X{\setminus}\pi^{-1}(\mathbb{P}^{2}{\setminus}B_{\Phi})$ consists of exceptional curves, whose images under $\Psi$ lie in a proper Zariski closed subset of $\Psi(X)$. Therefore, a generic point of $S_{\Phi}$ has exactly $k$ preimages in $\mathbb{P}^{2}{\setminus}B_{\Phi}$. We will call the number $k$ the topological degree of $\Phi:\mathbb{P}^{2}{\setminus}B_{\Phi}\to S_{\Phi}$.

The following proposition is classical and well known but we recall the proof for completeness.

As before, we consider a cubic rational map $\Phi$. We now assume that $\Phi$ is a planarization, i.e., the $\Phi$-image of every line in $\mathbb{P}^{2}$ is a subset of some plane in $\mathbb{P}^{3}$. We say that $\Phi$ is strictly cubic if there is no Zariski open subset $U\subset\mathbb{P}^{2}$ such that the restriction of $\Phi$ to $U$ coincides with some quadratic rational map. It follows from Lemma 2.1 that for every strictly cubic planarization $\Phi$, the set $B_{\Phi}$ is finite.

The next proposition is needed to formulate the theorem we will prove in this section.

To prove this proposition we need the following lemma.

The following theorem is proved in this section.

Note that, under the assumptions of Theorem 3.3, the topological degree of $\Phi:\mathbb{P}^{2}{\setminus}B_{\Phi}\to S_{\Phi}$ is well defined by Proposition 3.1.

First, we examine general properties of strictly cubic planarizations that are needed to prove Theorem 3.3.

In this section, we give a complete description of cubic planarizations thus completing the description of all planarizations. We will assume throughout this section that $\Phi$ is a strictly cubic planarization that is neither trivial nor co-trivial. Then $S_{\Phi}$ has dimension two, and the set $B_{\Phi}$ is finite. By Theorem 3.3, the topological degree of the map $\Phi:\mathbb{P}^{2}{\setminus}B_{\Phi}\to S_{\Phi}$ is at least two. We can now make this result stronger.

We also need the following lemma.

Throughout this section, we suppose that $\Phi:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{3}$ is a quadratic rational map such that the image of $\Phi$ lies in some quadratic surface $S$ but does not lie in a plane. We will classify all such quadratic maps up to projective equivalence. The classification must be classical but we failed to find a modern reference. The following theorem describes the classification over $\mathbb{C}$.

The corresponding real classification differs only in that the complex equivalence class of $\Phi_{1}$ splits into two real equivalence classes $\Phi_{1a}$ and $\Phi_{1b}$.

In Sect. 5.1, we prove Theorem 5.1, and in Sect. 5.2, we prove Theorem 5.2.

In this section, we give a complete list of local normal forms of planarizations. We assume that the ground field is $\mathbb{R}$.

In Theorem 6.1, the form $(T)$ represents all trivial planarizations, and the form $(CT)$ represents all co-trivial planarizations that are not trivial. These items correspond to infinitely many projective equivalence classes. However, note that there are finitely many classes of nontrivial non-co-trivial planarizations.

The equations of the surfaces parameterized by dual-quadratic planarizations $(C_{1})$–$(C_{6})$ are

$(C_{1})$: $4t^{3}-t(u^{2}+v^{2}+w^{2})+uvw=0$

$(C_{2})$: $4t^{3}+t(u^{2}-v^{2}+w^{2})+uvw=0$

$(C_{3})$: $4vu^{2}+u(t^{2}-4v^{2}+w^{2})-vw^{2}=0$

$(C_{4})$: $4tu^{2}-uw^{2}+tv^{2}=0$

$(C_{5})$: $u(vw-t^{2})+vt^{2}=0$

$(C_{6})$: $u(4tv-w^{2})+v^{3}=0$, where $[u:v:w:t]$ are homogeneous coordinates in $\mathbb{P}^{3}$. These equations are obtained by eliminating the three variables $x$, $y$, $z$ from the four equations