Recognizing implicitly given rational canal surfaces

In this paper we will pay an attention to the so called canal surfaces. These surfaces, which are defined as envelopes of moving spheres in 3-space, are very popular in Computer-Aided Design as they are often used as blending surfaces between the parts with circular ends. It was proved in [1] that any canal surface with a rational spine curve (a set of all centers of moving spheres) and a rational radius function possesses a rational parameterization. An algorithm for generating rational parameterizations of canal surfaces was developed and investigated in [2]. The class of rational canal surfaces with a rational spine curve and a rational radius function is a proper subset of the class of rational canal surfaces – all canal surfaces with a rational spine curve and rational squared radius function admit a rational parameterization, cf. [3, 4]. The reverse problem (a rational parametric description of a curve or a surface is given, find the corresponding implicit equation) is called the implicitization problem. An algorithm for computing the implicit equation of a canal surface generated by a rational family of spheres is presented in [5].

However our goal is different – we start with an implicit representation (i.e., with some polynomial in $x,y,z$) and want to decide if the corresponding algebraic surface is a rational canal surface or not. This is still a challenging problem, only partially solved in the recent past for a subfamily of canal surfaces, namely for the surfaces of revolution – see [6]. Moreover, in case of the positive answer we want to compute the equation of the spine curve and also the radius function. We would like to emphasize that this study is interesting not only from the theoretical point of view but it also reflects a need of the real-world applications as the results of many geometric operations are often described only implicitly. Then it is a challenging task to recognize the type of the obtained surface, find its characteristics and for the rational surfaces compute also their parameterizations. This is needed e.g. when the implicit blend surfaces (often of the canal-surface type) are constructed, see [7, 8, 9, 10].

Now, we start with short recalling some elementary notions. Let $\mathbb{E}^{3}_{\mathbb{R}}$ be Euclidean 3-space equipped with the Cartesian coordinates. A point $\mathbf{x}$ is represented with respect to a coordinate system by a vector $(x,y,z)$, and we do not distinguish between the point and its coordinate vector. Points in the projective closure of $\mathbb{E}^{3}_{\mathbb{R}}$ will be described using standard homogeneous coordinates

The equation $W=0$ describes the ideal plane as the set of all asymptotic directions, i.e., of points at infinity. The subset of the ideal plane which is invariant with respect to all similarities is called the absolute conic section $\Omega$ and characterized by

consisting of solely imaginary points. Circles are such conic sections which intersect $\Omega$ in two distinct points. Moreover, circles lying in parallel planes possess the same points on $\Omega$.

A canal surface is defined as the envelope of a one-parameter family of spheres $\Sigma(t)$ whose centers trace a curve $\mathcal{S}$ in 3-dimensional space parameterized by $\mathbf{s}(t)$ and possess radii $r(t)$, i.e.,

The curve $\mathcal{S}$ is called the spine curve and $r(t)$ the radius function of the canal surface. For constant $r(t)$ we obtain a pipe surface, for $\mathcal{S}$ being a straight line we arrive at a surface of revolution.

By appending the corresponding sphere radii $r$ to the points of the spine curve (or the skeleton, or the medial axis) we obtain the medial axis transform (shortly MAT), i.e., a curve $\mathcal{M}$ in 4-dimensional space. In addition, we define the curve $\mathcal{M}^{2}$ (and called a squared MAT) which is obtained by appending the squared sphere radii $R(t)$ (i.e., $r(t)=\sqrt{R(t)}$) to the points of the spine curve. The rationality of canal surfaces is described by the following proposition, cf. [4]:

The implicit equation $f(x,y,z)=0$ of the canal surface given by (3) can be obtained by eliminating the parameter $t$ from the defining equations

where $\Sigma^{\prime}$ is the derivative of $\Sigma$ with respect to $t$. The equation $\Sigma^{\prime}(t)=0$ describes the plane with the normal vector ${\mathbf{s}^{\prime}(t)}$, i.e., perpendicular to the spine curve $\mathbf{s}(t)$. Thus the canal surface contains a one-parameter set of the so called characteristic circles $\mathcal{C}_{\mathbf{s}(t)}=\Sigma(t)\cap\Sigma^{\prime}(t)$, which shows that the canal surfaces are generated by one-parameter family of circles.

We recall a fundamental property of canal surfaces

This enables us to formulate a naive approach to the recognition of rational canal surfaces. It suffices to find parameterizations of two rational curves corresponding in parameter (this means that for any choice of the parameter, the corresponding points lie on the same characteristic circle). Hence the intersection of the normals yields the rational parameterization of the spine curve and thus also of the squared radius function. Nevertheless it must be stressed out that finding rational curves on an algebraic surface is a hard problem and moreover computing their parameterizations corresponding in parameter is another very difficult task. Hence we will develop a different method, based on square-root parameterizations of plane sections of a given surface.

Finally, throughout the paper we assume that a given real algebraic surface $\mathcal{X}_{\mathbb{R}}$ is defined by the real polynomial $f(x,y,z)$. When homogenizing it we arrive at the polynomial $F(W,X,Y,Z)=0$ which describes the projective closure of the surface. In addition, we assume that the surface is considered in the complex extension, i.e., $\mathcal{X}\subset\mathbb{P}^{3}_{\mathbb{C}}$.

The simplest instances of canal surfaces are surfaces of revolution – in this case the spine curve is just a straight line. An algorithm for recognizing surfaces of revolution and deciding on their rationality has been recently presented in [6]. So, throughout this paper it is assumed that that $\mathcal{X}$ is not a surface of revolution.

As recalled above, the canal surfaces contain a one-parameter family of characteristic circles. Each of these circles intersects the absolute conic section $\Omega$ in two points. As the spine curve is not a straight line, all these intersection points cover the whole absolute conic and thus $\Omega\subset\mathcal{X}$. The multiplicity of $\Omega$ in $\mathcal{X}$ is called the cyclicity of $\mathcal{X}$ and denoted by $\mathrm{c}(\mathcal{X})$.

This simple criterion eliminates most of the candidates on canal surfaces because a generic surface has $\mathrm{c}(\mathcal{X})=0$. Now the remaining part of the method will be devoted to the reconstruction of the curve $\mathcal{M}^{2}$ from the given polynomial $f(x,y,z)$. If it is not possible to reconstruct $\mathcal{M}^{2}$ (see the details in Section 3), or the canal surface $\mathcal{Y}$ corresponding to the computed $\mathcal{M}^{2}$ differs from $\mathcal{X}$ then we can state that the given $\mathcal{X}$ is not a rational canal surface. Otherwise the output of the algorithm will be positive and $\mathcal{M}^{2}$ may be then used to parameterize the surface, see [2].

Let $\mathcal{H}$ be a generic irreducible plane section. Then the mapping $\xi$ from Lemma 2.4 is a double cover of a rational curve. As known $\mathcal{H}$ has to be a rational, elliptic or hyperelliptic curve. These curves are birational to plane curves in the so called Weierstrass form $\mathcal{E}:y^{2}-P(x)=0$, where $P(x)$ is a square-free polynomial. The algorithms for finding the Weierstrass form of elliptic and hyperelliptic curves, together with a birational morphism $\sigma:\mathcal{E\rightarrow H}$, were formulated in [11] and [12], respectively.

A curve $\mathcal{E}$ can be directly parameterized using square roots as $\mathbf{e}^{\pm}(t)=(t,\pm\sqrt{P(t)})$ and via $\sigma$ we arrive at the square root parameterization $\mathbf{h}^{\pm}=\sigma\circ\mathbf{e}^{\pm}$ of $\mathcal{H}$. Clearly, this is not a mapping as one parameter $t$ leads to two points $\sigma(t,\pm\sqrt{P(t)})$.

Two regular points on the canal surface are called corresponding whenever they lie on the same characteristic circle. If $\sigma(t,\pm\sqrt{P})$ are corresponding points for a generic $t$ then this square-root parameterization may be used for computing the squared MAT. However this is not necessarily satisfied and thus we one has to be very careful when constructing required square-roots parameterizations.

Recall that any mapping $\mathcal{H}\dashrightarrow\mathbb{P}^{1}_{\mathbb{C}}$ of degree two, which glues two corresponding points (i.e., points from $\mathcal{H}\cap\mathcal{C}_{\mathbf{s}}$), is called coherent. Such a mapping may be viewed as a kind of inverse to the desired square-root parameterization. Therefore in what follows we will go through all the three possible cases and we show how to construct this mapping.

Let $\mu:\hat{\mathcal{H}}\rightarrow\mathcal{H}$ be the desingularization of $\mathcal{H}$ and let $\mathbf{s}(t)$ be a proper parameterization of the spine curve $\mathcal{S}$. Then we have two mappings $\hat{\mathcal{H}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ of degree two, namely $\mathbf{s}^{-1}\circ\xi\circ\mu$ and $\pi\circ\sigma^{-1}\circ\mu$, where $\pi\circ\sigma^{-1}$ is the coherent mapping constructed in the previous section and $\xi$ is the mapping from Lemma 2.4. Since these two mappings glue the pairs of corresponding points they have to differ only by an automorphism $\phi:\mathbb{P}^{1}_{\mathbb{C}}\rightarrow\mathbb{P}^{1}_{\mathbb{C}}$ (c.f. the proof of Theorem 2.6 and Proposition 2.8 for elliptic and hyperelliptic curves, respectively). The structure of above mappings is summarized in the diagram

The mapping $\pi$ enabled us to parameterize the curve $\mathcal{E}$ in the Weierstrass form by a square-root parameterization $\mathbf{e}^{\pm}(t)$. Then via $\sigma$ we arrive at the square-root parameterization $\mathbf{h}^{\pm}(t)$ of $\mathcal{H}$ that correspond in parameter (which means that for any $t_{0}$ the points $\mathbf{h}^{+}(t_{0})$ and $\mathbf{h}^{-}(t_{0})$ are lying on the same characteristic circle of $\mathcal{X}$). Intersecting the normals constructed at these corresponding points yields a parameterization of the rational spine curve $\mathcal{S}$. This parameterization, cf. (9), can be expressed as $\mathbf{s}\circ\phi$, which is a proper rational parameterization of $\mathcal{S}$. The computation of the squared radius easily follows and we obtain a parameterization of $\mathcal{M}^{2}$. Finally, using [4] we compute from $\mathcal{M}^{2}$ a parameterization $\mathbf{x}(t,s)$ of the associated canal surface and if $f(\mathbf{x}(t,s))=0$ we can conclude that $\mathcal{X}$ is a canal surface with the squared medial axis transform $\mathcal{M}^{2}$.

The whole procedure is summarized in Diagram 1. The methods and approaches studied in this paper will be now presented in detail in the following two particular examples.

In this contribution we studied an interesting (and till now unsolved) theoretical problem, motivated by some technical applications (e.g. when implicit blends are constructed and then shall be characterized), i.e., how to recognize a rational canal surface from its defining polynomial equation. We designed an algorithm returning for rational canal surfaces also their squared medial axis transform which can be used for computing a rational parameterization of the given surface. The methods and approaches were presented on two commented examples. The investigation may be considered as another step towards the recognition of other implicitly given surfaces which play an important role in technical practice.

The first author was supported by the project NEXLIZ, CZ.1.07/2.3.00/30.0038, which is co-financed by the European Social Fund and the state budget of the Czech Republic. The work on this paper was supported by the European Regional Development Fund, project “NTIS – New Technologies for the Information Society”, European Centre of Excellence, CZ.1.05/1.1.00/02.0090.