Bijective Mapping Analysis to Extend
the Theory of Functional Connections
to Non-rectangular 2-dimensional Domains

Given a set of $n$ complex control points, $w_{i}$ with $i\in\{1,\dots,n\}$, in the $W$ domain associated to $n$ corresponding boundary reference points, $z_{j}$ with $j\in\{1,\dots,n\}$, in the unit-square $Z$ domain, the transformation,

is a $n$-point complex conformal mapping. In this equation, $\phi_{i}(z)$ are a set of switching functions related to the reference points, $z_{j}$. In particular, Let select $n=4$ control points (red markers on top plots of Fig. 2) defined by,

The switching functions, $\phi_{i}(z)$, are expressed as a linear combination of $n$ complex support functions. For the sake of simplicity, let the support functions be selected as complex monomials. The main reason for that is the simple closed-form expressions of derivatives and integrals for this set of functions. Note that, since any set of $n$ linearly independent functions can be selected as support functions, this provides additional degrees of freedom that are not analyzed in this study. Therefore, using monomial, the switching functions are expressed as,

where $c_{ij}$ are a set of constants. Since the switching property states that $\phi_{i}(z_{j})=\delta_{ij}$, we can derive the following expression,

which provides the coefficients needed for the direct (from $Z$ to $W$) mapping given in Eq. (3).

It is important to outline that in general complex mapping may not be bjiective, which makes impossible to find an inverse transformation for the whole domain. However, due to the switching functions, the inverse mapping always exists for the control points. When the complex mapping is bijective, then a least-squares approximated inverse mapping using a basis of orthogonal polynomials and presented in Section 5, can be used.

We can generalize the previous result by using the Theory of Functional Connections. Particularly, the functional,

where $g(z)$ is a free complex function, can be seen as the most general expression that defines the conformal mapping between the control points in $Z$ and $W$. In fact, Eq. (5) represents the TFC functional generalization of Eq. (3).

Figure 2 shows one effect of the free function in the complex mapping defined by the control points from Eq. (4). The free function modifies the domain while the conformal property is still preserved. The top-left plots of Fig. 2 shows a grid of orthogonal lines in the $Z$ domain, while the top-center plot shows the lines mapped in the corresponding $W$ domain using $g(z)=0$, that is, using Eq. (3), and the control points given in Eq. (4). The free function does not change the mapping of the control points. These points are constraints for the $W$ domain. The top-right plot shows the mapped $W$ domain obtained using the free function $g(z)=\sin(3z/2)+\cos(3z/2)$, that is, applying Eq. (5). This shows the free function “rubber” effect on the $W$ domain. The bottom plots are obtained for 8 control points distributed along a circle and using for the free function, the expression $g(z)=\sin(z^{2})\,e^{-z^{2}}$. This example highlights that the selection of control points is not completely free because they often produce non-bjiective mapping (points in the $W$ domain may be associated to multiple points in the $Z$ domain).

The idea of this methodology is to perform a linear transformation in polar coordinates with respect to a projection point that lies inside the domain. While the projection mapping can be applied to any dimensional domains, to be consistent with the complex conformal mapping, the analysis is here restricted to bivariate domains.

Let $\{r,\theta\}$ be the polar coordinates in the space $Z$, and let $\{d,\phi\}$ the polar coordinates in the space $W$. We know that for any point inside the square boundary of $Z$, the boundary can be defined as $r_{b}(\theta)=f(\theta)$ where $f$ is a continuous but not differentiable function. In the same way, if the boundaries in $W$ can be expressed as $d_{b}(\phi)=g(\phi)$ from a given projection point, then, the following direct transformation can be defined:

whose inverse transformation is:

Nevertheless, in some applications it is useful to define the boundary in $W$ as a given polygon defined by a set of control points. Therefore, we include in this section the algorithm to generate a transformation mapping between a set of $n$ control points. In this case, and since we are using the control points as the reference to perform the transformation, the angles in both mappings are not equal as in the previous equations. Particularly, a linear interpolation in the angle between the control points and the projection point is performed in order to maintain the continuity of the transformation.

We define $a$ and $b$ to the coordinates in $Z$ of a given point inside the original domain that is required to be transformed, and $x$ and $y$ to the coordinates in $W$ of the transformed point. Moreover, ${\bm{c}}=\{c_{a},c_{b}\}$, ${\bm{\upsilon}}=\{\upsilon_{x},\upsilon_{y}\}$ are the coordinates of the projections points in $Z$ and $W$ respectively, while $\{p_{a}(j),p_{b}(j)\}$, and $\{q_{x}(j),q_{y}(j)\}$, with $j\in\{1,\dots,n+1\}$, are the coordinates of the $n$ control points defined in $Z$ and $W$ respectively plus the first one repeated in order to close the polygon. Note that the control points have to be consequtive, that is, it must be possible to generate the polygon by linking the control points in the order provided. Then the transformation of $\{a,b\}$ into $\{x,y\}$ can be performed using Algorithm 3.1. Finally, the inverse transformation is equivalent to the algorithm presented.

As an example of this method, Fig.3 (right) presents the result of applying the transformation (direct and inverse) to the control points provided by Table 1 and the projecting points $c=\{0,0\}$ and $\upsilon=\{0,0\}$. As it can be seen, the polynomial in $W$ resembles a “T” shape, where each consecutive pair of control points generates a region where the transformation is continuous and differentiable. However, at the boundaries between regions the transformation stops to be differentiable, although it mantains the property of continuity.

There are cases in which it is not possible to generate a function $d_{b}(\phi)=g(\phi)$ as in the previous case due to the shape of the domain in $W$. In these situations, it is possible to solve these problems using a set of projection points instead of just one, however, some considerations regarding the continuity of the transformation must to be considered first. In that regard, and as a first approach to deal with this situation, we could think on finding the smallest number of projection points required to cover the whole domain. This represents in effect the so called “art gallery problem”.

The “art gallery problem” (known also as the museum problem) is a problem in computational geometry providing the solution for the minimum number of required points to cover the complete domain. Its name comes from the real-world problem of placing the minimum number of guards in an art gallery such that, all together, can observe the whole gallery. This problem was first solved by Chvátal in 1975 [20], but Fisk, in 1978, provided such a brilliant and short proof [21] that Paul Erdős decided to include it in “The BOOK,” an imaginary book in which God keeps the most elegant proof of each mathematical theorem. This book was then published after Erdős dead [22]. The solution to this problem states that $\left\lfloor n/3\right\rfloor$ guards are always sufficient and (sometimes) necessary to guard a simple polygon with $n$ vertices. In general these points are in the boundary of the polygon and thus, although the theorem is interesting, it is not really practical for most applications since it distributes points more densily in the boundaries of the domain in $W$ rather than obtaining a more uniform distribution of points.

Therefore, instead of focusing on identifying the smallest number of projection points required, in this work we deal with the problem on how to separate both domains (in $Z$ and $W$) in a set of regions, one for each projecting point, in such a way that the property of continuity is preserved through the transformation. Once these regions are defined, we proceed as in the case of single-point projection mapping. Note that by doing that, each time that a point is transformed, the method requires to determine the region, from the ones defined, to which it belongs. This can be done, for instance, by using the transformation for each of these regions chequing if the point is inside the region in the $Z$ space.

Let $\delta\Omega$ be the boundary in $W$ of a given domain. Then, connecting with a line two points in $\delta\Omega$ implies connecting two points in the boundary of $Z$ since during the transformation the boundary in one space transforms into the boundary on the other space. For instance, from Figure 4, if we select the points A and B from $W$, we know that both points belong to the boundary in $Z$ and also, both points can be connected through a curve in $Z$ and $W$. Then, if we use just straight lines as curves, this means that A and B in $Z$ must be in a different side of the square boundary. The same can be said with the points $C$ and $D$. Following this procedure, it is possible to generate 4 regions in which each projection point can cover completely its own region. At this point, it is important to note that the boundary between regions 1 and 4 is the curve between A and B in both spaces. In the same way, regions 3 and 4 have a boundary in the curve between A and C, while in regions 2 and 4 the boundary in common is defined by the curve between B and D. This means that since the transformation is continuous inside the regions and at the boundaries of these regions, then, the whole transformation is continuous in the whole domain. This process can be continued even using the boundaries between these defined regions. This means that, for instance, if region 4 is split in half vertically, a point in the curve between A and B will be part of the new boundary created between regions 4’ and 4”, while another point in the boundary should be in $\delta\Omega$ between C and D.

In order to present an example of application for the case of multiple projection points, we apply this methodology to the “C” and “I” beam transformations shown in Fig. 3 . To that end, we know that for these cases we can cover all the domain with two projection points. Moreover, and since both figures are symmetric with respect $b=0$ and $y=0$ repectively, we use this line as the boundary betwwen the two regions in which to apply Algorithm 3.1. In particular, for the case of the “C” beam shape, we define the control points for the two defined regions in Table 2, where the projecting points used are ${\bm{c}_{1}}=\{0,0.5\}$, ${\bm{c}_{1}}=\{0,-0.5\}$, ${\bm{\upsilon}_{1}}=\{0,3i\}$, and ${\bm{\upsilon}_{2}}=\{0,-3i\}$. Figure 3 shows the result of the transformation.

In some applications it is of interest to maintain the density of points through the transformation between $W$ and $Z$. This can be achieved easily using the projection mapping transformation proposed by defining appropriately the control points in the square domain in $Z$.

Let $n$ be the number of uniformly distributed points in $W$ that are inside the problem domain of area $A$. Let $n_{p}$ be the number of points in a given region in $W$ of area $A_{w}$ that is defined by two consecutive control points and the projection point. Then, since the points are uniformly distributed: $n_{p}/n=A_{w}/A$. This argument can be also applied to the square domain in $Z$ leading to: $n_{p}/n=A_{z}/A_{s}$ where $A_{z}$ is the area defined by the transformed control points and the projection point in $Z$, and $A_{s}$ is the area of the square. This means that, since $n_{p}/n$ is a fixed value in the transformation, we have to impose that $A_{z}=A_{s}\,A_{w}/A$ in order to maintain the density of points. In other words, the proportion of the area or each region with respect with the area of the domain must be equal in both domains. Therefore, it is poisible to select the two control points in the boundary of $Z$ in such a way that this area is maintained. For instance, if the area of a given region in $W$ is 10% of the domain area, and one of the control points in $Z$ was already in the position $\{1,0\}$, this means that the second control point for that region should be $\{1,0.8\}$. It is important to note that in general, these new control points will not coincide with the corners of the square, and thus, additional control points in the square (and their transformed equivalents in $W$) must be generated in order to define completely the domain in $Z$.

The strength of the projection mapping consists of providing the direct and inverse transformations with the same formalism and with no approximation. However, for generic polygon domains this leads the TFC problem to be subject to boundary conditions made of a sequence of piecewise functions that in general can even be discontinuous. In this subsection using the Heaviside step function,

we show how to replace, with no approximation, a set of $n$ contiguous piecewise functions by a single function that can be used in the TFC framework. In particular, the possibility of discontinuities between subsequent piecewise functions (no $C^{\,0}$ continuity) is also included.

Consider, for example, the boundary constraints described by the following set of four piecewise functions, each defined in contiguous ranges,

This piecewise function is shown in the top plot of Fig. 5; while the single function, $f(x)$, which is derived by the general equation,

applied for $n=4$ piecewise functions, is shown in the bottom plot of Fig. 5. In particular, this example includes, $C^{\,0}$ continuity (black marker) at $x_{2}=-1/2$, discontinuity (blue dashed line) at $x_{3}=0$, and $C^{\,0}$ and $C^{\,1}$ continuity (red marker) at $x_{2}=1/2$.

The function, $f(x)$, computed using Eq. (8) can be used, for instance, to describe a boundary constraint at $y=-1$ in the 2-dimensional TFC matrix formulation [2, 3],

where the matrix tensor is simply,

and the switching vectors are,

Therefore, the TFC functional is,

The single function, $f(x)$, can be used in the TFC framework as a boundary constraint. Figure 6 shows two TFC surfaces using the single constraint boundary function, $f(x)$, shown in re. This has been done for two distinct expressions of the free functions: $g(x,y)=0$ (left figure, most simple interpolating surface) and $g(x)=\sin(5\,x)\,e^{x\,\cos(5\,x)}$ (right figure).

The polynomial mapping using $n=4$ points on bi-variate domains (quadrangular polygons) is nonlinear. However, an inverse mapping is still possible to derive, as shown in subsection 4.2. This means that this non-linear mapping is bijective.

Again, the mapping is performed between the unit-square domain, $(a,b)\in(-1,+1)\times(-1,+1)$, shown in top-left of Fig. 8, and the quadrangular polygonal domain, defined by the four points, $x_{k}=[0,\,5,\,6,\,-2]$ and $y_{k}=[0,\,-2,\,8,\,4]$, shown in top-right of Fig. 8. In particular, the mapping has been performed using a grid of $400$ points.

The direct mapping, $(a,b)\to(x,y)$, is,

where, the quadratic switching polynomial functions functions, $f_{k},(a,b)$, have the following expressions,

Figure 8 shows the direct mapping to the quadrangular polygonal domain defined by the four points (center figure), $x_{k}=[0,\,1,\,0,\,-1]$ and $y_{k}=[-1,\,0,\,1,\,0]$, and to a domain obtained by rotating clockwise by $30$ deg. the $Z$ control points. This has been done using a grid of $400$ points.

Since this mapping transform lines to lines, the generic polygon side can be obtained simply by,

The inverse mapping, $(x,y)\to(a,b)$, can be obtained using the four boundary lines,

to obtain the two lines passing through the point $(x,y)$,

or in a scalar form,

which can be written as,

Setting,

allows to write the solution in terms of a quadratic equation,

and finally,

Constrained expressions for cubic quadrangular domains can be obtained by mapping the domain $(a,b)\in(-1,1)\times(-1,1)$ unit-square domain to a cubic quadrangular domain $(x,y)$, where each quadrangular side is identified by three points. This non-linear mapping is given by,

where, the cubic polynomials functions $f_{k},(a,b)$ are the switching ($f_{i}(a_{j},b_{j})=\delta_{ij}$) functions,