Surface Aligned Ray Networks and a General Method for Their Construction

B. F. Krutoi

(Presented by the scientific seminar of the departments of mine surveying and geodesy)

The method of angular (ray) intersections is now applied only to the construction of networks of a very special kind—from triangular cells (triangulation). The purpose of this article is to show that by this method networks of a considerably more general kind can be constructed; we shall call them ray networks. In geodesy such a problem is posed and solved here for the first time [1].

1. General properties of ray networks

The principal feature of a ray network is the following. If, in such a network, one specifies a very limited number of fixed vertices, sides, and rays, then the position of the remaining vertices to be determined is found in the ray network by means of a series of direct ray intersections, performed from neighboring fixed and previously constructed determined vertices of the network. It follows that the main part of the initial data of a ray network consists of ray bundles—ray fans—whose sources are placed at most or all of the vertices, while their rays are directed toward neighboring vertices of the network.

In what follows we shall consider only surface aligned ray networks. By this we shall mean ray networks constructed on some smooth surface—for example, on a plane or a spheroid—by means of surface ray fans with aligned rays. By aligned we shall mean such surface rays as possess the property that they are the shortest between any two neighboring points of the surface lying on these rays. It follows from this that the sides (edges) of the indicated networks will be segments of aligned surface threads¹ between the corresponding vertices.

From the very method of constructing ray networks by successive direct intersections it follows that if, in a surface aligned ray network, there are \(k\) vertices to be determined, then the corresponding totality of ray fans must ensure the formation from it of exactly \(2k\) independent angles.

The requirement just stated is only a necessary condition for the constructibility of aligned ray networks on a surface. A sufficient condition for their constructibility is established below (Section 5, item 7).

¹ A thread is a one-dimensional continuous set of points (a line).

It is clear that the necessary condition for constructibility of surface-aligned ray networks in no way restricts the form of their cells. Therefore the cells of a surface-aligned ray network may be of any kind: triangular, quadrilateral, pentagonal, etc., provided only that the indicated necessary condition of constructibility is observed. However, it will be more desirable to construct surface-aligned ray networks from mixed, triangular and quadrilateral cells, since in this case the checking of observations is carried out more uniformly.

2. Some auxiliary concepts

Let us consider some auxiliary concepts needed for what follows. First of all, we shall refine and develop the concept of a surface ray.

We shall call a surface ray \(L_{ch}\) a thread of the given surface issuing from some point \(C\) of this surface, passing through another point \(H\) of the same surface, and continuing without bound further on the surface. We shall call the point \(C\) the vertex (station point), and the point \(H\) the sighting point of the given surface ray \(L_{ch}\). In accordance with our aims, we shall confine ourselves to considering only aligned surface rays formed by aligned threads of the surface.

We shall call a surface ray \(L_{ch}\) free and denote it, as before, by \(L_{ch}\), if it is known only that the points \(C, H\) lie on the given surface, but the position of these points on the surface is not specified.

We shall call a free surface ray \(L_{ch}\) directed and denote it by \(\uparrow L_{ch}\), if the position of the points \(C, H\) on the surface is not specified, but the direction \(\uparrow_{ch}\) of this ray is given. Here by the direction \(\uparrow_{ch}\) of the ray \(\uparrow L_{ch}\) we mean the position of this ray relative to a certain given axial ray \(oL_c\) on the surface and passing through its vertex \(C\).

We shall call a free surface ray \(L_{ch}\) movably pinned and denote it by \(\female L_{ch}\), if only its sighting point \(H\) is specified; if only the vertex \(C\) is specified, the free ray \(L_{ch}\) shall be called fixedly pinned and denoted by \(\male L_{ch}\).

Finally, surface pinned rays \(\female L_{ch}\) and \(\male L_{ch}\), for which their directions \(\uparrow_{ch}\) and \(\downarrow_{ch}\) are specified, shall be called respectively a movably fixed ray \(\female L_{ch}\) and a fixedly fixed ray \(\male L_{ch}\). Note that the ray \(\female L_{ch}\), in contrast to the ray \(\male L_{ch}\), is not completely determined on the surface, since the position of its vertex \(C\) is unknown.

Let us now refine the composition of the initial data in a surface ray network. We shall call a composition of initial data necessary if, with this composition, the position of all determined vertices of the network is obtained uniquely, independently of the magnitude of possible errors in these initial data. We shall call a composition of initial data extended if from this composition there can be selected a number of necessary sets of initial data. The construction of a surface ray network from initial data of extended composition is always ambiguous because of their errors. Hence arises the problem of mutual adjustment, equalization, of the initial data in a ray network when they are of extended composition.

In conclusion, we shall call the underlying surface that surface on which the given surface ray network is situated.

3. Principal Types of Surface Aligned Ray Networks

We shall now consider the principal types of surface aligned ray networks, whose subdivision can be carried out in several ways—depending on the choice of the generalizing criteria.

The first method of subdivision. Among a number of generalizing criteria, one may first of all choose the most essential one: the presence or absence in the surface aligned ray network of directed rays \( \uparrow L_{cn} \). Depending on this basic criterion, we shall distinguish the following two subsets of the indicated networks:

Fig. 1. Directly connected directed ray networks

a) directed (Fig. 1), b) undirected (Figs. 2, 3). From what follows it will be seen that the construction of directed ray networks is carried out more simply than that of undirected networks.

The second method of subdivision. Thus, the presence or absence of orientation in surface-aligned ray networks is their first and most important generalizing feature, according to which they may be divided into the two subsets indicated above. But it is easy to see that these networks can be subdivided still more naturally according to another generalizing feature: depending on the method of connection in them of the part being determined with the fixed vertices.

Fig. 2. Directly connected undirected ray networks

According to this, less essential but nevertheless characteristic feature, we shall distinguish the following three subdivisions of surface-aligned ray networks: a) directly connected, b) inversely connected, c) mixed-connected.

Below, a definition is given for each of these subdivisions, and examples are presented of the corresponding networks belonging to both subsets considered above.

We shall call directly connected those surface ray networks in which the rays \(L_c\) necessary for their construction are specified not only at the vertices being determined, but also at all fixed vertices (Figs. 1, 2). To this subdivision belongs the majority of the previously proposed ...

of these kinds of surface ray networks—with triangular cells. In Fig. 1 examples are given of directed, and in Fig. 2 of undirected ray networks of directly connected subdivision.

We shall call inversely connected those surface ray networks in which the corresponding set of rays \(\mathcal{L}_c\) is specified only at certain vertices (Fig. 3). At the present time

Fig. 3. Inversely connected undirected ray networks

only two very special cases of networks of this subdivision have been found: a single inverse resection (Pothenot’s problem) and a double inverse resection (Hansen’s problem). In Fig. 3 examples are given only of undirected inversely connected ray networks, since directed inversely connected networks differ only in the assignment of the direction \(\uparrow_{cn}\) for one of their free rays \(\mathcal{L}_{cn}\).

Finally, as mixed-connected we shall classify those intermediate surface ray networks in which the ray pencils \(\mathcal{L}_c\) are specified on the determined part and on some of the fixed vertices (not shown in the figure). Of the varieties of surface ray networks proposed so far, this category includes a single mixed resection and rare cases of networks of triangles with mixed attachment of the determined part to fixed vertices.

The third method of subdivision. In addition to the subdivision of surface aligned ray networks described above into subsets and subdivisions, one more method is possible for dividing these networks into basic types—depending on the composition of the initial data according to which they are constructed. According to this last generalizing feature, we shall distinguish the following two unions of surface aligned ray networks: (a) free, (b) nonfree.

If the initial data of a surface ray network include, in addition to ray pencils, only such fixed rays, sides, and vertices without the specification of which the construction of all determined vertices of the network is impossible, then we shall call such a ray network free.

If, however, the initial data of a surface ray network include, in addition to ray pencils and the necessary fixed rays, sides, and vertices, also redundant fixed rays, sides, and vertices, then we shall call such a ray network nonfree.

It is not difficult to see that all surface ray networks depicted in Figures 1–3 are nonfree.

The division of surface ray networks into free and nonfree is necessary for judging the degree of accumulation in them of observational errors, and also when adjusting ray networks according to extended (Section 2) initial data.

4. Further subdivision of directed and undirected surface aligned ray networks into the corresponding genera

The subdivision of surface aligned ray networks into two subsets—directed and undirected—is, as noted above, fundamental, since the presence or absence of directionality determines, to a certain degree, the greater or lesser complexity of constructing these networks.

However, further study of the question reveals a number of additional features in the construction of various ray networks belonging to one and the same directed or undirected subset. This gives grounds for subdividing each of the two indicated subsets of surface aligned ray networks into three intermediate genera. The immediate basis for such an additional subdivision of the two basic subsets into genera is the following noteworthy feature of surface aligned ray networks.

Although surface aligned ray networks are completely determined by the corresponding sets of necessary (Section 2) initial data, in most cases these networks cannot be constructed at once by direct resections from the indicated initial data. To carry out such a construction, it is required first to find certain additional data, computed from the same necessary initial data. Such additional data may be, for example, the direction \(\uparrow_{cn}\) or \(\downarrow_{cn}\) of one of the pinned rays \(\male \mathcal{L}_{ck}\) or \(\female \mathcal{L}_{cn}\), passing through the vertices \(C, H\) of the network; half-

...the position of one of the determined, $C$ or $H$, vertices of the network on the fixed rays $\hat{\phi}L_{cn}$, $\hat{\varphi}L_{cn}$, or on the underlying surface, etc.

Depending on the indicated characteristic, i.e., depending on the number $s$ and the composition of the additional data, we shall distinguish the following genera of surface-aligned ray networks in each of the two principal subsets indicated above.

SUBSET I. DIRECTED NETWORKS (Fig. 1).

Genus 0 (Fig. 1a): $s=0$. No additional data are needed.

Genus 1 (Fig. 1b): $s=1$. The additional datum is the position of one of the determined vertices $H$ of the network on the fixed ray $\hat{\phi}L_{cn}$; for example, in Fig. 1b, the position of the determined vertex 3 on the fixed ray $\hat{\phi}L_{\overline{18}.3}$.

Genus 2 (Fig. 1c): $s=2$. The additional datum is the two-dimensional position on the underlying surface of that one of the determined vertices of the network from which the construction of the remaining determined vertices of the network can be begun by straight intersections. For example, in Fig. 1c, the two-dimensional position of the determined vertex 3 on the underlying surface.

SUBSET II. UNDIRECTED NETWORKS (Figs. 2, 3).

Genus 1 (Figs. 2a, 3a): $s=1$. The additional datum is the directedness $\downarrow_{cn}$ or $\uparrow_{cn}$ of one of the pinned $\hat{\varphi}L_{cn}$ or $\hat{\phi}L_{cn}$ rays of the network. For example, in Fig. 3a, the reverse directedness $\downarrow_{\overline{17}.2}$ of the movably pinned ray $\hat{\varphi}L_{1.17}$; in Fig. 2a, the direct directedness $\uparrow_{51.2}$ of the immovably pinned ray $\hat{\phi}L_{51.2}$.

Genus 2 (Figs. 2b, 3b): $s=2$. The additional data are the directedness of one of the pinned rays of the network and the position of the initial determined vertex on this ray. For example, in Fig. 3b, the directedness $\downarrow_{1.\overline{16}}$ of the pinned ray $\hat{\varphi}L_{1.\overline{16}}$ and the position of the determined vertex 1 on this ray.

Genus 3 (Figs. 2c, 3c): $s=3$. The additional data are the directedness of one of the pinned rays of the network and the two-dimensional position on the underlying surface of that one of the determined vertices of the network from which the construction of the remaining determined vertices of the network can be begun. For example, in Fig. 3c, the directedness $\downarrow_{2.25}$ of the pinned ray $\hat{\varphi}L_{2.25}$ and the two-dimensional position of the determined vertex 1 on the underlying surface.

It follows from what has been said, therefore, that in networks of subset II, in comparison with networks of subset I, it is necessary in all three genera to specify one extra additional datum—the directedness $\uparrow_{cn}$ or $\downarrow_{cn}$ of one of the pinned $\hat{\phi}L_{cn}$ or $\hat{\varphi}L_{cn}$ rays of the network.

5. General method for finding additional data in surface ray networks and the condition for a definite solution of this problem

In Section 1 it was noted that the construction of ray networks, including surface networks, is carried out by successive straight intersections. However, already in Section 4 it was indicated that by this method a surface ray network can be constructed at once only when $s=0$. If $s>0$ (we have established that $0 \leq s \leq 3$), then in this case it is necessary, using the initial data of the ray network, first to find its $s$ additional data and only then to construct this network by successive straight intersections with the aid of the entire set of initial and additional data.

The second problem is very simple in essence and breaks down into a number of uniform operations. Therefore we shall deal with the solution only of the first, considerably more difficult problem. At the same time we shall try to establish those conditions under which the indicated first problem will have a definite solution, provided that the initial data are of the necessary composition.

Thus we have arrived at the formulation of the following important problem.

For a given set of necessary initial data, determine \(s\) additional data for an arbitrary surface-aligned ray network of an \(s\)-gon, and also clarify the condition for a definite solution of this problem.

At the cost of long and persistent searches I succeeded in finding a solution of the indicated problem, very simple in conception and at the same time quite general, for \(s>0\) (for \(s=0\) such a problem does not arise). This general solution is obtained by a sequence of arguments and the conclusions following from them, on which we shall now dwell.

1. Suppose that, in a surface ray network of genus \(s\), we have established in some way the values \(\nu_1, \nu_2,\ldots,\nu_s\) of the additional data, consistent with the adopted set of necessary initial data. Then, using the indicated necessary and additional data, we can construct, by successive straight-line intersections, the corresponding positions of all the determined vertices of the network. Since in this construction, in addition to the necessary initial data, \(s\) additional data were also used, while the total number of data needed to construct the network in question must remain unchanged, we shall find that \(s\) of certain necessary data turn out to be, as it were, superfluous, not needed for the indicated construction. Externally this circumstance will manifest itself in the apparent superfluity of \(s\) of certain necessary rays \(\mathcal{L}_{ch}\) of the network. Such rays, seemingly superfluous but in fact absolutely necessary for the construction of the network, we shall call residual and denote by \(\hat{\mathcal{L}}_{ch}\). These residual rays \(\hat{\mathcal{L}}_{ch}\), for consistent values of the necessary and additional data, will pass perfectly exactly through the corresponding terminal vertices \(\hat{H}\).

The question of which rays in the network will turn out to be residual \(\hat{\mathcal{L}}_{ch}\) always depends on the order of construction of this network adopted by us. Therefore one can always ensure that only rays of the network specified in advance, internal or adjacent to solid vertices, will be residual. However, the determination of the additional data is usually performed somewhat more simply if boundary rays are taken as residual. An example of such a choice of residual rays is presented in Fig. 1в. If, in the ray network of genus 2 shown in this figure, the data \(\nu_1, \nu_2\) are taken as additional data, together with the coordinates \(x_3, y_3\) of vertex 3 in some surface geodetic control, then in this case it is desirable to obtain as residual the rays

\[ \hat{\mathcal{L}}_{18.22} \quad \text{and} \quad \hat{\mathcal{L}}_{17.21}. \]

1. Suppose now that, instead of the exact additional data \(\nu_1, \nu_2,\ldots,\nu_s\), we have taken in a ray network of genus \(s\) some approximate, erroneous values \(\nu_1^{(0)}, \nu_2^{(0)},\ldots,\nu_s^{(0)}\) of these quantities, poorly consistent with the adopted necessary data. Error—

…the use of additional data \(v_1^{(0)}, v_2^{(0)}, \ldots, v_s^{(0)}\) will lead to the fact that the approximate residual rays \(\delta \widetilde L_{c k}^{(0)}\), constructed from \(v_1^{(0)}, v_2^{(0)}, \ldots, v_s^{(0)}\) and from the adopted set of necessary initial data, will in general not pass through the corresponding prescribed positions \(\widetilde H\) of their terminal vertices \(H\). It is easy to see that, depending on the kind of residual rays, such prescribed terminal vertices \(\widetilde H\) for the rays \(\delta \widetilde L_{c k}^{(0)}\) will be:

a) solid vertices \(\widetilde H \equiv T\) of the net, if the residual ray is a nodal ray \(\delta \widetilde L_{PT}\), going from a nodally determined vertex \(P\) to the solid vertex \(T\) (in Fig. 4—the solid vertices 22 and 21 for the rays \(\delta \widetilde L_{18,22}^{(0)}\) and \(\delta \widetilde L_{17,21}^{(0)}\));

Fig. 4. Construction of approximate residual rays \(\delta \widetilde L_{c k}^{(0)}\) in a directed ray net of the 2nd kind

b) previously constructed approximate positions \(\widetilde H^{(0)}\) of the determined vertices \(H\) of the net, if the residual ray is an internal ray \(\delta \widetilde L_{c k}\), going from a determined vertex \(C\) of the net to the same determined vertex \(H\) adjacent to it (in Fig. 5—the approximately constructed determined vertex \(\widetilde 1^{(0)}\) for the ray \(\delta \widetilde L_{19,1}^{(0)}\)).

1. The deviation of the approximate residual ray \(\delta \widetilde L_{c k}^{(0)}\) relative to its prescribed terminal vertex \(\widetilde H\) will be determined by the segment directed transverse to \(\delta \widetilde L_{c k}^{(0)}\) (the arrow) \(\uparrow \lambda_{c k}^{(0)}\) of the surface-aligned string from the vertex \(\widetilde H\) to the ray \(\delta \widetilde L_{c k}^{(0)}\) (Fig. 6, and also Figs. 4 and 5). We shall call the indicated directed segment \(\uparrow \lambda_{c k}^{(0)}\) the transverse displacement of the ray \(\delta \widetilde L_{c k}^{(0)}\) relative to the corresponding terminal vertex \(\widetilde H\).

The lengths \(\lambda_{c k}^{(0)}\) of the transverse displacements \(\uparrow \lambda_{c k}^{(0)}\) will also be regarded as directed quantities, and for this purpose we shall agree (Fig. 6) that…

\[ \tag{1} \begin{cases} \lambda_{sn}^{(0)} > 0, & \text{if the ray } \delta \widetilde{L}_{sn}^{(0)} \text{ is to the right of the ray } C^{(0)}\widetilde{H},\\[4pt] \lambda_{sn}^{(0)} < 0, & \text{if the ray } \delta \widetilde{L}_{sn}^{(0)} \text{ is to the left of the ray } C^{(0)}\widetilde{H}. \end{cases} \]

These directed lengths \(\lambda_{sn}^{(0)}\) of the arrows \(\uparrow \lambda_{sn}^{(0)}\), in the graphical

Fig. 5. Construction of approximate residual rays \(\uparrow L_{sn}^{(0)}\), adjoining and internal, in a directed ray network of the \(l\)-th kind

construction of a surface ray network, are found, with their sign according to (1), directly from the drawing of the network, composed from the approx-

Fig. 6. Determination of the directed lengths \(\lambda_{sn}^{(0)}\) at the arrows \(\uparrow \lambda_{sn}^{(0)}\)

imate \(\varphi_1^{(0)}, \varphi_2^{(0)}, \ldots, \varphi_s^{(0)}\), the additional data and the necessary accepted initial data. In the case of a numerical method, or of a mixed, graphical-numerical, method of constructing a ray network,

the quantities \(\lambda_{cn}^{(0)}\), with the sign according to (1), are found, in view of their smallness, with sufficient accuracy as follows (Fig. 7):

\[ \lambda_{cn}^{(0)}=D_{cn}^{(0)}\sin l_{cn}^{(0)} \simeq D_{cn}^{(0)}\,\frac{l_{cn}^{(0)\prime\prime}}{\rho^{\prime\prime}}, \tag{2} \]

where \(\rho=206265\), \(D_{cn}^{(0)}=C^{(0)}\widetilde H^{(0)}\), \(l_{cn}^{(0)}=\uparrow_{cn}^{(0)}-\uparrow'_{cn}\), and \(\uparrow_{cn}^{(0)}\) and \(\uparrow'_{cn}\) are, respectively, the directions of the ray \(\sigma\widetilde L_{cn}^{(0)}\) and of the reference segment \(C^{(0)}\widetilde H^{(0)}\).

Fig. 7. Computation of directed lengths \(\lambda_{cn}^{(0)}\) and \(\uparrow\lambda_{cn}^{(0)}\)

4. From what has been said above it follows that the transverse displacements \(\uparrow\lambda_{cn}^{(0)}\) of the residual rays \(\sigma\widetilde L_{cn}^{(0)}\), relative to their terminal vertices \(\widetilde H\), are, in a surface ray network of the \(s\)-th kind, functions of the corresponding approximate additional data \(\nu_1^{(0)}, \nu_2^{(0)}, \ldots, \nu_j^{(0)}, \ldots, \nu_s^{(0)}\) and of that part \(\sigma'\) of the adopted necessary data which participates in determining the quantities \(\nu_j\).² Therefore, if we denote by \(\lambda_i^{(0)}\) the directed length of the transverse displacement \(\uparrow\lambda_{cn_i}^{(0)}=\uparrow\lambda_i^{(0)}\) for some \(i\)-th residual ray \(\sigma\widetilde L_{(cn)_i}^{(0)}=\sigma\widetilde L_i^{(0)}\), then we may write that

\[ \lambda_i^{(0)} = F_i\left(\nu_1^{(0)},\nu_2^{(0)},\ldots,\nu_j^{(0)},\ldots,\nu_s^{(0)};\sigma'\right) \qquad (i=1,2,\ldots,s). \tag{3} \]

At the same time it is obvious that, for exact additional data \(\nu_1,\nu_2,\ldots,\nu_s\),

\[ \lambda_i = F_i\left(\nu_1,\nu_2,\ldots,\nu_j,\ldots,\nu_s;\sigma'\right)=0 \qquad (i=1,2,\ldots,s). \tag{4} \]

Let

\[ \nu_j=\nu_j^{(0)}+\varepsilon_j \qquad (j=1,2,\ldots,s). \tag{5} \]

Assuming the corrections \(\varepsilon_j\) to be sufficiently small, we expand (4) in a Taylor series. Then, instead of (4), we shall have the more expanded

Definition

of the additional datum \(\xi_1\)

\[ \lambda^{(0)}_{19,\overline{21}}=-41;\qquad \lambda^{(1)}_{19,\overline{21}}=+25 \]

\[ l^{0,1}=+16 \]

\[ a_{11}= \frac{\lambda^{(1)}_{19,\overline{21}}-\lambda^{(0)}_{19,\overline{21}}}{l^{0,1}} =+4.0 \]

Equation (8):

\[ a_{11}\xi_1+\lambda^{(0)}_{19,\overline{20}}=0 \]

\[ 4.0\xi_1+41=0 \]

\[ \xi_1=l^{(0)}=+16 \]

Measure

\[ 0\qquad 10\qquad 20 \]

Fig. 8. Construction of a directed ray network of the first kind.

2) Determination of \(\xi_1,\xi_2\)

by the identity mapping

a) Ob—

b) Mapping

\[ \xi_1=\frac{3^{(0)}3^{(1)}}{u^{(0)}u^{(1)}}=+4.5 \]

\[ \xi_2=\frac{3^{(0)}3^{(2)}}{u^{(0)}u^{(2)}}=+7.5 \]

Determination of additional data

\[ \xi_1,\xi_2 \text{ from the summary (8)} \]

\[ \lambda_{14,20}^{(0)}=+7.5 \qquad \lambda_{17,21}^{(0)}=+2.0 \]

\[ \lambda_{14,20}^{(1)}=-6.0 \qquad \lambda_{17,21}^{(1)}=+7.0 \]

\[ \lambda_{14,20}^{(2)}=+5.0 \qquad \lambda_{17,21}^{(2)}=-4.5 \]

\[ 3^{(0)}3^{(1)}=+11.0 \qquad 3^{(0)}3^{(2)}=-11.0 \]

\[ \alpha_{1.1} = \frac{\lambda_{14,20}^{(1)}-\lambda_{14,20}^{(0)}}{3^{(0)}3^{(1)}} = -1.2 \]

\[ \alpha_{1.2} = \frac{\lambda_{14,20}^{(2)}-\lambda_{14,20}^{(0)}}{3^{(0)}3^{(2)}} = -0.2 \]

\[ \alpha_{2.1} = \frac{\lambda_{17,21}^{(1)}-\lambda_{17,21}^{(0)}}{3^{(0)}3^{(1)}} = +0.5 \]

\[ \alpha_{2.2} = \frac{\lambda_{17,21}^{(2)}-\lambda_{17,21}^{(0)}}{3^{(0)}3^{(2)}} = -0.6 \]

Summary (8)

\[ \begin{cases} -1.2\,\xi_1-0.2\,\xi_2+7.5=0,\\ +0.5\,\xi_1-0.6\,\xi_2+2.0=0. \end{cases} \]

\[ D=+0.82 \]

\[ \xi_1=+5.0 \qquad \xi_2=+7.6 \]

Scale

0 10 20

Fig. 9. Construction of a directed ray network of the 2nd order

1) Determination

\[ \delta \text{ relative to the given } \xi_i \]

\[ \lambda^{(0)}_{8,19}=-6 \qquad \lambda^{(1)}_{8,19}=+4 \]

\[ \varepsilon_i=l^{(0)}l^{(1)}=+8 \]

\[ a_{11}=\frac{\lambda^{(1)}_{8,19}-\lambda^{(0)}_{8,19}}{\varepsilon_i}=+1.3 \]

Equation (8):

\[ a_{11}\xi_i+\lambda^{(0)}_{8,19}=0 \]

\[ +1.3\,\xi_i-6=0 \]

\[ \xi_i=l^{(0)}=+4.6 \]

2) Rotation of the network \(\mathfrak R\)

\[ \omega=\angle l^{(0)}\overline{l}_1^{(1)} \text{ is given} \]

\[ \vartheta=\angle l^{(0)}\overline{l}_1=\frac{\xi_i}{\varepsilon_i}\,\omega \]

Scale

\[ 0 \qquad 10 \qquad 20 \]

Fig. 10. Construction of an undirected ray network of the first kind.

Determination
of the additional data \(\xi_1, \xi_2\).

\[ \lambda_{8,18}^{(0)}=-17;\qquad \lambda_{8,18}^{(1)}=+.5;\qquad \lambda_{8,18}^{(2)}=-5 \]

\[ \lambda_{7,18}^{(0)}=-6.5;\qquad \lambda_{7,18}^{(1)}=+15;\qquad \lambda_{7,18}^{(2)}=+11.5 \]

\[ l^{(0)(1)}=+8 \qquad l^{(0)(2)}=+11 \]

\[ \alpha_{11}= \frac{\lambda_{8,18}^{(1)}-\lambda_{8,18}^{(0)}}{l^{(0)(1)}}=+1.9 \]

\[ \alpha_{12}= \frac{\lambda_{8,18}^{(2)}-\lambda_{8,18}^{(0)}}{l^{(0)(2)}}=+1.1 \]

\[ \alpha_{21}= \frac{\lambda_{7,18}^{(1)}-\lambda_{7,18}^{(0)}}{l^{(0)(1)}}=+2.8 \]

\[ \alpha_{22}= \frac{\lambda_{7,18}^{(2)}-\lambda_{7,18}^{(1)}}{l^{(1)(2)}}=-0.5 \]

\[ \text{System }(8) \]

\[ \begin{cases} +1.9\xi_1+1.1\xi_2-17=0,\\ +2.8\xi_1-0.5\xi_2-6.5=0. \end{cases} \]

\[ D=-4.0 \]

\[ \xi_1=l^{(0)1}=+3.8 \qquad \xi_2=l^{1\to 2}=+7.4 \]

Angle of rotation \(\vartheta\)

\[ \omega=\angle 1^{(0)}.16.1^{(1)}=\text{given} \]

\[ \vartheta=\angle 1^{(0)}.16.1 = \frac{\xi_1}{l^{(0)(1)}}\,\omega \]

Scale

\[ 0\qquad 10\qquad 20 \]

Fig. 11. Construction of an undirected ray network of the 2nd kind

Determination of the supplementary data \(\xi_1,\xi_2,\xi_3\).

\[ \lambda^{(0)}_{5,24}=-12.5;\quad \lambda^{(1)}_{5,24}=-21;\quad \lambda^{(2)}_{5,24}=-3;\quad \lambda^{(3)}_{5,24}=+11 \]

\[ \lambda^{(0)}_{8,26}=-9.5;\quad \lambda^{(1)}_{8,26}=+3.5;\quad \lambda^{(2)}_{8,26}=-20;\quad \lambda^{(3)}_{8,26}=-1.5 \]

\[ \lambda^{(0)}_{9,27}=-23;\quad \lambda^{(1)}_{9,27}=33;\quad \lambda^{(2)}_{9,27}=-15.5;\quad \lambda^{(3)}_{9,27}=+22 \]

\[ l^{(0)}l^{(1)}=+10;\quad l^{(0)}l^{(2)}=+10;\quad 2^{(0)}2^{(3)}=+14. \]

\[ a_{11}=\frac{\lambda^{(1)}_{5,24}-\lambda^{(0)}_{5,24}}{l^{(0)}l^{(1)}}=-0.85;\qquad a_{12}=\frac{\lambda^{(2)}_{5,24}-\lambda^{(0)}_{5,24}}{l^{(0)}l^{(2)}}=+0.95 \]

\[ a_{13}=\frac{\lambda^{(3)}_{5,24}-\lambda^{(0)}_{5,24}}{2^{(0)}2^{(3)}}=+1.68 \]

\[ a_{21}=\frac{\lambda^{(1)}_{8,26}-\lambda^{(0)}_{8,26}}{l^{(0)}l^{(1)}}=+1.30;\qquad a_{22}=\frac{\lambda^{(2)}_{8,26}-\lambda^{(0)}_{8,26}}{l^{(0)}l^{(2)}}=-1.05 \]

\[ a_{23}=\frac{\lambda^{(3)}_{8,26}-\lambda^{(0)}_{8,26}}{2^{(0)}2^{(3)}}=+0.57 \]

\[ a_{31}=\frac{\lambda^{(1)}_{9,27}-\lambda^{(0)}_{9,27}}{l^{(0)}l^{(1)}}=-1.00;\qquad a_{32}=\frac{\lambda^{(2)}_{9,27}-\lambda^{(0)}_{9,27}}{l^{(0)}l^{(2)}}=\ [[unclear: value]] \]

\[ a_{33}=\frac{\lambda^{(3)}_{9,27}-\lambda^{(0)}_{9,27}}{2^{(0)}2^{(3)}}=+3.14 \]

Summary \(\Theta\):

\[ \begin{aligned} -0.85\xi_1+0.95\xi_2+1.68\xi_3-12.5&=0,\\ +1.30\xi_1-1.05\xi_2+0.57\xi_3-9.5&=0,\\ -1.00\xi_1+0.75\xi_2+3.14\xi_3-23&=0. \end{aligned} \]

\[ \Theta=-1.38 \]

\[ \Theta_{\xi_1}=-9.71,\qquad \Theta_{\xi_2}=-6.17,\qquad \Theta_{\xi_3}=-11.79 \]

\[ \xi_1=\left|\Theta^{(0)}\Theta^{(1)}\right|=+7.1,\qquad \xi_2=\left|\Theta^{(0)}\Theta^{(2)}\right|=+4.5,\qquad \xi_3=\left|\Theta^{(0)}\Theta^{(3)}\right|=+8.3 \]

Angle of rotation \(\vartheta\)

\[ \omega=L\,2^{(0)}\overline{25}\,2^{(3)}-90^\circ20 \]

\[ \vartheta=L\,2^{(0)}\overline{25}\,2^{(0)3} +\frac{\xi_3}{2^{(0)}2^{(3)}}\,\omega \]

Scale

\[ 0\qquad 10\qquad 20 \]

Fig. 12. Construction of an undirected ray network of the 3rd kind

\[ F_i\left(\nu_1^{(0)}, \nu_2^{(0)}, \ldots, \nu_j^{(0)}, \ldots, \nu_s^{(0)}; \sigma'\right) + \sum_{i=1}^{s} \left(\frac{\partial F_i}{\partial \nu_j}\right)_0 \cdot \varepsilon_j =0, \tag{6} \]

\[ (i=1,2,\ldots,s), \]

where the partial derivatives \(\dfrac{\partial F_i}{\partial \nu_j}\) are computed for \((\nu_1^{(0)}, \nu_2^{(0)}, \ldots,\)

\(\nu_j^{(0)}, \ldots, \nu_s^{(0)}; \sigma')\). Taking now (3) into account and introducing the notation

\[ \left(\frac{\partial F_i}{\partial \nu_j}\right)_0 =a_{ij}, \qquad (i,j=1,2,\ldots,s) \tag{7} \]

we reduce (6) to the following final form:

\[ \sum_{j=1}^{s} a_{ij}\xi_j+\lambda_i^{(0)}=0, \qquad (i=1,2,\ldots,s), \tag{8} \]

where we have denoted the required corrections \(\varepsilon_j\) (constant quantities) by \(\xi_j\).

Expression (8) is a system of \(s\) plane¹ equations with \(s\) unknown corrections \(\xi_1,\xi_2,\ldots,\xi_j,\ldots,\xi_s\) to the approximate additional data \(\nu_1^{(0)}, \nu_2^{(0)},\ldots,\nu_j^{(0)},\ldots,\nu_s^{(0)}\). Thus we have established that the determination, for a surface-aligned ray network of the \(s\)-th kind, of additional data \(\nu_1,\nu_2,\ldots,\nu_j,\ldots,\nu_s\) consistent with the accepted initial data of this network reduces to solving the system (8) and to computations according to (5).

1. Note now that the calculation of the \(s^2\) coefficients \(a_{ij}\) of the system (8) by means of relations (7) is unacceptable from a production point of view, since it requires the construction and differentiation of the functions \(F_i\), which have a very complicated and entangled form, especially for spherical ray networks. Moreover, all these computations have no checks, which is inadmissible. For the same reasons, the application of the indicated method of calculating the coefficients \(a_{ij}\) when constructing a ray network graphically is entirely inappropriate, when what is required above all is speed and simplicity of work.

The considerations presented led me to devise another method for finding the coefficients \(a_{ij}\) in the system (8), which consists of a number of uniform, easily checked operations and, in the graphical construction of a surface ray network, is also carried out graphically. This method is set forth below.

1. The determination of the coefficients \(a_{ij}\) in the system (8) by the method proposed by me is based on the consideration that, in general,

\[ a_{ij}=\left(\frac{\partial F_i}{\partial v_j}\right)_0 = \]

\[ =\lim_{\varepsilon_j\to 0} \frac{ F_i(v_1^{(0)},v_2^{(0)},\ldots,v_j^{(0)}+\varepsilon_j,\ldots,v_s^{(0)};\sigma') - F_i(v_1^{(0)},v_2^{(0)},\ldots,v_j^{(0)},\ldots,v_s^{(0)};\sigma') }{\varepsilon_j}, \]

or, more briefly,

\[ a_{ij}=\left(\frac{\partial F_i}{\partial v_j}\right)_0 = \lim_{\varepsilon_j\to 0}\frac{F_i^{(j)}-F_i^{(0)}}{\varepsilon_j} = \lim_{\varepsilon_j\to 0}\frac{\lambda_i^{(j)}-\lambda_i^{(0)}}{\varepsilon_j}, \tag{9} \]

\[ (i,j=1,2,\ldots,s), \]

where, in accordance with (3), we have set

\[ F_i^{(0)}=F_i(v_1^{(0)},v_2^{(0)},\ldots,v_j^{(0)},\ldots,v_s^{(0)};\sigma')=\lambda_i^{(0)} \]

\[ F_i^{(j)}=F_i(v_1^{(0)},v_2^{(0)},\ldots,v_j^{(0)}+\varepsilon_j,\ldots,v_s^{(0)};\sigma')=\lambda_i^{(j)}. \tag{10} \]

Here \(\lambda_i^{(j)}\) is the directed length of the transverse displacement \(\uparrow \lambda_{(cn)i}^{(j)}=\uparrow \lambda_i^{(j)}\) of the \(i\)-th residual ray \(\delta \widetilde{L}_{(cn)i}^{(j)}=\delta'\widetilde{L}_i^{(j)}\) relative to the corresponding terminal vertex \(\widetilde{H}_i^{(j)}\) for the case when the construction of the surface ray network of the \(s\)-th kind is carried out from the initial data \((v_1^{(0)},v_2^{(0)},\ldots,v_j^{(0)}+\varepsilon_j,\ldots,v_s^{(0)};\sigma')\).

But the arrows \(\uparrow\lambda_i^{(j)}\), for sufficiently small \(\varepsilon_j\), almost coincide with the initial arrows \(\uparrow\lambda_i^{(0)}\) and change their directed length \(\lambda_i^{(j)}\) almost proportionally to the corrections \(\varepsilon_j\). Hence it follows that, for small \(\varepsilon_j\), all \(s^2\) coefficients \(a_{ij}\) in the summary (8) can be found with a sufficient degree of accuracy from the relation, close to (9),

\[ a_{ij}=\frac{\lambda_i^{(j)}-\lambda_i^{(0)}}{\varepsilon_j} = \frac{\lambda_i^{(j)}-\lambda_i^{(0)}}{v_j^{(1)}-v_j^{(0)}}, \qquad (i,j=1,2,\ldots,s), \tag{11} \]

where it is set that

\[ v_j^{(1)}=v_j^{(0)}+\varepsilon_j. \tag{12} \]

Expression (11) is the desired one. Let us note that, for directed ray networks, in particular for plane ones, this expression will be not approximate, but perfectly exact.

1. To complete our problem of determining the \(s\) additional data \(v_1,v_2,\ldots,v_s\) in a surface aligned ray—

of an \(s\)-th order ray network, we must still establish the condition for the determined solution of this problem. Such a condition is, evidently, the requirement

\[ |a_{ij}| \ne 0, \tag{13} \]

where \(|a_{ij}|\) is the determinant of the summary matrix (8). If, however, \(|a_{ij}|=0\), or if \(|a_{ij}|\) is close to 0, then in this case the additional data of the surface ray network either cannot be determined at all from (8) and (5), or are determined from them very unreliably. Then it is necessary to modify somewhat the internal structure of the intended ray network, or else to take other residual rays.

It is not difficult to see that inequality (13) may also be regarded as the desired sufficient condition for the constructibility of a ray network of order \(s>0\) (Section 1).

6. Examples of finding, by the general method, additional data in surface aligned ray networks

Figures 8, 9 and 10, 11, 12 give examples of determining additional data in surface ray networks by the method proposed by me. These determinations were carried out graphically, which makes it possible to present most vividly and simply the essence of the indicated method and its features for surface ray networks of various kinds. Auxiliary computations and secondary constructions are also included in these figures.

Figures 8, 9 show the finding of additional data in directed surface ray networks of the first and second orders. The search for additional data in such networks is carried out especially simply, since in this case the directions \(\uparrow^{(h)}_{C H}\) of the intersecting rays \(\uparrow L^{h}_{C H}\), for given \(C, H\), remain unchanged for all \(h=0,1,\ldots,s\), i.e., in all \(s+1\) reconstructions of the network. As residual rays \(\sigma \widetilde L_{(C H)_i}\) in these examples only adjacent rays \(\sigma \widetilde L_{(P T)_i}\) are taken. It is not difficult to realize that the reconstructions of the networks carried out in Figs. 8, 9 are, respectively, their one-dimensional and two-dimensional parallel related transformations.

Figures 10, 11, 12 explain the determination of additional data in undirected surface ray networks of orders 1–3. The additional data in these networks are found with greater difficulty, since in one of the network reconstructions a different direction of the original adjacent ray is taken than in the remaining \(s\) reconstructions. In all three examples only adjacent rays \(\sigma \widetilde L_{(P T)_i}\) are taken as residual ones.

It is easy to see that the network reconstructions carried out in the indicated drawings are related transformations of the following kinds: a) a segmented rotation (Fig. 10), b) a segmented rotation and a one-dimensional parallel transformation (Fig. 11), c) a segmented rotation and a two-dimensional parallel transformation (Fig. 12).

7. Comparison of surface aligned ray networks of general form with triangulation

The investigation we have carried out of surface aligned ray networks of general form makes it possible to establish the following observa-

distinctive features of these networks in comparison with their special case—triangulation.

1. The average number of observed rays at the vertices of a triangulation is 6–7, whereas in a ray network of the general type it can easily be reduced to 3–5 (Figs. 1–3). This substantially facilitates and reduces the cost of field work in such ray networks.

2. Free directed ray networks of the general type, constructed on a long fixed side, make it possible to pass very simply from the long side of the main network to arbitrarily small sides of the filling network.

Nonfree directed ray networks of the general type, constructed in the form of a double, triple, etc., chain of cells between fixed sides, make it possible immediately to reduce, by the required number of times, the length of intermediate sides. This ensures an easy transition from the main network to the filling network with sides of the required length.

The aforementioned qualities of directed ray networks of the general type, both free and nonfree, make them a convenient and flexible means for constructing a filling network of any density, allowing the transition from the main network to be carried out with the smallest number of stages of development.

1. Unlike triangulation, surface ray networks of the general type can be constructed in such a way that no conditions arise in them other than angular and additional ones; or else there is only a very limited number of side and vertex conditions (Figs. 1–3). This makes it possible to adjust such simplified-ray networks fairly simply by conditional observations, and still better—by Aner’s conditional indirect observations [2].

2. In view of the smaller number of observed rays in simplified-ray networks as compared with triangulation, their vertices are determined with somewhat lower accuracy than in triangulation. However, comparison for the networks being compared of the corresponding \(n\)-dimensional unit ellipsoids of errors of the form

\[ a_{(nn)} x_{(n)} x_{(n)} = 1, \tag{14} \]

where \(a_{(nn)}\) is the matrix of the indirect normal equations, shows that the difference in the sizes of like axes of the triangulation and simplified-ray ellipsoids (14) does not exceed on the average 10–20%. Therefore, with sufficiently high accuracy of angular observations, there is every reason to replace triangulation by cheaper simplified-ray networks.

REFERENCES

1. Berichte zur X. Generalversammlung der J. U. G. G.—Assoziation für Geodäsie—in September 1954 in Rom 1955, Stuttgart.

2. Aner. Ausgleichung durch Anwendung des arithmetischen Mittels. Z. f. Vermessungswesen, H. 3, 1926, Stuttgart.

1. Plane—instead of the unfortunate word “linear.” ↩

2. In particular, \(\sigma'=\sigma\), where \(\sigma\) is the complete set of necessary initial data of the surface ray network. ↩