Geometry and Geodesy on the Primary Visual Cortex as a Surface of Revolution

Abstract

Biological mapping of the visual field from the eye retina to the primary visual cortex, also known as occipital area $V1$, is central to vision and eye movement phenomena and research. That mapping is critically dependent on the existence of cortical magnification factors. Once unfolded, $V1$ has a convex three-dimensional shape, which can be mathematically modeled as a surface of revolution embedded in three-dimensional Euclidean space. Thus, we solve the problem of differential geometry and geodesy for the mapping of the visual field to $V1$, involving both isotropic and non-isotropic cortical magnification factors of a most general form. We provide illustrations of our technique and results that apply to $V1$ surfaces with curve profiles relevant to vision research in general and to visual phenomena such as ‘crowding’ effects and eye movement guidance in particular. From a mathematical perspective, we also find intriguing and unexpected differential geometry properties of $V1$ surfaces, discovering that geodesic orbits have alternative prograde and retrograde characteristics, depending on the interplay between local curvature and global topology.

1. Introduction

Since ground-breaking work of Hubel and Wiesel [1], and Daniel and Whitteridge [2], consideration of mapping of the visual field from the eye retina to visual cortex has become increasingly relevant in psychophysics studies of vision in general and eye movements in particular. Daniel and Whitteridge showed that retinotopic mapping is critically dependent on the existence of cortical magnification factors [2]. Human or primate primary visual cortex, or occipital area $V1$, is of foremost concern. Daniel and Whitteridge also showed that $V1$ has a convex three-dimensional shape once it is unfolded [2], a discovery confirmed by, and consistent with, other major investigations [3,4,5]. Our paper is premised on this pioneering work. We shall posit that unfolded $V1$ can be modeled as a surface of revolution embedded in three-dimensional Euclidean space, at least with a certain degree of mathematical precision.

In many fields, the study of geodesy has always been and still is vital to the development of science, technology, and engineering. Geodesic curves are mathematically defined as parallel-transporting their own tangent vectors on any smooth manifold. Extremal paths connecting points on smooth manifolds are typically provided by geodesic arcs.

Pursuing general relativity studies [6,7,8,9], we developed geodetic procedures that result in first-order differential equations obeyed by geodesic curves over regular two-dimensional (2D) surfaces of revolution, S, embedded in ordinary three-dimensional (3D) Euclidean space, $E^3$.

With the prospect of applications to vision and eye movement research, we provide in this paper an essential mathematical tool for any such application. That is a complete formulation and solution of the differential geometry and geodesy problem pertaining to conformal diffeomorphisms between the unit sphere, $S^2$, representing the visual field, and surfaces of revolutions, such as $V1$. We consider both isotropic and non-isotropic CMF’s. Isotropic CMF’s are presumed to properly account for most experimental characterizations of $V1$. Non-isotropic CMF’s are required to account for stereographic projections to planar geometry or displays. Spheroidal geometry may also be formulated in terms of non-isotropic CMF’s and thus compared with the CMF-isotropic geometry of $V1$.

In this paper, we consider in particular the model framework provided in Ref. [2]. However, our geodetic technique applies generally to any regular surface of revolution, derived from a plane profile curve rigidly rotated around a z-axis, thus providing azimuthal symmetry. It produces first-order geodesic-orbit differential equations that can be readily solved by quadrature or numerical integration. Remarkably, we rely exclusively on ordinary calculus, rigorously bypassing any more substantive use of Ricci calculus. Most notably, our geodetic technique does not require any determination of Christoffel symbols for any surface of revolution.

No geodetic characterization of $V1$ has been rigorously obtained heretofore. Our technique thus provides an essential tool for mathematical modelling and description of correlations between eccentricity and acuity among items in their $V1$ representation. That is bound to be critical for studies involving visual ‘crowding’ effects [10,11,12]. Geodetic patterns and their analysis on $V1$ may also critically inform studies of eye movement guidance in visual search [13,14,15,16,17,18,19,20,21,22,23,24,25,26].

2. Visual Cortical Surfaces

Physically, light rays from external sources pass through the eye’s lens system into the retina, a sheet of photoreceptor cells lining the curved inner surface of each eye. The visual axis (or line of sight) corresponds to the light ray that passes through the center of the pupil and hits the center of an area within the retina called the fovea. The fovea has the highest concentration of retinal receptors. The spatial density of ganglion cells, which receive visual information from photoreceptors and transmit it along the optic nerve, decreases in all directions from the center of the fovea. Consequently, the highest visual resolution occurs in the fovea and decreases peripherally with increasing visual angle.

The primary visual cortex, or area $V1$ in the back of the brain, consists of a thin sheet of cells that, for our purposes, can be described as a 2D surface with a homogeneous and isotropic density of neurons. With good approximation, the projection of retinal units into $V1$, via the lateral geniculate nucleus, accomplishes a biological remapping that provides each retinal processing unit with an equal area of $V1$ for further processing [27,28]. Consequently, the projection of dense foveal concentration of retinal receptors occupies a much larger proportion of $V1$ than do more peripheral retinal regions.

Daniel and Whitteridge [2], described a cortical magnification factor (CMF) in terms of the number of millimeters ($mm$) along meridians of $V1$ concerned with one angular degree of ‘eccentricity’, w, in the visual field. Mathematically, one can model the visual field as the hemisphere of the 2D unit sphere, $S^2$, having $0\le w\le \pi /2$ as colatitude, relative to the visual axis. Daniel and Whitteridge [2], found that the CMF decreases with increasing eccentricity w on $S^2$, but the CMF hardly varies with the azimuthal angle $\varphi$ of revolution around the visual axis. Subsequent research has confirmed those findings [3,5,10,29,30].

The CMF is an experimentally determined function. Since the total area of $V1$ varies from individual to individual and from species to species, the CMF is individually peculiar, although it is commonly expressed as an average across adults of a given species.

In typical vision experiments, the minimum size that a symbol or a chart letter drawn on a screen must have in order to be recognized by an observer increases with eccentricity w in the visual field, away from the visual axis. The farther out, the bigger that chart letter must be drawn on the screen. The size threshold for perceptual identification is commensurate with the CMF. It means that, if we visualize threshold-sized letters on a screen as mapped on $V1$, each has about the same size on $V1$: see Figure 8. For the reverse situation, i.e., equally sized chart letters on a screen, their size as mapped into $V1$ shrinks with increasing eccentricity, as shown in Figure 1 of Ref. [10].

With regard to measurements of distance between points on $V1$, previous numerical estimates could at best approximate geodetic separations at relatively short range, but hardly so for more distant points, lying on distant meridians of $V1$, for example. This problem provided our initial motivation to develop a mathematically rigorous method to perform geodetic calculations on $V1$ at all distances, which is required to account for magnification of visual interactions throughout the entire mapping of items on $V1$ from the visual field.

It has been previously demonstrated that physiological measures in other visual cortical areas and behavioral performance measures are invariant under relatively simple transformations of the configuration of stimuli as they are mapped in $V1$ space [10,11,12,18]. This does not imply that those perceptual phenomena necessarily result from processing within $V1$. It does mean, however that eccentricity-dependent scaling is basically established by the CMF, as laid out in $V1$ and then inherited by further connections with other cortical areas. It is thus plausible that spatially based perceptual phenomena may be further understood or revealed when studied directly within the $V1$ representation of the visual field, which in turn requires a precise geodetic account of that mapping.

All such considerations essentially apply to a single fixation during which visual space is mapped from the eye retina to the primary visual cortex. Let us now consider the even more complex problem of the evolution of that mapping from one fixation to the next. Visual search generally proceeds through a series of eye movements consisting of saccades and fixations that bring sequentially different areas of the scene under scrutiny by the eye’s fovea. Affecting that pattern is a complex interplay between the scene’s display and boundaries and the observer’s perceptual, attentional, and mnemonic resources, whether deployed automatically or intentionally, overtly or covertly. Perceptual constructs such as the area of conspicuity (AC) or attentional focus, inhibition of return (IOR) or attentional momentum, scanpaths or other search strategies have been quantitatively investigated and further simulated with computational models [13,14,15,16,17,18,19,20,21,22,23,24,25,26].

Such research, however, is typically based on the externally fixed visual scene. The presence of cortical magnification implies that, after each saccade or eye movement, that visual scene is entirely transformed as remapped in the primary visual cortex, with its major accompanying changes in cortical object sizes and distances relative to other items originally present at each fixation. All such complex reconstructions derive from the biological remapping of the visual field to the primary visual cortex. Geodetic remapping of neighboring and distant items directly on a $V1$ model surface may thus provide further insight into complex perceptual constructs such as AC, IOR or scanpath features, complementing investigations of visual search patterns and guidance based exclusively or primarily on the appearance of the external scenery.

The mapping of visual space on the primary visual cortex is suitable for an enlightening application of our general mathematical formulation of geodesy to diffeomorphic surfaces of revolution. This requires, however, a considerable degree of mathematical idealization and the following simplifying assumptions. We model the neural mapping as a conformal diffeomorphism between the unit sphere, $S^2$, and its representation on $V1$. We must assume such diffeomorphism to be axially, i.e., azimuthally symmetric, since only that pertains to surfaces of revolution about that z-axis. We thus set aside for the moment some experimental evidence that further suggests somewhat different CMF’s along different meridians of $V1$, e.g., in the upper versus the lower visual hemifields [31]. We also assume that the left and right visual hemifields are seamlessly joined, ignoring their physical separation in the brain.

Beyond that, we consider an independent hypothesis of local isotropic magnification, positing locally equal magnification factors, latitudinally and longitudinally. That hypothesis is very restrictive, whereas it is not at all mathematically required in our geodetic theory. Thus, we will also provide more general equations and illustrations of diffeomorphisms that involve local anisotropic magnification, latitudinally and longitudinally. Nevertheless, for specific illustrations of geodesy on $V1$, we will focus on classical literature that makes the assumption of local isotropic magnification, using a basic expression of isotropic CMF with parameter values determined experimentally [2,5,10,29].

There have been, of course, many studies of neuroanatomy and modelling of visual cortex and retinotopy. Besides studies to which we specifically refer, other important works include those of Boscain, Bosking, Bressloff and Cowan, Citti and Sarti, Duits and Mashtakov, Hoffman, Petitot, Schwartz, Zucker, as quoted in Refs. [32,33,34,35,36,37,38,39,40], for example. Some of those studies apply analytic and differential geometry techniques comparable or superior to ours.

Let us consider, for example, the spherical extension of Mashtakov and Duits [40], of the Petitot–Citti–Sarti model [37,38,39]. Unlike the earlier planar approximation, the eye retina is approximated as a hemisphere, envisioned as a $S^2$ base manifold. A retinotopic map of $S^2$ to $V1$, involving a CMF, may then be modeled as a fiber bundle. More specifically, the lift of an image by the cortex can be interpreted as a map from $S^2$, i.e., the retina, to $S^3$, representing the cortex as a double cover of the $SO\left(3\right)$ group. Then, the Hopf fibration naturally appears as a fiber bundle, where the base manifold is $S^2$ and a fiber is $S^1$: see Figure 4 in Ref. [40]. This construction allows for modeling the summation of the output of neural cells along vertical connections on $V1$, i.e., inside $V1$ orientational hypercolumns, as a real-valued function. Introducing in this fiber bundle an appropriate sub-Riemannian manifold and metric, corresponding geodesics that obey certain variational principles of ‘cost optimization’ can be devised to approximate ‘perceptual completion’ of contours stored as neural information on $V1$ [38].

Although there must ultimately be important connections, both the approach and the technique that we develop in this paper are quite different from those that we just mentioned. Here, we do not question nor investigate either the neurological structure or the perceptual reconstruction of the $V1$ retinotopy. We just take at face value the $V1$ model derived from Refs. [2,3,4,5]. Then, we solve mathematically the geodesy problem for that model, or any other surface of revolution, for that matter, generated from $S^2$ diffeomorphisms with any isotropic or non-isotropic CMF’s. It turns out that our exact solution of that problem technically circumvents any need of Ricci calculus, let alone any use of sub-Riemannian fiber bundle models designed to account for $V1$ architecture and connections between cells [37,38,39,40], which we cannot investigate at this time.

3. From Ricci Calculus to Geodesic Orbits on Surfaces of Revolution

A two-dimensional (2D) regular surface of revolution, S, is generally obtained by rigidly rotating a one-dimensional (1D) smooth plane curve, $\gamma \left(\mu \right)$, called the profile curve, around a coplanar z-axis [41]. The azimuthal angle of axial rotation and symmetry, $\varphi$, varies continuously between 0 and $2\pi$ radians. For the sake of simplicity, we may further assume that the profile curve does not intersect itself.

It is natural to transform from Cartesian to cylindrical coordinates, $(r,\varphi ,z)$, to identify points, $\overline{p}$, of the three-dimensional (3D) Euclidean space, $E^3$, in which S is embedded. Namely,

$$\begin{array}{c}\hfill \overline{p}=(x,y,z)=(rcos\varphi ,rsin\varphi ,z).\end{array}$$

(1)

We may set $\varphi =0$ for all points lying on the profile curve, which is thus parametrized with $\mu$ values through coordinates $\gamma \left(\mu \right)=\left(r\right(\mu ),z(\mu \left)\right)$. Then, upon revolution of $\gamma \left(\mu \right)$, points on S can be identified in terms of $(\mu ,\varphi )$ values.

Now, the path of the profile curve $\gamma \left(\mu \right)$ may be obtained from elimination of the $\mu$-parameter, which generates a functional relation between r and z. For our applications, it is convenient to express $z=z\left(r\right)$ as that functional relation. Alternatively, we may derive $r=r\left(z\right)$. Under conditions for which the inverse function theorem applies [41], those relations are equivalent and we may assume that both the function and its inverse are locally single-valued. Globally, however, $r=r\left(z\right)$ is more typically single-valued. As Figure 2 later exemplifies, there are typically two different z-values, one above and one below the equatorial plane that correspond to the same positive r-value.

In cylindrical coordinates, the infinitesimal arc element of Euclidean distance in $E^3$ is differentially derived from Equation (1) as

$${\left(dl\right)}^2={\left(dx\right)}^2+{\left(dy\right)}^2+{\left(dz\right)}^2={\left(dr\right)}^2+r^2{\left(d\varphi \right)}^2+{\left(dz\right)}^2.$$

(2)

When $\overline{p}$ points of $E^3$ are further constrained to lie on the 2D surface, S, the functional relation, $z=z\left(r\right)$, defining the profile curve path, requires that $dz=\frac{dz}{dr}dr$. That reduces the arc element along S to

$${\left(d{l}_S\right)}^2=g_{ij}d{x}^{i}d{x}^j=g_{rr}{\left(dr\right)}^2+g_{\varphi \varphi }{\left(d\varphi \right)}^2,$$

(3)

where $d{x}^i=(dr,d\varphi )$, summation is implied over repeated dummy indices $(i,j)$ and the metric tensor components are obtained as

$$g_{rr}=1+{\left(\frac{dz}{dr}\right)}^2,$$

(4)

$$g_{\varphi \varphi }=r^2.$$

(5)

In general, a smooth curve $\Gamma \left(\nu \right)$ lying on S can be parametrized with $\nu$ values upon which its two coordinates depend, i.e., $x^i=(r\left(\nu \right),\varphi \left(\nu \right))$. Tangent vectors, $\overline{U}$, to $\Gamma \left(\nu \right)$ on S thus have two corresponding components, $U^i=\frac{d{x}^i}{d\nu }$.

A geodesic curve $\Gamma \left(\lambda \right)$ on S has this defining characteristics: it transports its own tangent vector, $\overline{V}$, with $V^i=\frac{d{x}^i}{d\lambda }$ components, parallel to itself, as a curve parameter, $\lambda$, called affine, increases continuously. Intuitively, this means that a geodesic curve advances as ‘straight’ as it possibly can along the curved S surface. Typically, either an extremal or possibly the shortest arc length between two neighboring points on S is measured by a geodetic path connecting those two neighboring points.

Masters of the 19th and 20th centuries developed powerful mathematical theories of differential geometry on manifolds. Currently, we refer to some of that as Ricci calculus, providing rules to perform ‘absolute’ differentiation of tensor fields. In particular, geodesic curves are defined as having zero covariant derivative, meaning ‘no change’ of their tangent vectors along their advancing direction on the manifold. In other words, the tangent vector remains ‘covariantly constant’ along a geodesic curve on the manifold, implying ‘no intrinsic acceleration’ along it.

The basic differential equations obeyed by the contravariant components, $V^i=\frac{d{x}^i}{d\lambda }$, of the vector, $\overline{V}$, tangent to a geodesic curve, or, equivalently, by the coordinates, $x^i\left(\lambda \right)$, of the geodesic curve, have the form

$${\left({\nabla }_{\overline{V}}\overline{V}\right)}^i=\frac{d{V}^i}{d\lambda }+{\Gamma }_{jk}^i{V}^j{V}^k=\frac{d^2{x}^i}{d{\lambda }^2}+{\Gamma }_{jk}^i\frac{d{x}^j}{d\lambda }\frac{d{x}^k}{d\lambda }=0.$$

(6)

Given a vector basis, $\left\{{\overline{e}}_l\right\}$, for the expansion of vectors tangent to the manifold, Christoffel symbols, ${\Gamma }_{jk}^i$, represent the i-component of the covariant derivative of the ${\overline{e}}_j$ basis vector along the ${\overline{e}}_k$ basis vector [42]. Most notably, Equation (6) are second-order differential equations obeyed by the $x^i\left(\lambda \right)$ coordinates, which determine the geodesic curve, $\Gamma \left(\lambda \right)$.

Introducing covariant vector components,

$$V_i=g_{ij}{V}^j,$$

(7)

equivalent geodesic equations

$$\frac{d{V}_k}{d\lambda }=\frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^k}\right)V^i{V}^j$$

(8)

can be generally derived [43]. Both forms of geodesic equations inherently define classes of affine parameters, $\lambda$, linearly related to each other.

When there are metric symmetries, Equation (8) is most suitable to provide corresponding conserved quantities. For surfaces of revolution S with axial, i.e., azimuthal symmetry, $g_{ij}$ components are independent of $\varphi$: see Equations (4) and (5). Therefore, a first integration of a $\varphi$-component in Equation (8) yields immediately a conserved quantity

$$V_{\varphi }=g_{\varphi \varphi }{V}^{\varphi }=r^2\left(\frac{d\varphi }{d\lambda }\right)=L=\mathrm{const}.$$

(9)

In a mechanical analog, Equation (9) represents conservation of angular momentum along the symmetry axis. In elementary differential geometry, the same result can be obtained more laboriously and it is known as Clairaut’s relation (1733, 1735) or theorem [41], which should not be confused with other important theorems due to Clairaut. Notice in particular that a geodetic curve profile or any other $\varphi$-meridian have zero ‘angular velocity’, hence $L=0$.

Parallel transport by geodesic curves of their tangent vectors further requires conservation or constancy of the vector’s norm or length. For geodesics on S, we thus obtain

$$V_i{V}^i=g_{ij}{V}^i{V}^j=g_{rr}{\left(\frac{dr}{d\lambda }\right)}^2+\frac{L^2}{r^2}=C^2>0,$$

(10)

where Equation (9) has been introduced in Equation (3), divided by ${\left(d\lambda \right)}^2$. In a mechanical analog, the positive constant $C^2$ plays a role similar to that of energy conservation. In particular, setting $C^2=1$ amounts to setting the curve affine parameter equal to the arc length itself, namely, $\lambda =l_S$.

Thus, without any need to further integrate either geodesic Equation (6) or Equation (8), we are able to express Equation (10) as a first-order decoupled differential equation

$${\left(\frac{dr}{d\lambda }\right)}^2=\left(\frac{1}{g_{rr}}\right)\left(C^2-\frac{L^2}{r^2}\right),$$

(11)

which can be directly integrated by separation of variables.

Surprisingly, elementary books in differential geometry remain ultimately stuck with Equation (6). As a result, Pressley [41], states on p. 227 that “geodesic equations for a surface of revolution cannot usually be solved explicitly.” Likewise, the opening paragraph of Sec. 7.6, p. 174, of Ref. [44] states that for “surfaces of revolution... it is usually still not possible to solve the geodesic equations explicitly...” In fact that is precisely what we can do, using Equation (11). Do Carmo arrives at a geodesic Equation (6) on p. 261 of Ref. [45] that is equivalent to our Equation (11). He does that by laboriously reworking through Christoffel symbols and Equation (6). That is redundant, in our opinion. Nevertheless, do Carmo’s derivation is certainly correct, formally, and further instructive, mathematically.

Further progress can be made by relating initial conditions to the study of turning points, having

$${\left(\frac{dr}{d\lambda }\right)}_{r_t}=0.$$

(12)

Thus,

$${r_t}^2\equiv \frac{L^2}{C^2}$$

(13)

may provide a turning point of Equation (11). In a mechanical analog, such $r_t$ may play a role similar to that of a periapsis.

On the other hand, if $1/g_{rr}$ vanishes for a greater $r_t^{\prime }$-value that may provide either the actual turning point, as in Equation (28) of Ref. [7], or another turning point further away, which may then play a role similar to that of an apoapsis in a mechanical analog. The purple line, representing a geodetic equatorial parallel in Figure 1, provides an example of the latter, i.e., a ‘maximal apoapsis’.

In terms of $r_t$ defined by Equation (13), by taking the ratio between first integrals in Equations (9) and (11) for $L\ne 0$, we can eliminate the presence of any affine parameter and obtain a geodesic orbit equation

$${\left(\frac{dr}{d\varphi }\right)}^2=\left(\frac{r^2}{g_{rr}}\right)\left(\frac{r^2}{r_t^2}-1\right).$$

(14)

This differential Equation (14) represents a central result of ours. It can be directly solved for any regular surface S of revolution by quadrature or numerical integration, virtually by-passing any more substantive Ricci calculus. Just like Equation (11), indeed, Equation (14) requires no determination of any Christoffel symbol for any regular surface of revolution!

Corresponding to Clairaut’s relation [41], we can also show that

$$L/C=\pm r_t=r_{0}sin{\psi }_0=rsin\psi =\mathrm{const}$$

(15)

is conserved along a geodesic. Here, $\psi$ represents the angle that a parallel-transported vector $\overline{V}$, tangent to the geodesic curve, forms with the local meridian, passing through a point at $(r,\varphi )$ on S. Thus, parallel transport from an initial vector ${\overline{V}}_0$ tangent to S at an initial point $(r_0,{\varphi }_0)$ yields a geodesic orbit equation

$${\left(\frac{dr}{d\varphi }\right)}^2=\left(\frac{r^2}{g_{rr}}\right)\left(\frac{r^2}{{[r_{0}sin{\psi }_0]}^2}-1\right).$$

(16)

Although differently parametrized, Equations (14) and (16) are equivalent via Equation (15). One may use one geodesic orbit equation or the other, depending on one’s choice of initial or search conditions.

Furthermore, Equation (12) means that $r_t$ is an extremal point. Correspondingly, Equation (15) implies that, at $r=r_t$, we must have $\psi ={\psi }_t=\pm \pi /2$, meaning that a geodesic orbit must become extremally tangent to a local parallel at a minimal $r_t$ turning point. Parallels themselves are not geodetic, however, with the following exception. When $z=z\left(r\right)$ becomes also extremal, we have $dz/dr=\infty$ and $g_{rr}=\infty$ in Equation (4). That generates a geodetic equatorial parallel, reducing Equation (11) to a maximal turning point: $r\left(\lambda \right)=r_t^{\prime }=\mathrm{const}$. The purple line in Figure 1 provides an example of that ‘apoapsis’.

If one wishes to use a typically single-valued function $r=r\left(z\right)$ to globally define the profile curve, the geodesic orbit equation, corresponding to Equation (14), becomes

$${\left(\frac{dz}{d\varphi }\right)}^2=\left(\frac{{\left[r\left(z\right)\right]}^2}{g_{zz}}\right)\left(\frac{{\left[r\left(z\right)\right]}^2}{{\left[r\left(z_t\right)\right]}^2}-1\right),$$

(17)

where

$$g_{zz}=1+{\left(\frac{dr}{dz}\right)}^2,$$

(18)

and

$${\left[r\left(z_t\right)\right]}^2={r_t}^2=\frac{L^2}{C^2}.$$

(19)

Typically, $r_t$ is unique, while $z_t$ may or may not be. For example, on an ellipsoid of revolution with a geodesic equatorial circle in the $x-y$ plane of reflection symmetry, typical geodesics wind around an equatorial band, cycling between its two symmetrically opposite $z_t=\pm r_t$ bounds, or ‘vertex latitudes’, derived from the same minimal $r_t$ value [46]. On each cycle, these typical geodesic curves also cross twice the maximal $r_t^{\prime }$ equatorial value, at ‘nodes’ where $|sin\psi |$ is correspondingly minimal on account of Equation (15).

Let us also recall that fundamental second-order differential ‘equations of motion’ such as Equation (6) or Equation (8) do not explicitly contain any initial conditions or parameters. Our first-order differential equations explicitly contain conserved quantities that reflect initial conditions or parameters, such as L, C, $r_t$ or $r_{0}sin{\psi }_0$ because they already involve first integrations of more fundamental ‘equations of motion’.

4. Geodesy on the Primary Visual Cortex and Diffeomorphic Surfaces of Revolution

In our mathematical formulation, we use cylindrical coordinates $(r,\varphi ,z)$ to describe $V1$ as a surface of revolution embedded in 3D Euclidean space, $E^3$. We consider the 2D unit sphere $S^2$, parametrized in $(w,\varphi )$ spherical-polar coordinates. When we refer to the visual field more specifically, we limit to $0\le w\le \pi /2$ the range of the visual angle or ‘eccentricity’, but let us still denote as $S^2$ the corresponding 2D unit hemisphere. Now, we wish to construct a magnified conformal diffeomorphism from $S^2$ to $V1$, which we denote as $\Delta :S^2\to V1$.

Most generally, latitudinal and longitudinal dilations of $V1$ from $S^2$ involve arbitrarily different $M_{\varphi }(w,\varphi )$ and $M_w(w,\varphi )$ magnification functions. However, azimuthal or axial symmetry requires that both latitudinal and longitudinal magnification functions become independent of $\varphi$. Since in this paper we are dealing exclusively with surfaces of revolution, we shall assume that $\varphi$-independence.

Vice versa, loss of axial symmetry requires consideration of metric symmetries that may still exist. Triaxial ellipsoids provide a prototypical example [46]. Jacobi famously discovered (1839) that geodesic equations, expressed in ellipsoidal coordinates, are still separable, although it was later proved by Cohn-Vossen that there are no Killing vector fields for smooth convex surfaces embedded in $E^3$. That would also be the case if there are different CMF’s along different meridians of $V1$, e.g., in the upper versus the lower visual hemifields [31].

Setting that aside, we assume a latitudinal CMF such that

$$r\left(w\right)=M_{\varphi }\left(w\right)sinw.$$

(20)

Thus, parallel circles on $V1$ have circumferences of length $2\pi r=2\pi M_{\varphi }\left(w\right)sinw$, corresponding to parallel circles on $S^2$ with circumferences of length $2\pi sinw$.

Since azimuthal symmetry also requires that all meridians must be equally magnified longitudinally, an arc length $dw$ along any meridian on $S^2$ is mapped to an arc length $M_w\left(w\right)dw$ along the corresponding meridian on $V1$, involving such a longitudinal CMF at any given w.

Now, local isotropic magnification generally means that $M_{\varphi }(w,\varphi )=M_w(w,\varphi )$. Thus, assumptions of azimuthal symmetry and local isotropic magnification are completely independent conditions, neither necessary nor sufficient to each other. Since in this paper we always require azimuthal symmetry, making the further assumption of local isotropic magnification implies the existence of a single isotropic cortical magnification factor such that

$$M_{\varphi }\left(w\right)=M_w\left(w\right)=M\left(w\right).$$

(21)

Such restrictive assumption is inessential to our mathematical formulation, as Equation (27) will demonstrate. Nevertheless, we will make the assumption of Equation (21) when referring to classical experimental studies on the primary visual cortex [2,3,5,10,29,30]. Other studies, however, have considered possibilities of non-isotropic cortical magnification and proposed alternative models and expressions for corresponding CMF’s [32,33,34].

While the arc element on $S^2$ is expressed as

$${\left(d{l}_{S^2}\right)}^2={\left(dw\right)}^2+{sin}^2\left(w\right){\left(d\varphi \right)}^2,$$

(22)

the arc element on $V1$ can be derived from that of Equation (22) by using Equations (2) and (3). Assuming local isotropic magnification, Equation (21), we thus express the arc element on $V1$ in terms of $(w,\varphi )$ coordinates on $S^2$ as

$${\left(d{l}_{V1}\right)}^2={\left[M\left(w\right)\right]}^2{\left(d{l}_{S^2}\right)}^2=g_{ww}{\left(dw\right)}^2+g_{\varphi \varphi }{\left(d\varphi \right)}^2,$$

(23)

where

$$g_{ww}={\left[M\left(w\right)\right]}^2=g_{rr}{\left(\frac{dr}{dw}\right)}^2,$$

(24)

$$g_{\varphi \varphi }=r^2\left(w\right)={\left[M\left(w\right)\right]}^2{sin}^2\left(w\right).$$

(25)

Originally, Rovamo and Virsu illustrated such a mapping in Figure 1 of Ref. [29]. We reproduce their results and the essential features of the $\Delta :S^2\to V1$ diffeomorphism in our Figure 1.

Given Equation (24), Equation (23) corresponds to Equation (3), to which our general geodesic results apply. Using Equations (13) and (14), we may then express the geodesic orbit equation on $V1$ as

$${\left(\frac{dw}{d\varphi }\right)}^2={sin}^2\left(w\right)\left(\frac{{\left[M\left(w\right)\right]}^2}{r_t^2}{sin}^2\left(w\right)-1\right).$$

(26)

This first-order differential equation represents our main result for application to $V1$, assuming an isotropic CMF of general form. Here, the conserved quantity is the minimal $r_t$ value at a turning point. According to Equation (15), we may equivalently replace $r_t$ with $\pm r_{0}sin{\psi }_0$ at the initial point.

It is straightforward to derive a generalization of the geodesic orbit Equation (26) that does not require the assumption of isotropic magnification, as expressed in Equation (21). All we need to do is to replace $M_w\left(w\right)$ for $M\left(w\right)$ in Equation (24) and $M_{\varphi }\left(w\right)$ for $M\left(w\right)$ in Equation (25), according to the more general latitudinal CMF representation of $r\left(w\right)$ in Equation (20). Thus, following the same procedure that led us to Equation (26) for isotropic magnification, we obtain for non-isotropic magnification that

$${\left(\frac{dw}{d\varphi }\right)}^2=\frac{{\left[M_{\varphi }\left(w\right)\right]}^2}{{\left[M_w\left(w\right)\right]}^2}{sin}^2\left(w\right)\left(\frac{{\left[M_{\varphi }\left(w\right)\right]}^2}{r_t^2}{sin}^2\left(w\right)-1\right).$$

(27)

These first-order differential equations can be directly integrated by separation of variables. Assuming that the coordinates of the initial and final points have $w_i<w_f$ and ${\varphi }_i<{\varphi }_f$, a geodesic orbit with $L>0$ develops. If we take the positive square root of Equation (26), the unknown angle of the final point is given by

$$\varphi \left(w_f\right)={\varphi }_i+r_t{\int }_{w_i}^{w_f}\frac{dw}{sin\left(w\right)\sqrt{{\left[M\left(w\right)\right]}^2{sin}^2\left(w\right)-r_t^2}}.$$

(28)

Being expressed as an integral area, Equation (28) provides a complete solution by ‘quadrature’ of the differential Equation (26). However, Equation (28) applies only when no turning point is traversed by the geodesic arc that connects the two points. An alternative expression will be given later in Equation (34) when a turning point must be traversed along the way.

Depending on the functional form assumed for $M\left(w\right)$, it may or may not be possible to further express Equation (28) in terms of special integrals of mathematical physics that are currently provided as standard functions in computer algebra packages. That was the case for similar work of ours [7], involving Jacobi elliptic integrals and functions [47,48]. That is not so for the form of $M\left(w\right)$ most basically considered in vision research [2,3,5,10,29,30]. Nevertheless, accurate solutions of Equation (28) can always be obtained numerically, as we will demonstrate in Section 5. Particular care must be taken, however, if a minimal turning point at $r_t$ is attained because the integrand in Equation (28) diverges therein, although numerical integration must still converge.

It is instructive at this point to check the structure and verify the correctness of Equation (28) by reproducing geodesic orbits on $S^2$, which must be arcs of great circles. Of course, we can do that by simply setting $M\left(w\right)=1$ in Equation (28). That integral can now be performed analytically, yielding

$$\begin{array}{cc}\hfill \varphi \left(w_f\right)& ={\varphi }_i+r_t{\int }_{w_i}^{w_f}\frac{dw}{sin\left(w\right)\sqrt{{sin}^2\left(w\right)-r_t^2}}\hfill \\ & ={\varphi }_i+{tan}^{-1}\left(\frac{\sqrt{2}{r}_{t}cos\left(w_i\right)}{\sqrt{1-2r_t^2-cos\left(2w_i\right)}}\right)\hfill \\ & -{tan}^{-1}\left(\frac{\sqrt{2}{r}_{t}cos\left(w_f\right)}{\sqrt{1-2r_t^2-cos\left(2w_f\right)}}\right).\hfill \end{array}$$

(29)

The integrand and its result are indeed real in the region of integration, where ${sin}^{-1}{r}_t\le w_i\le w_f\le \pi -{sin}^{-1}{r}_t$. Any great circle on $S^2$ correspondingly cycles between two parallel circles symmetrically placed above and below the equator, becoming tangent to such ‘vertex latitudes’ with minimal distance $r_t$ of closest approach to the z-axis.

We have also reproduced classical results for geodesics on ellipsoids of revolutions [46], by integrating Equation (27) with spheroidal non-isotropic CMF’s, expressing such quadratures analytically in terms of Jacobi elliptic integrals and functions [47,48].

Experimentally, it has been found that the cortical magnification factor is basically isotropic and monotonically decreasing with w. A basic expression for

$$M\left(w\right)=\frac{A}{1+Bw}$$

(30)

has been provided and variously parametrized [2,3,5,10,29,30].

The CMF, hence A in Equation (30), is a length measured in $mm$, while $Bw$ in Equation (30) must be a pure number, to maintain unit coherency. Trigonometric functions and corresponding calculus operations are most readily performed by measuring angles in radians. As we do that, B in Equation (30) must be expressed in units of the inverse of a radian. After calculus operations are completed, angular w-values may also be converted to degrees by $180/\pi \simeq 57.2958$ multiplication. In that case, numerical B-values must be correspondingly converted by a 57.2958 division. We typically consider all w-values between 0 and $\pi /2$ radians in the visual field, though a zone of particular interest experimentally lies below w-values of about $\pi /20\simeq 0.157$ radians or 9 degrees from the center of the fovea: see Figure 1.

The curve profile for the $V1$ surface of revolution can be obtained by combining Equation (4) with Equation (24) to yield

$${\left(\frac{dz}{dw}\right)}^2={\left[M\left(w\right)\right]}^2-{\left(\frac{dr}{dw}\right)}^2\equiv {\left[f\left(w\right)\right]}^2.$$

(31)

The right-hand side of Equation (31) must be expressed exclusively as a function of eccentricity, taking the w-derivative of $r\left(w\right)$ from Equation (25). After $\int f\left(w\right)dw$ integration of the positive square root of Equation (31), the curve profile can be expressed parametrically in terms of w as $\gamma \left(w\right)=\left(r\right(w),z(w\left)\right)$.

Any arc length along the curve profile or any meridian is immediately obtained by analytic integration of Equation (30), yielding

$$l(w_f,w_i)={\int }_{w_i}^{w_f}M\left(w\right)dw=\frac{A}{B}log\left(\frac{1+B{w}_f}{1+B{w}_i}\right),$$

(32)

where log denotes the natural logarithm.

In Figure 2, we plot the curve profile for $M\left(w\right)$ as given in Equation (30), using parameter values of $A=525.6$ and $B=33.48$ for rhesus monkeys [5,29]. Alternative values can be found in the literature, e.g., $A=1061.03$ and $B=68.97$ for humans [10,30].

Upon revolution, the shape of the $V1$ surface, as shown in Figure 1b, is qualitatively similar to that of a spheroid, for which analytic solutions of geodesic orbit equations can be obtained using either our technique or traditional methods. Thus, geodesics on $V1$ share some global or qualitative similarities with geodesics on spheroids [46]. However, $V1$ lacks the reflection symmetry about the equatorial plane that spheroids have. More critically, the negative slope of $M\left(w\right)$ that persists at $w=0$ for the particular form of Equation (30) is responsible for an infinite Gaussian curvature at that umbilical point singularly. To smooth that out, $M\left(w\right)$ should at least be rounded as to attain a zero-derivative at $w=0$. Based on available experimental evidence, we have no particular reason nor means to meaningfully remove the cusp of $M\left(w\right)$ at the foveal point that is responsible for the umbilical singularity of $V1$, assuming Equation (30). This means that geodesics on $V1$ that reach to very low w eccentricities, rather than the foveal origin exactly, may display some peculiar behavior therein. This must be examined with even greater numerical accuracy in the integration of Equation (28) when small or vanishing $r_t$ values are attained.

Parallel latitudinal circles are not geodetic on $V1$, except for the equatorial parallel. That can be determined by setting to zero $d/d\varphi =(dw/d\varphi )(d/dw$) derivatives of the left- and right-hand sides of Equation (26), respectively. Equivalently, one may set to zero the $d/dw$ derivative of the square-root of Equation (25), thus determining the largest turning point, $r_{t,max}^{\prime }$. For the $M\left(w\right)$ form assumed in Equation (30) with our chosen parameters [5,29], we obtain $w_{t,max}^{\prime }\simeq 0.4358rad\simeq 24.{97}^{\circ }$. In Figure 1 that geodetic equator, colored in purple, lies a bit below the third parallel circle up from the bottom, which has $w=3\pi /20\simeq 0.4712rad\simeq {27}^{\circ }$.

If we relax the assumption of local isotropic magnification, Equation (21), we can still readily obtain the curve profile for any surface of revolution in terms of its non-isotropic CMF’s relative to $S^2$ as

$${\left(\frac{dz}{dw}\right)}^2={\left[M_w\left(w\right)\right]}^2-{\left(\frac{dr}{dw}\right)}^2,$$

(33)

where the w-derivative of $r\left(w\right)$ must now be taken from Equation (20).

5. Illustrations

In principle, we have thus solved completely the problem of geodesy on surfaces of revolutions in general and on a basic model of the primary visual cortex, or area $V1$, in particular. In this section, we perform some numerical integrations of our central differential equations and we generate corresponding illustrative figures. These integrations and figures are meant to demonstrate as simply as possible our mathematical techniques and further depict main features of geodesics on models of the primary visual cortex, visual field and their stereographic projections, all connected by various types of axially symmetric conformal diffeomorphisms. Such illustrations are not meant to be either exhaustive nor predictive of any specific visual phenomenon or experiment. At this stage, our geodetic techniques have been developed with the purpose of providing a basic tool to further investigate any such phenomenon and develop a deeper understanding or more advanced theories of vision.

In Figure 3a, we show an example of a geodesic orbit on a model of $V1$, based on the isotropic CMF given in Equation (30). That geodesic orbit connects a red point at coordinates ($w_0=0.25$, ${\varphi }_0=0$) with a blue point at coordinates ($w_f=\frac{13\pi }{32}\simeq 1.2763$, ${\varphi }_f=\frac{7\pi }{8}$). In order to find the geodesic arc that connects those two points, we must solve a so called ‘inverse geodesic problem’. Namely, we must find the ‘right’ tangent-vector direction at the red point, in terms of conserved $r_t$ or ${\psi }_0$ quantities in Equation (15), which sets the geodesic on a course that arrives ‘ballistically’ at the intended destination. To that end, we use a root solver program that finds $r_t$ through converging iterations. Namely, we vary $r_t$ time and again after numerical integrations of Equation (28) until a geodesic orbit starting at ($w_0$, ${\varphi }_0$) arrives precisely at ($w_f$, ${\varphi }_f$) ultimately. In this example, the sought parameter value is $r_t=11.1808$. Given Equations (15) and (25), we have equivalently $r_0=13.877$ and ${\psi }_0=0.9368$ radians $=53.68$ degrees at the red point.

This geodesic orbit wraps around $V1$ along the shortest path between the red and blue points, as a string pulled tight against a physical realization or 3D printing of the $V1$ surface would. We should mention, however, that there are infinitely many more geodesic orbits that also connect the same two points. All those geodesic orbits still parallel-transport their own tangent vectors, but wind around $V1$ any number of times. Similar results for spheroids, paraboloids, hyperboloids, and other quadric surfaces of revolutions further depend on characteristics of their global topology [7].

Notice that $r_0>r_t$ in this example. This means that the ‘vertex latitude’ of this geodesic lies beneath or ‘south’ of the ‘launching’ red point. After its northbound ‘launch’ with ${\psi }_0<\pi /2$, the geodesic crosses the equatorial parallel at a ‘node’ with its smallest ${\psi }_t^{\prime }=0.903747$ radians at $r_{t,max}^{\prime }=14.2313$. The final blue point also lies beneath or ‘south’ of the other northern ‘vertex latitude’, where this geodesic would become again tangent to a parallel having the same minimal radial turning-point $r_t=11.1808$ value. However, the geodesic arc that connects the red and the blue points never reaches any minimal turning point at either of those ‘vertex latitudes’ in Figure 3a.

In Figure 3b, we show the diffeomorphic image through $\Sigma \circ {\Delta }^{-1}:V1\to E^2$ of the same geodesic orbit on $V1$ stereographically projected from the $S^2$ visual field to its tangent polar plane $E^2$ at $z=0$. On $E^2$, the geodesic arc that connects the red with the blue point is of course a Euclidean straight segment. That illustrates most vividly how profoundly different are geodesic arcs that connect distant item representations either on $V1$ or their projecting images on a flat screen. Mathematical details on how to perform the precise stereographic projection, $\Sigma :S^2\to E^2$, which is required for visual representation of a plane screen are provided in Appendix A, which may be read at this point.

In Figure 4, we show the explicit construction in 3D of the stereographic projection $\Sigma$ from $S^2$ onto its tangent polar plane $E^2$, resulting in the $\Sigma \circ {\Delta }^{-1}$ mapping of the original geodesic curve on $V1$ to its ultimate image on the $x-y$ plane $E^2$. Neither of the $V1$-geodesic-curve images on $S^2$ and $E^2$ are geodetic in those latter two manifolds: starkly far from geodetic in fact.

Now, suppose that, in Figure 3a, we steadily lower the $w_f$ eccentricity of the blue point. The $r_t$ value must correspondingly increase, until it matches precisely $r_t=r_0=13.877$. Then, the red point has become a turning point. That occurs for $w_f=0.3347$ and ${\psi }_0=\pi /2\simeq 1.571$. We may lower even further the $w_f$ eccentricity of the blue point. Then, we expect the $r_t$ value to decrease once again, as the geodesic ‘launched’ from the red point must initially ‘dip’ southward with ${\psi }_0>\pi /2$, then reach its turning point at $r_t$, before climbing again northward and ultimately reach the blue point. In this situation, the negative square root of Equation (26) for the first branch of the ‘dipping’ geodesic must be taken, whereas the positive square root of Equation (26) must still be taken for the subsequent climbing branch of the geodesic. Thus, Equation (28) must be replaced by

$$\begin{array}{ccc}\hfill \varphi \left(w_f\right)& =\hfill & \hfill {\varphi }_i+r_t{\int }_{w_t}^{w_i}\frac{dw}{sin\left(w\right)\sqrt{M{\left(w\right)}^{2}sin{\left(w\right)}^2-r_t^2}}\\ & \phantom{=}\hfill & \hfill +r_t{\int }_{w_t}^{w_f}\frac{dw}{sin\left(w\right)\sqrt{M{\left(w\right)}^{2}sin{\left(w\right)}^2-r_t^2}}.\end{array}$$

(34)

Ultimately, the eccentricity of the blue point may decrease to equate $w_f=w_0=0.25$. In that case, the turning point lies along the meridian that bisects the latitudinal arc that now connects the equilatitudinal red and blue points. In that case, we find that $r_t=13.7645$ or ${\psi }_0=1.69871>\pi /2\simeq 1.571$. In fact, symmetrically, ${\psi }_f=1.44288$ at the blue point has the same sine as ${\psi }_0=1.69871$ at the red point, i.e., 0.99183.

In all cases where the geodesic must first ‘dip’ south and traverse a turning point in order to ultimately connect two distant points, it is necessary to break the solution of the ‘inverse geodesic problem’ into two parts, smoothly connecting the two arcs that separately join the initial and final points to the same turning point. The root solver technique that we have outlined with regard to the drawing of Figure 3a can still be applied to both terms in Equation (34) with no fundamental change. Indeed, only a single parameter requires a ‘root solution’, since $r_t$ and $w_t$ in Equation (34) are related via Equation (25). For any two points initially given, it is relatively simple to determine first of all whether Equation (28) or Equation (34) should apply. Complete and efficient numerical solutions of this ‘technical’ problem are of great ‘practical’ concern, since data from psychophysics experiments may contain hundreds or thousands of items having similar eccentricities near the fovea. We have further generated routines that automatically find the geodesic arc connecting any two points, lying either below or above the geodesic equator, involving whichever ‘southern’ or ‘northern’ vertex latitude that may be traversed at turning points.

There are various alternatives to approach this type of problems most generally. Consider, for example, a ‘cluster’ of items on $V1$ and suppose that you wish to determine with a given accuracy their geodetic distances from an initial point, $(w_0,{\varphi }_0)$, outside that ‘cluster’, say. You may then ‘launch’ from the corresponding $r_0$ a geodesic Equation (16) with any ${\psi }_0$ that generally ‘aims’ at the cluster. Then, by increasing and decreasing ${\psi }_0$ by multiples of a small increment, $\delta {\psi }_0$, you may generate a family of radial geodesics, all emanating from the same initial point that covers the cluster, ‘from side to side’. You can then generate another family of ‘geodesic circles’, orthogonal to the radial geodesic family, according to a Gauss Lemma [41,45]. You may thus cover the whole cluster with this coordinate-grid patch. Depending on the size that items may have on $V1$, you may refine your grid until each item tightly fits within a single or just a few quadrangles of your grid. The radial length from that quadrangle on $V1$ to the initial point thus provides with good approximation the geodetic distance of its item from the original point. This procedure can be very effective and easy to program, since it does not require solution of any ‘inverse geodesic problem’. Notice, however, that geodesic ‘angular’ distances between items within the clusters are not provided by geodesic-circle arcs, since those are not geodesic curves. Of course, there are other ways to generate ‘normal coordinate’ patches that are fully geodesic [41,42,45], but those are more elaborate and may or may not be needed for ‘practical concerns’ on $V1$.

In Figure 5, we show the explicit construction in 3D of the stereographic projection $\Sigma$ on $E^2$ of the geodesic arc of great circle on $S^2$, connecting the red point with the blue point. This projection is a Euclidean straight segment, i.e., a geodesic path on $E^2$, mapping a geodesic path of great circle on $S^2$ through a geodesic diffeomorphism $\Sigma$.

In Figure 7a, we draw two geodesic circles on $V1$, both centered at the point ($w=0.25$, $\varphi =0$). Their geodetic radii have lengths of 5 and 10 units (mm). In addition, geodetic ‘spokes’ of equal length are drawn for each circle. In the stereographic projection of the visual field on $E^2$, shown in Figure 7b, images of three geodesic circles, having originally geodetic radii of 5, 10, and 20 units on $V1$, show their increasing distortion caused by the lack of isometry in their $\Sigma \circ {\Delta }^{-1}:V1\to E^2$ composite mapping.

In Figure 8a, we begin by drawing on $V1$ four geodesic circles with equal radii of five units, arrayed tangentially to one another along the $\varphi =0$ meridian. Thus, the areas of these four circles are approximately equal on $V1$. In Figure 8b, we first show the stereographic projections of these four circles on $V1$ mapped with $\Sigma \circ {\Delta }^{-1}$ through the $S^2$ visual field to the $E^2$ plane. Images of the circle centers on $V1$ are marked as red points on $E^2$. Thus, geodesic circles with equal radii and approximately equal areas on $V1$ map as increasingly magnified areas of stretched circles on $E^2$ as their visual eccentricities increase, as prescribed by cortical magnification. Further explanations of the lettering and corresponding surface orientations through stereographic projections are provided in Appendix B.

Last but not least, we solve some cases of a ‘direct geodesic problem’ with remarkable consequences. Consider a geodesic starting at a turning point with a pre-assigned $r_t=12$ value, for example. According to Equation (25), an initial point at coordinates ($w_0=0.0976$, ${\varphi }_0=0$) has that $r_0=r_t=12$ value. This initial turning point is marked by a ‘fat’ red dot in Figure 9. By parallel-transporting its own tangent vector from there on, we develop the unique geodesic orbit that solves Equation (28) through a continuum of final points ($w_f$, ${\varphi }_f$), all covariantly evolved from that initial red point with those initial conditions. The first semi-cycle of the geodesic, drawn in red in Figure 9, thus begins tangentially at the ‘southern vertex latitude’ and arrives tangentially at the ‘northern vertex latitude’. The second semi-cycle of the geodesics, drawn in green in Figure 9, does the converse, arriving again tangentially at the ‘southern vertex latitude’. However, the geodesic orbit has advanced longitudinally through this full cycle by an angular increment of $\Delta \varphi =0.5621$ radians, relative to the initial red-point angle ${\varphi }_0=0$. The third semi-cycle of the geodesic, drawn in purple in Figure 9, climbs northward just as the first semi-cycle did, while maintaining the same angular shift of $\Delta \varphi$ all along, relative to the first semi-cycle. That shift is maintained at the ‘northern vertex latitude’, where the ascending purple line becomes blue, turning southward in its fourth semi-cycle. That blue line then arrives tangentially at the ‘southern vertex latitude’ with a longitudinal shift of $2\Delta \varphi$, marked by a ‘fat’ blue dot in Figure 9, relative to ${\varphi }_0=0$ of the initial red dot. After that, the geodesic orbit will advance indefinitely through $\Delta \varphi$ rigidly shifted cycles, unless $\Delta \varphi$ is exactly commensurate to $2\pi$, in which case the geodesic will become a closed periodic orbit. Typically, however, our open geodesic orbit will fill the entire band between its two ‘vertex latitudes’. This prograde geodetic shift, i.e., further advancing in the geodesic direction, is similar to that occurring on prolate ellipsoids [46]. This is of course a direct consequence of azimuthal symmetry. It is also worth noting in Figure 9 that crossings between geodetic branches accumulate around the ‘vertex latitudes’, where those branches become more and more latitudinally tilted.

However, a more complex pattern of longitudinal shifts emerges on $V1$. Namely, there is a critical turning-point value, $r_{tc}=11.194$, for which $\Delta \varphi =0$ and the geodesic orbit closes precisely after one cycle between ‘vertex latitudes’. No such kind of closed orbits exists on spheroids. A critical closed geodesic orbit is shown in Figure 10.

For $r_t>r_{tc}$, the longitudinal shift is positive, or prograde, as shown in Figure 9 for $r_t=12$ and in Figure 11 for $r_t=11.5$, more extensively. However, for $r_t<r_{tc}$, the longitudinal shift becomes negative, or retrograde, i.e., lagging behind along the geodesic direction, as shown in Figure 12 for $r_t=10.875$. That behavior is similar to longitudinal deficits of geodetic cycles occurring on oblate ellipsoids [46].

This complex behavior originates from the interplay between local curvature and global topology, which is more complex on isotropically magnified surfaces than it is on equatorially symmetric spheroids. We shall not pursue any more advanced inquiry on this matter. Current computer packages allow extremely fast and accurate numerical integration of ‘simple’ equations such as Equation (28) for all cases of ‘practical’ concern, such as ours in this paper. However, further analytic characterization of primitives of Equation (28) for isotropically magnified surfaces may lead to discoveries of deeper connections with, or generalizations of, beautiful theories such as those developed for elliptic integrals and functions [47,48].

6. Conclusions

During foveation and an eye fixation, the probability of detecting a peripheral target is a function not only of target eccentricity, but also of the proximity of distracting objects from that target, which contributes to a ‘crowding’ limiting condition [11,12]. Visual crowding derives from the breakdown of one’s ability to identify peripheral objects in the presence of other nearby objects. In turn, this critically depends on cortical magnification of the visual field in the primary visual cortex, or $V1$ area [2,3,5,10,27,28,29,30].

Following eye movements, mapping of items just recorded on the eye retina must be sequentially updated in a coordinate frame intrinsic to $V1$. Studies of vision or visual search did not possess heretofore precise mathematical tools, hence the ability, to take into account cortical magnification directly based on the $V1$ representation of the visual field. Geodesy on $V1$ was surmised, but hardly computable within a rigorous and complete mathematical framework.

In this paper, we have provided that framework and all the basic tools of differential geometry and geodesy that are needed to analyze and compute whatever pertains to any model of $V1$ as a surface of revolution and its corresponding projections.

In Section 2, we summarized basic elements of retinotopy of the visual field on the primary visual cortex, emphasizing the cortical magnification involved in that biological mapping. We referred (a) to ‘crowding’ effects and (b) to constructs involving perceptual, attentional and mnemonic resources that guide or inform vision or visual search performed with eye movements and fixations.

Based on general relativity studies, we further developed our geodetic technique that applies to any regular surface of revolution embedded in Euclidean space. That leads to first-order differential geodesic equations that can be solved by quadrature or numerical integration. We rely exclusively on ordinary calculus, bypassing any more complex use of Ricci calculus and Christoffel symbols. A geodesic orbit first-order differential Equation (14) represents a central result that we provide in Section 3.

In Section 4, we applied our geodetic technique to a conformal diffeomorphism that represents the mapping of the visual field, represented as the unit hemisphere, $S^2$, into $V1$, as illustrated in Figure 1. A geodesic orbit first-order differential Equation (26) provides our main result for application to $V1$ surfaces of revolutions, expressed in terms of isotropic cortical magnification factors, or CMF’s, having a most general form. Integral solutions are provided in Equations (28) and (34) and applied to a CMF, Equation (30) that is typically considered in vision research.

The study of turning points is central to our geodetic formulation and the classification of geometries to which it applies. We applied that study particularly to the elliptic-like geometry to which $V1$ belongs. ‘Direct’ or ‘inverse geodesic problems’ are thus formulated in terms of the prescription or search of a minimal $r_t$ parameter upon which our geodesic orbit solution in Equation (28) most critically depends. That $r_t$ is in turn related to initial conditions and consequent conservation of an angular momentum analog prescribed in Equation (15), corresponding to a theorem of Clairaut.

Most generally, latitudinal and longitudinal dilations of diffeomorphisms from $S^2$ involve different magnification functions. However, axial symmetry requires both latitudinal and longitudinal magnification functions to be independent of the azimuthal angle $\varphi$. We must assume that symmetry, since we are dealing exclusively with surfaces of revolution in this paper.

A further assumption of local isotropic magnification requires equality of latitudinal and longitudinal CMF’s. That is a very restrictive assumption, which is inessential in our mathematical formulation. Thus, we provide a generalized geodesic orbit Equation (27) that may apply more broadly in vision research. Besides that, Equation (27) is definitely needed to account for geodesy of general diffeomorphisms between the unit sphere and all kinds of surfaces of revolution, including spheroids and stereographic projections.

In practice or experiment, items may appear to be, or they are actually located or depicted on a flat surface or screen. We thus studied a corresponding stereographic projection of the visual hemisphere onto its tangent polar plane. That geodesic diffeomorphism maps geodesic paths of $S^2$ to geodesic paths of $E^2$, but in terms of non-isotropic CMF’s. We provided more technical details of this type of stereographic projection and other diffeomorphisms involving non-isotropic CMF’s in an Appendix A. Further details regarding surface orientations and their preservation through stereographic projections have been provided in an Appendix B.

In Section 5, we presented and discussed various prototypical illustrations of our geodetic results on diffeomorphic surfaces $S^2$, $V1$ and their tangent $E^2$ plane. Those include solutions of ‘inverse’ and ‘direct’ geodesic problems with corresponding magnifications (see Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7), projections of right-handed sides of orientable surfaces (see Figure 8), and cycling of either space-filling or periodic geodesic orbits between their ‘vertex latitudes’, with either prograde or retrograde longitudinal shifts on $V1$ (see Figure 9, Figure 10, Figure 11 and Figure 12). Deeper connections between primitives of Equation (28) and theories of elliptic integrals and functions may emerge from these findings.

We did not report in this paper further results regarding, for example, geodesic curvature, normal curvature, or relative torsion of general curves on $V1$. Nor did we present calculations of principal curvatures or other differential geometry characteristics of $V1$. In fact, our technique allows either analytical expression or accurate numerical evaluation of any such quantity, as well as precise calculation of any length or area of whatever figure on $V1$.

Both in principle and in practice the problem of differential geometry and geodesy on $V1$ or any other surface of revolution with either isotropic or non-isotropic cortical magnifications from $S^2$ is thus fundamentally solved or soluble with our techniques.

This provides mathematical and computational tools to investigate to which extent geodesy may biologically affect or perceptually inform reconstruction of visual scenery in certain brain areas or connections. Of course, this is only an entry point to a wide range of inquiries that lie beyond the scope of this paper. However, there are at least some indications that a geodetic account of cortical magnification of the visual field from the eye retina to the $V1$ area is a prerequisite for a more quantitative account and understanding of visual ‘crowding’ effects. Further questions of guidance of eye movements and fixations may also be better understood when grounded on a geodetic account of the $V1$ sequential mapping of the visual field, beyond mere consideration of the external scenery.