Remarks on the Geometry of Visibles

Gordon BelotNew York University

AbstractAn explication is offered of Reids claim (discussed recently by Yaffe and others) that thegeometry of the visual field is spherical geometry. It is shown that the sphere is the only surfacewhose geometry coincides, in a certain strong sense, with the geometry of visibles.

1. Introduction.

If you lie on your back in the centre of a square room and examine the ceiling, each of the

four corners will appear to form an obtuse angle. It seems to follow that these angles sum to more

than 2 radiansand hence that a square ceiling as it appears in our visual field cannot be viewed

as a Euclidean quadrilateral. What, then, is the geometry of visual figures?

In his discussion of the geometry of visibles, Thomas Reid appears to argue that this

geometry coincides with the simplest of the many geometries in which the sum of the interior

angles of a plane figure exceeds the sum for Euclidean figuresnamely, spherical geometry.1 In a

recent paper in this journal, Gideon Yaffe provides both an insightful commentary on Reids

discussion and a suggestive reconstruction of Reids argument.2

There are, I think, some infelicities in Yaffes discussion. (i) His presentation serves two

aims: the interpretation of Reid, and the construction of a valid proof of the equivalence of visual

and spherical geometry. As a result, it is not always easy to discern the precise structure of the

proof offered. (ii) Indeed, it appears to me that there remain gaps in Yaffes version of the

1 T. Reid, An Inquiry into the Human Mind on the Principles of Common Sense, ed. D. Brookes (Edinburgh UniversityPress, 1997), VI.VII and VI.IX.2 G. Yaffe, Reconsidering Reids Geometry of Visibles, The Philosophical Quarterly, 52 (2002), pp. 602620. Inwhat follows, I am indebted to this paper and to a talk, Thomas Reids Geometry of Visibles, given by James VanCleve at New York University in November 2001.

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argument. (iii) Yaffes construal of the claim of equivalence as involving a proof-theoretic

component is problematic.

In 2 below, I revisit the question of the equivalence of visual and spherical geometry, and

describe how this equivalence can be established quickly and cleanly, drawing upon precisely the

sort of considerations that Yaffe finds in Reid. 3 situates this approach with respect to Yaffes. In

4, I turn to another questionone which is close to the surface in Yaffes paper and in related

work by Van Clevenamely, Is the sphere the only surface whose geometry coincides with the

geometry of visibles?

2. Visual Geometry is Spherical Geometry.

It is traditional in this context to employ the following highly idealised model of vision. The

eye is treated as a point located in Euclidean space. For convenience, we equip this space with

Cartesian coordinates, and locate the eye at the origin. The eye can see in all directions, and for

each direction is able to determine whether an opaque body lies in that direction. In effect, the eye

sees in black and white, and does not register information about the distance to visible objectsa

small object located close to the eye will have the same appearance as a large object located far

away, so long as the two objects intersect the same rays from the eye.3 So in order to specify the

content of visual experience at an instant, it suffices to specify the set of rays from the origin which

intersect visible bodies.

Let us stipulate meanings for the terms appearing in the geometry of visibles. A visual

pointthat is, a point in the visual fieldis just a ray through the origin (in this paragraph and the

next, unmodified geometrical terms have their normal Euclidean sense). A visual line is a set of

3 A ray through the origin is a set of points in Euclidean space of the form {(tx,ty,tz): t>0, (x,y,z)R3/(0,0,0)}.

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rays which arises as the set of rays lying on a plane through the origin. A visual point visually lies

on a visual line if the ray in question belongs to the set of rays in question. The visual distance

between two visual points is the lesser of the two angles subtended by the two rays (this will be

measured in radians, so the visual distance between two points is always between zero and ). The

visual angle between two (distinct) visual lines coincides with the dihedral angle between the two

planes corresponding to them. This angle can be calculated as follows. Let our two planes be and

. Let the ray c lie in their intersection (there will be two such rays). Choose a plane that

intersects c orthogonally. Then each of and intersects in a line segments; the dihedral angle

between and is just the lesser of the two angles formed by these two line segments (this angle is

independent of the choice of c and ).

The italicized terms in the above paragraph name the basic concepts of the geometry of

visibles. The corresponding terms in spherical geometry have standard meanings. A spherical-

geometric point is a point on the surface of the unit sphere. A spherical-geometric line is a great

circle on the unit sphere (a great circle arises as the intersection of the sphere with a plane passing

through its centre). A spherical-geometric point spherical-geometrically lies on a spherical-

geometric line if the point is a member of the set of points constituting the line. The spherical-

geometric distance between two spherical-geometric points is just the Euclidean length of the

shortest great-circular arc on the unit sphere that joins the two points. The spherical-geometric

angle between two (distinct) spherical-geometric lines coincides with the angle between their

tangents at their point of intersection.

We fix for our consideration the unit sphere with the eye at its centre. Note that there is a

natural one-to-one and onto correspondence between the set of visual points and the set of

spherical-geometric points of the unit spherenamely, that according to which each ray

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corresponds to the unique point at which it intersects the sphere. Of course, this also gives us a way

of associating sets of visual points with sets of spherical-geometric points. So there is a bijective

correspondence between possible contents of the visual field and sets of points of spherical

geometry.

We call a spherical-geometric point or line the projection of a visual-geometric point or line

if the two correspond to one another in the sense of the preceding paragraph. It is not difficult to

convince yourself of the following:

(i) The projection of a visual point is a spherical-geometric point.

(ii) The projection of a visual line is a spherical-geometric line.

(iii) A visual point visually lies on a visual line iff the projection of the visual point spherical-

geometrically lies in the projection of the visual line.

(iv) The visual distance between two visual points is equal to the spherical-geometric distance

between their projections (here the fact that we are projecting to the unit sphere is crucial).

(v) The visual angle between two visual lines is equal to the spherical-geometric angle between

their projections (choose the tangent plane to the sphere at a point of intersection of the spherical

lines as the plane to be used in calculating the dihedral angle corresponding to the visual angle).

This shows that visual geometry and spherical geometry are isomorphic with respect to the

geometric notions in playi.e., that the projection map preserves the applicability of our

geometrical vocabulary. This provides a clear sense in which spherical geometry and visual

geometry are the same. This sense will be expanded upon in the next section. For now, note two

consequences of this isomorphism of structure: (1) any claim involving the notions of points, lines,

incidence, distance, and angle that can be interpreted as making a claim about visual geometry can

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be interpreted as making a corresponding claim about the geometry of the sphere, and vice versa;

(2) any two such claims will have the same truth value.

3. Relation to Yaffes Approach.

This section contains three remarks regarding the relation between the present approach and that of

Yaffes paper.

(1) Where I take a visual point to be a ray through the origin, Yaffe takes a visual point to be an

arbitrarily selected point lying on such a ray. Where I take the visual figure corresponding to a

region of Euclidean space to be the set of rays intersecting that region, Yaffe takes it to be the

image of an arbitrary continuous map which selects for each ray in that set a point lying on that ray.

These differences are immaterial: my stipulations strike me as being neater; but the results

corresponding to (i)(v) above go through under Yaffes stipulations.

(2) The considerations leading to (i)(v) are of the sort that Yaffe and Van Cleve find in Reid.

(3) Yaffe interprets Reids assertion of the equivalence between visual and spherical geometry as

involving the following three claims: (a) we can associate with any sentence, G, making an

assertion regarding spherical-geometric points, lines, distances, angles, and so on, a sentence, V(G),

that makes a corresponding assertion regarding the corresponding objects of visual geometry; (b)

any such G and V(G) will have the same truth value; and (c) any proof of an assertion G in

spherical geometry can be transformed into a proof of the corresponding assertion, V(G), in the

geometry of visibles by applying the operator V to each of the assertions in the proof of G.

What does (c) add to the conjunction of (a) and (b)? Suppose that we have a putative proof,

cast in terms of our geometric vocabulary. By (a), we can interpret its premises and conclusion in

either spherical-geometric terms or visual-geometric terms. By (b), the argument will either be

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sound under both interpretations, or unsound under both interpretations. So the only way it could

count as a proof under one interpretation but not under the other is if its premises come out as

geometrical axioms under one interpretation but not under the other. But given a set of axioms for

one of our geometrical systems, nothing prevents us from taking their reinterpretations as providing

axioms for the other system. So (c) adds nothing to (a) and (b).

It is, however, it is natural to worry that the conjunction of (a) and (b) is not enough to

guarantee that our two geometries are really the same. For suppose that we work in a (two-sorted)

first-order language, in which incidence, distance, and angle are the only non-logical primitives.4

Then the conjunction of (a) and (b) just says that our two geometries are elementarily equivalent as

models for our languagethey render all of the same sentences true. But we know, by the

Lwenheim-Skolem theorem, that there will be models elementarily equivalent to ours which we

would not think of us having the same structure as our models (since the differ as to the cardinality

of the set of points).

So elementary equivalence with respect to such a language is too coarse a notion to capture

the notion of sameness that we seek. The natural move is to supplement our resources. We could

extend our language by adding enough constants to name the points and lines of our models and

focus on the diagrams of our models.5 Or we could shift to a second-order language, and look for a

categorical set of axioms. In either case, we will be led to count models as being the same when

they are related by an isomorphism. And any such isomorphism will guarantee elementary

equivalence. 6

4 These primitives would lead to an appalling axiom system. In fact, a three-place order relation on points suffices forthe treatment of spherical geometry; see J. R. Kline, Double Elliptic Geometry in Terms of Point and Order Alone,Annals of Mathematics 18 (1917), pp. 3144.5 See W. Hodges, A Shorter Model Theory (Cambridge University Press, 1997), 1.46 See, W. Hodges, A Shorter Model Theory, Theorem 2.4.3.

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So these paths lead us back to where we were at the end of 2counting visual geometry

and spherical geometry as the same in virtue of fact that the projection map is an isomorphism

between the two (for any reasonable geometrical vocabulary).

4. Is the Sphere Special?

The geometry of the unit sphere coincides with visual geometry via the projection map

carrying visual points to points on the sphere. It is natural to ask whether there are other surfaces

whose geometries have this feature.

Let us call a surface (i.e., a two dimensional submanifold) in Euclidean space projectable if

it intersects each ray through the origin exactly once. We can associate a projection map with such

a surfacemapping rays to their points of intersection, and so on.

As a submanifold of Euclidean space, a projectable surface is endowed with a natural

geometry: points are points; lines are geodesics (i.e., straightest curves); incidence has its usual

sense; the distance between two points is given by the length in the ambient Euclidean space of the

shortest geodesic joining them; and angles of intersection are given by looking at the angles formed

by the tangents at the point of intersection.

We are interested in the question: Which surfaces have a natural geometry that coincides

with visual geometry, in the sense that the projection map is an isomorphism?

I make two remarks, by way of making clear how much is being demanded here.

First remark. Normally, we do not think of the choice of a unit for measuring length as

being intrinsic to geometry. So we would be tempted to count any sphere, of any diameter, as

having the same geometry as visual geometrysince the only effect of shifting from the unit

sphere to an arbitrary sphere is that the projection map re-scales distances by a constant factor. For

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the sake of simplicity of exposition, I resist this temptation here. But this choice makes no material

difference to the argument below, since for every surface satisfying the weaker criterion which

permits re-scaling of length there is a surface satisfying my stronger criterion (the two surfaces

being related by a dilatation corresponding to the factor by which length is re-scaled).

Second remark. Let a visual figure be a finite set of visual points and lines. Let us say that

the projection of a figure to a projectable surface is distortion-free if the analogs of (i)(v) of 2 go

through when restricted to the points and lines of the geometrical figure. It is automatic, of course,

that the visual points of the figure project to points of the geometry of the surface. What we have to

worry about is whether the distances and angles are preserved and whether the visual lines of the

figure project to lines in the geometry of the surface (if this holds, then relations of incidence are

preserved as well). For any given visual figure, it is always possible to find a non-spherical

projectable surface that carries a distortion-free projection of it. We can do this by starting with the

unit sphere, then denting it gently inwards or outwards; such dents are always permitted if they are

far from the projected figure; they are highly constrained but still possible even if they affect points

in this figure.

Now to say that the projection map to a surface is an isomorphism is to say that the surface

carries distortion-free projections of every visible figure. Our question is whether any non-spherical

surfaces satisfy this stringent condition.

The answer is: No. Intuitively, this is because the geometry of visibles is invariant under

rotations of Euclidean space about the origin; so any surface whose geometry coincides with that of

the geometry of visibles will have to be similarly invariant.

Here is a more careful argument. Choose a projectable surface that carries distortion-free

projections of every visual figure. The visual line consisting of rays lying in the x-y plane projects

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to the intersection of our surface with this plane. By assumption, this set corresponds to a geodesic

of the natural geometry of our surface. In terms of polar coordinates in the x-y plane, this set is of

the form {(r(),)}, where 0

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I have observed that the projection map, which takes a ray from the eye to the point at

which that ray intersects the unit sphere surrounding the eye, provides an isomorphism between

visual geometry and spherical geometrythat is, a map from the elements of one geometry to those

of the other which preserves the applicability of the corresponding geometric vocabulary. I have

suggested that this observation provides the neatest explication of Reids claim that the geometry of

visibles is spherical geometry. Finally, I have argued that the sphere is specialit is the only

surface whose geometry coincides with that of visual geometry, in the sense that the natural

projection map provides an isomorphism of the necessary kind.7

7 I would like to thank Gideon Yaffe and the editors of this journal for helpful suggestions. This paper is based uponwork supported by the National Science Foundation under Grant No. SES-0135445.