Is the sphere the only surface with circular projections? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are circular?

Question

Several ancient arguments suggest a curved Earth, such as the observation that ships disappear mast-last over the horizon, and Eratosthenes' surprisingly accurate calculation of the size of the Earth by measuring a difference in shadow length between Alexandria and Syene. These observations, however, suggest merely a curved Earth rather than a spherical one.

Another ancient argument specifically suggesting a spherical Earth is the fact that the shadow of the Earth on the moon during a Lunar eclipse is circular.

My question is: is it true that the sphere is the only surface all of whose projections are disks?

It surely seems to be true. The corresponding fact, however, is not true in two dimensions. The Reuleaux triangle pictured below is a figure of constant width, meaning that every projection of it in the plane is a line segment of the same length.

There are also surfaces of constant width in higher dimensions, meaning that any two parallel bounding set of hyperplanes (touching the boundary) have constant separation. But all of the non-spherical examples of such surfaces I have seen have obviously non-circular projections.

It also seems clear that finitely many circular projections is insufficient, since intersecting finitely many cylinders would produce a surface having corners and containing some straight line segments.

The fact that you can spin such a surface with all circular projections inside any bounding cylinder is suggestive, but it is also true that you can spin the Reuleaux triangle inside a square, even though it isn't circular.

Further questions would include:

• To what extent are other surfaces determined by their projections? That is, which other shapes can we recognize by the set of their shadows?
• In particular, can we recognize the cube and other regular solids by their shadows?
• Which sets of shadows are realizable as projections of a surface? Is there some way to characterize these sets? Clearly they must be continuously deformable to one another and obey several other obvious conditions.

We had a great time discussing the question after our logic seminar here in New York this week, when our speaker Maryanthe Malliaris asked the spherical Earth question.

December 20, 2010: In light (or dark, as it were) of the lunar eclipse tonight, I am bumping this question, with the remark also that despite the truly outstanding answers we have received, several of the further questions stated above are not fully answered.

Answer 1

The answer to the title question is yes (well, I assume that by a "surface" you mean something reasonable, like a boundary of a convex set).

Let $AB$ be the longest segment with endpoints on the surface. We may assume that its length equals 2 and its midpoint is the origin. Consider projections to the planes that contain $AB$. Since projections do not increase distances, $AB$ is a diameter of each projection. Hence all projections to this family of planes are unit discs centered at the origin. The intersection of the corresponding cylinders is the unit ball, hence the result.

Added. In general, we cannot determine a convex body from the set of shadows (if we don't know the correspondence between shadows and directions of projections).

Take a unit ball and cut off three identical tiny caps whose centers form a regular triangle on the sphere and are not on one great circle. Looking at shadows, you cannot tell whether all three or only two caps are removed, because each projection shows you no more than two of them.

The same construction works for polyhedra if you start with an icosahedron rather than a ball.

Answer 2

Warm-up

Let's start in the plane.

A curve of constant width can be thought of in terms of a slab in the plane of the given width that moves around as a function of angle. A moving slab has an envelope, which is the curved form by the points where the boundary lines are instantaneously rotating (i.e., the limit of where nearby points cross). The pair of curves touch the slab at the ends of a perpendicular to the slab. The midpoint of this perpendicular segment moves forward or backward along the axis of the slab.

To put it another way: the (3-dimensional) tangent line bundle to the plane has a contact structure, which is a 2-plane field giving directions of "allowed" motions. It's like an ice skate: it can move forward or backward, and it can rotate. A curve following these rules is called a Legendrian curve, and curves of constant width correspond to Legendrian sections of the fibration (tangent line bundle of plane) --> (sets of parallel planes = $\mathbb {RP}^1$.

The Legendrian sections also correspond to differentiable curves in the plane, generically with an odd number of cusps (unless the singularities are degenerate), whose tangent line turns consistently by 180 degrees in one direction. For any such curve and any $w$, draw the figure swept out by the perpendiculars in each direction a distance $w$: it is a curve of constant width.

Three dimensions

Added: See Ivanov's answer for a more efficient route for deducing it's a sphere.

If every projection of a surface $FS$ (fake sphere) in 3 dimensions is a circle, then we can think about the family of cylinders that enclose it. This is codified as a map from $\mathbb {RP}^2 = $ the set of parallelism classes of lines to the set of lines in $\mathbb R^3$ that is a section of the parallelism equivalence relation.

From the two-dimensional picture, we can visualize what happens in a slice: For each tangent direction to $\mathbb {RP}^2$, there is an axis about which the cylinder is instantaneously rotating. The common perpendicular to the axis of rotation and the axis of the cylinder intersects the cylinder at two points where the surface is tangent to the cylinder. This line segment is inside $FS$.

Now think about a different tangent direction to the same point of $\mathbb {RP}^2$. We get another axis of rotation, describing what happens in that direction. But, if the new line segment did not intersect the first, the first line segment would escape from the moving cylinder.

We conclude that all axes for any point in $\mathbb{RP}^2$ coincide, and $FS$ contains a round disk passing through the common point of these axes. This defines a map from $\mathbb {RP}^2$ to the tangent line bundle of $\mathbb{R}^3$, where $FS$ is swept out by circles centered at the tangent lines in the plane perpendicular to the line.

However, the base point for the tangent line must be constant, because if it were to move, then the sideways view would contain a constant width strip about the image, and this is not contained in a disk unless the map is constant.

Conclusion: $FS$ is not fake. It is a genuine $S^2$. You can go ahead and buy it.

What can you learn from a set of Shadows?

Suppose we have a collection of shadows of an unseen object that is moving in unknown ways---we don't know the projections corresponding to which shadow. What can we deduce?

I don't have an answer, but here are some things that can be done as a start:

The set of closed subsets in the Euclidean plane of any given bounded diameter has a topology and metric, the Hausdorff topology and metric. (More precisely, we're taking the quotient space of the Hausdorff topology by a compact equivalence relation, and using minimum Haudorff distance between equivalence classes to induce a metric on the quotient.) The profile of a projection is a smooth function of the projection, so we get a also get a smooth structure on this space: a diffeomorphism to $\mathbb{RP}^2$.

Let's think first about a generic, smooth convex shape, for which every projection is different, so the set of our shadows is homeomorphic to $\mathbb{RP}^2$. We can try to reconstruct successively more information: the projective structure, the metric structure, and finally the set of solid cylinders in space which enclose the shape.

A line, in the projective structure, consists of a set of profiles that are perpendicular projections to a collection of planes that share a line.

For any profile, there is a circle's worth of 1-dimensional projection, with an invariant, the width. Critical points of width are projections for which the line between points mapping to extremes of the projection are perpendicular to the surface. Think of the pair of planes tangent to the surface at the endpoints of such a line segment.

As one moves around in $\mathbb{RP}^2$, the multiset of critical widths changes. Consider a profile where one of these critical widths is is a critical point with respect to the space of profiles. This happens when the pair of tangent planes that project to tangent lines of the profile are perpendicular to the line segment connecting the points of tangency. At any such point: for instance, when the diameter is maximal or minimal--- we can spin the projection around the axis to get an entire circle's worth of profiles sharing the same critical width.

A better way to think of these globally critical widths is to imagine a pair parallel planes squeezing down on the surface, a kind of caliper. This makes width a function on $\mathbb{RP}^2$. A Morse function on $\mathbb{RP}^2$ has at least 3 critical points, so there are at least 3 globally critical widths.

In this way, we get an initial network of lines for the projective structure we're seeking. Any two lines in $\mathbb{RP}^2$ intersect. Furthermore, we know the angles between these projective lines, since any two lines intersect, and in the profile corresponding to the intersection, we see two diameters at once with angle equal to their angle in space.

Once we have a projective line identified together with its axis, we can deduce the profile in the projection that maps the axis to a point, up to diffeomorphisms of $\mathbb R^2 \setminus 0$ that take lines to lines and act as rotations on any one line. The information, in other words, is a curve in the positive orthant that tells the pair of distances of intersection points of the curve with lines through the origin. Generically, there is only one profile in our collection that matches, so we can deduce what it is. This gives the information we need to get the angle parametrization of our projective line.

I think we're well on the way to complete identification of a generic smooth convex shape, but, I'll leave it here for now. Feel free to add, refine, or streamline..

Answer 3

Might I also point out the unstated assumptions in your question and in the fine answers thus far by Bill Thurston and Sergei Ivanov?

The unstated assumption is that

we are speaking about opaque objects casting opaque shadows,

and that all of the information content we obtain is merely in the outer boundary / envelope of the shadow, whether we are talking about a line segment in $\mathbb{R}$ projected by an object in $\mathbb{R}^2$, a 2-dimensional shadow in $\mathbb{R}^2$ cast by an object in $\mathbb{R}^3$, or by a 3-d shadow cast onto $\mathbb{R}^3$ by an object in $\mathbb{R}^4$, etc.

If we start talking about allowing for transparent and translucent objects which pass (or scatter) graded amounts of light based on their local density distribution, we can allow for non-constant shadows. These non-constant shadow projections can allow the observor to use many different algorithms to infer the interior distribution density function of the shadow-casting object.

This type of shadow analysis is used in Computerized Axial Tomography, also known as CAT scanning or CT scanning. Multiple 2-d images are acquired as an x-ray receiver (CCD) is rotated around the object to be probed while an x-ray source is also rotated around the object synchronously at a position on the opposite side of the object. The multiple acquired 2-d shadows of X-rays passing through the body can be used to reconstruct a fairly accurate rendition of the density distribution of the body using a Radon transform.

Prior to the axial rotation type of scanning, a linear type of scanning was used in a technique called Tomography. The subject is placed at the origin in 3-space, a single piece of x-ray film is placed at position ($-x$, $+y$, 0) at time $t_0$ in the $xz$ plane and translated to position ($+x$, $+y$, 0) at time $t_1$. The x-ray source is placed at position ($+x$, $-y$, 0) at time $t_0$ and is translated to position ($-x$,$-y$,0) at time $t_1$. This is akin to taking a photograph with a long exposure time, keeping it pointed at one point in space as the camera moves along a path in space. The resulting photographic image will have the sharpest focus at the "focal plane" $y=0$, while the objects at further distances along the $y$-axis from the $x$-axis will be progressively blurred.

There is a lot of very interesting mathematics involved in the signal acquisition and signal analysis of Axial Tomography and in the image reconstruction algorithms, as well as in MRI (magnetic resonance imaging).

Also, as an aside, no one has specified whether the "light sources" casting the shadows are point sources in the near-field at a particular distance $d$ within a few orders of magnitude of the size of the objects, or whether we are assuming that the light source is effectively a point source at infinity casting isometric projections. Also, the specific region receiving the shadow was not clearly defined. I only mention this because of the recent questions on Mathoverflow concerning the importance of rigour in mathematics, and because I do indeed believe in the importance of rigour in the formulation of the questions and, thus, in defining the restricted domains in which the succeeding mathematical calculations can be applied.

Allowing for non-opaque shadows makes it much more difficult to find another object in $\mathbb{R}^3$ whose projected shadow would match the intensity distribution of the projected shadow of a translucent sphere.

Answer 4

The first two questions (and others) are discussed in the following two papers of Kuzminykh, A. V.:

• An isoprojection property of the sphere (1973)
• Reconstructibility of a convex body from the set of its projections (1984)