History of Topics in Special Relativity/Penrose–Terrell

History of Topics in Special Relativity () Multiple chapter topics History of Lorentz transformations History of relativistic four vectors History of relativistic tensors Single page topics History of twin paradox History of Penrose–Terrell effect History of time dilation History of three-acceleration transformation Müller's history of relativity

Early history of the Penrose–Terrell effect

Date of article creation: 19 July 2025; Last major revision: 12 December 2025

If the length of an object is measured exactly at the locations of its endpoints, then its w:rest length $l'$ is related to its length $l$ in a relatively moving frame by the w:length contraction formula:

(1) $\quad l=l'{\sqrt {1-v^{2}/c^{2}}}$

However, it turns out that the visual appearance of that object when photographed from a distance leads to quite different results, because optical aberration and time-of-flight effects have to be considered as well. In particular, light rays arriving simultaneously at the camera weren't emitted simultaneously from all locations at the source.

1D rods

The visual length $L$ of a 1D rod that is both oriented and moving along the x-axis, and which approaches or recedes a camera that is located on the same x-axis, is given by

(2) $\quad {\begin{matrix}L_{\rm {approach}}={\frac {l}{1-v/c}}=l'{\sqrt {\frac {1+v/c}{1-v/c}}}\\L_{\rm {recede}}={\frac {l}{1+v/c}}=l'{\sqrt {\frac {1-v/c}{1+v/c}}}\end{matrix}}$

So if the 1D rod approaches it appears to be longer, if it recedes it appears to be shorter, and none of those lengths precisely matches the contraction formula (1) but rather depend on the w:relativistic Doppler factor that also appears in the w:relativistic aberration formula.

Formula (2) was introduced by § Petzoldt (1914) and in a more general way that also allows for motions along lines parallel to the x-axis by §§ Lorentz (1914)​ and Lampa (1924). Even before those authors, the visual position of a point was derived by § Mangoldt (1910) and related to photographic images by § Grünbaum (1911).

The visual length on a photograph directly corresponds to contraction formula (1) when the moving 1D rod is oriented perpendicular to the camera's line of sight and the rod's center is exactly in the camera's line of sight at the time of light emission, because in this case the light emanating simultaneously from both ends of the moving rod will travel the same distance to the camera and therefore arrive simultaneously as well, which can be compared to the visual length of a stationary rod that was momentarily at the same position as the moving rod when light was emanated from its endpoints.

This was demonstrated by § Lorentz (1922).

In case the rod's center is not exactly in the camera's line of sight at the time of light emission, Lorentz contraction can only be detected on photographs by resorting to approximations such as making the rod's length very small with respect to the rod's distance from the camera.

This was shown by §§ Mangoldt (1910), Grünbaum (1911), and Lorentz (1914).

3D objects

It becomes more complicated when 3D objects are considered. First we notice the conformal property of the w:relativistic aberration formula

(3) $\quad \tan {\frac {\phi }{2}}={\sqrt {\frac {1+v/c}{1-v/c}}}\tan {\frac {\phi '}{2}}$

because it only changes the scale but not the shape of spherical outlines, thus the visual appearance of spheres remains spherical on camera.

This was pointed out by § Plummer (1910) and rediscovered by § Penrose (1958/59) and Terrell (1959).

Furthermore, light emitted on the backside of arbitrary 3D objects such as spheres, rods and cubes can reach the camera, because the object's motion frees the path of light rays that were hidden otherwise. Consequently, if the object is very far away and subtends a very small angle, it doesn't appear contracted but rather rotated by $\phi -\phi '$.

This was pointed out by § Penrose (1958/59) and Terrell (1959).

The description of visual distortions is connected to the names of Penrose and Terrell in most papers, hence it is called w:Penrose–Terrell effect/rotation or Terrell effect/rotation or Lampa–Penrose–Terrell effect. Given the historical information provided above, the expression Penrose-Terrell effect is appropriate when it comes to rotations alone; yet when it comes to the visual appearance of moving bodies in general, one should should not forget the early contributions of Plummer, Petzoldt, Lorentz, Lampa.

Historical papers

Plummer (1910)

w:Henry Crozier Keating Plummer derived the following consequence from relativistic aberration:

In the first place we may consider the stereographic projection of the celestial sphere on the tangent plane perpendicular to the direction of motion. The form of the law of aberration

$\tan {\frac {1}{2}}\phi '={\sqrt {\frac {U-v}{U+v}}}\tan {\frac {1}{2}}\phi$

shows that the effect of aberration is simply to alter the scale of the projection. But the stereographic projection is a conformal representation of the sphere. Hence actual configurations on the sphere are only changed conformally by the effect of aberration, or in other words any small area is altered only in size and not in shape. We know that aberration merely changes the scale of a photograph of a small part of the sky, and the truth of this fact now becomes independent of the velocity (however large) of the observer. Also stars which appear to lie on a circle at any one time will continue to do so permanently.

Thus Plummer showed that independent of the velocity of the observer, the photograph of parts of the celestial sphere is changed by relativistic aberration only in terms of scale, not shape, thus spherical outlines remain spherical.

Mangoldt (1910)

w:Hans Carl Friedrich von Mangoldt demonstrated the constancy of light speed in different frames by using the case of a light ray emitted from the origin and then traversing distance $-l$ in the negative x-direction during time $t=l/c$. A moving marker momentarily co-located at position $-l$ at time $t$ will indicate the Lorentz transformed value

$x'=\beta (x-vt)=-\beta \left(l+v{\frac {l}{c}}\right)=-\beta l\left(1+{\frac {v}{c}}\right)$

and a momentarily co-located moving clock indicates

$t'=\beta \left(-{\frac {v}{c^{2}}}x+t\right)=\beta \left({\frac {v}{c^{2}}}l+{\frac {l}{c}}\right)=\beta {\frac {l}{c}}\left(1+{\frac {v}{c}}\right)$

where $\beta =1/{\sqrt {1-v^{2}/c^{2}}}$ was used by him as Lorentz factor, proving $x'/t'=-c$ as it should be.

At the end of his paper, he alluded to the possibility of using photographs of sufficiently small areas at sufficient distance from the camera, which should directly provide an approximately correct image of relativistic effects such as length contraction and time dilation as indicated by markers and clocks.

a) We see that his position equation that incorporates the time-of-flight of light (using modern notation $\gamma$ for the Lorentz factor)

$x'=-\gamma l\left(1+{\frac {v}{c}}\right)=l{\sqrt {\frac {1+v/c}{1-v/c}}}$,

is formally equivalent to equation (2), and it's easy to re-interpret his result in terms of visual lengths: We introduce a rod whose rest length $l'$ is defined by coordinate $x'$ as left endpoint and the origin as right endpoint, in which case Mangoldt's distance $l$ is identical with the rod's visual length $L$ as photographed at time $t=l/c$ by a camera located at position $-l$.

b) Ironically, even though he was in possession of the optical position formula that incorporates the time-of-flight of light, he only discussed an approximate case in order to directly demonstrate length contraction and time dilation on photographic images without considering time-of-flight delays, which was clarified by § Grünbaum (1911). Anyway, Mangoldt's restriction to an approximate case shows, that he was well aware that an exact treatment requires the consideration of time-of-flight delays.

Grünbaum (1911)

In a follow-up paper to § Mangoldt (1910), the physicist w:Fritz Grünbaum discussed the photographic image of distant points. He described a marker (indicating its spatial position) and a co-located clock (indicating its time) that are at relative rest with respect to a photographic apparatus, and another marker/clock pair in relative motion with respect to that apparatus. When both marker/clock pairs meet each other at point P, the apparatus located at point R makes a photo of them. In the rest frame of the apparatus, $l$ is the distance from P to R, and T is the time at which the image was created on the photographic plate. On that image, the resting marker indicates its spatial coordinate $x_{r}$ while the resting clock indicates its time $t_{r}=T-l/c$ since light took some time to traverse distance $l$. In order to find out which values are indicated on the photographic image of the moving marker/clock pair, he used the Lorentz transformation (with $\beta$ as Lorentz factor):

${\begin{matrix}x'_{r}=\beta \left(x_{r}-vt_{r}\right)\Rightarrow \beta \left[x-v\left(T-{\frac {l}{c}}\right)\right]\\t'_{r}=\beta \left(t_{r}-{\frac {v}{c^{2}}}x_{r}\right)\Rightarrow \beta \left[\left(T-{\frac {l}{c}}\right)-{\frac {v}{c^{2}}}x\right]\\\left[\beta ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}},\ x_{r}=x,\ t_{r}=T-{\frac {l}{c}}\right]\end{matrix}}$

which he summarized in the following table:

On the image of the apparatus:

Stationary marker: $x,\ y=0$ Stationary clock: $T-{\frac {l}{c}}$ Moving marker: $\beta \left[x-v\left(T-{\frac {l}{c}}\right)\right]$ Moving clock: $\beta \left[\left(T-{\frac {l}{c}}\right)-{\frac {v}{c^{2}}}x\right]$

He then described the symmetrical case when the photographic image of P is made by a moving observer B, discussed the special cases $T=T'=0$ , which together with $x=0$ and $l=y$ and by considering time-of-flight effects, allows to indirectly recover both length contraction and time dilation as well as their symmetry by analyzing the data given on the images.

Finally he clarified Mangoldt's claim according to whom one can directly demonstrate length contraction without considering time-of-flight effects: He showed that this can only be done if one strictly follows Mangoldt's requirement that the rod length $x$ must be very small compared to distance $y$, which indeed produces the Lorentz contraction formula $x'=\beta x$ because the differences in the emission times at the endpoints are so small that they approximately happened simultaneously.

a) His formula for the visual position of a photographed marker in motion easily allows us to derive the visual length between two points: We place R to the origin $x_{0}=0$ so that point P is at position $x=l$, thus we can interpret R and P as endpoints of a horizontally aligned and moving rod measured at time $T=0$, which gives (using modern notation $\gamma$ for the Lorentz factor):

$x'_{r}=\gamma \left[x-v\left(T-{\frac {l}{c}}\right)\right]{\underset {x=l \atop T=0}{=}}\gamma \left[l-{\frac {v}{c}}l\right]=l{\sqrt {\frac {1-v/c}{1+v/c}}}$

equivalent to equation (2) by identifying his length $l$ with visual length $L$ and coordinate $x'_{r}$ as the right endpoint of a rod of rest length $l'$ whose left endpoint is at the origin. So Grünbaum correctly incorporated time-of-flight effects in order to derive the visual position of points with respect to photographs. Yet he treated the visual length of moving rods only in approximation while clarifying Mangoldt's statements.

b) In a short review of Grünbaum's paper, w:Max von Laue (1911) described it as using photographic images to demonstrate Lorentz contraction.

Petzoldt (1914)

In an otherwise philosophical analysis of relativity, w:Joseph Petzoldt demonstrated the following consequence:

The perspectivistic shape changes of bodies whose velocity is comparable to that of light, do not appear to have been investigated yet. [..] The image of an object, which is generated on the retina of the eye or on the photographic plate - as a snapshot - is produced by light coming from all the points of the object in question, but it wasn't sent at the same time: all of the light that contributes to the production of the image arrives simultaneously with the light that originated from those points of the object that were closest to the eye or plate; the further the object's point is from the plate or eye, the sooner it must send its contribution of light. The result is that the image of a receding or approaching object having a large speed comparable to that of light, does not correspond to the image of the same object if it were at rest relative to the eye or plate, but rather gives the impression that it originated from a shorter or longer object within that radius of distance. In case the object is receding from the eye or plate, the light contributed from the most distant points would have been sent later, in the case of approach it would have been sent earlier than in the state of rest. A simple calculation that takes into account the Lorentz contraction, results in the fact that the object in question whose physical length is $l{\sqrt {1-{\tfrac {v^{2}}{c^{2}}}}}$, where $l$ is the length for the "co-moving" observer, would have to be assessed on the basis of this photographic momentary image as having the perspectivistic moving length:

$l{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\cdot {\frac {c}{c+v}}\qquad$ or $\qquad l{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\cdot {\frac {c}{c-v}}$,

depending on the whether the object recedes or approaches the eye or plate.

a) Petzoldt correctly derived the visual length of moving 1D rods, his results being equivalent to equation (2):

${\begin{matrix}l{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\cdot {\frac {c}{c-v}}=l{\sqrt {\frac {1+v/c}{1-v/c}}}\\l{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\cdot {\frac {c}{c+v}}=l{\sqrt {\frac {1-v/c}{1+v/c}}}\end{matrix}}$

b) As explained in Petzoldt's Wikipedia article, it was w:Albert Einstein himself who publicly recommended Petzoldt's 1914 paper in a newspaper article and privately praised it by letter, which implies that by 1914, Einstein should have been aware of the pitfalls that arise when describing photographs of Lorentz contracted bodies. However, in the s:first appendix to his popular book w:Relativity: The Special and the General Theory that was added to the third edition from 1918 and still remains in modern editions, he seems to have overlooked this fact. Einstein argued that $\Delta x=1/a$ is the length indicated on a "snapshot" or "instantaneous photograph" of a unit length at rest in K' that was taken from system K at time $t=0$, and that $\Delta x'=a\left(1-v^{2}/c^{2}\right)$ is the length indicated on a snapshot of a unit length at rest in K that was taken from system K' at time $t'=0$; since symmetry requires that $\Delta x=\Delta x'$, he concluded that $a^{2}=1/\left(1-v^{2}/c^{2}\right)$. If we plug his result into his snapshot length equations we get $\Delta x={\sqrt {1-v^{2}/c^{2}}}$ and symmetrically $\Delta x'={\sqrt {1-v^{2}/c^{2}}}$, identical to the Lorentz contraction of unit rest lengths. Yet while Einstein's derivation is correct, his choice of visualizing this result in terms of "snapshots" can be misleading since one has to consider time-of-flight effects which leads to visual deviations in most cases.

Lorentz (1914)

w:Hendrik Lorentz (March 1914) described two rulers along line EF, one at rest with respect to observer A having length $l'=l$ and the other moving with respect to A with velocity $v$ having contracted length $l'=l/a$ (where $a$ is the Lorentz factor). He then derived a general formula for the approximated visual length $s$ of the rods as a function of $l'$, using an photographic apparatus moving with velocity $w$. On the screen of the apparatus there is a small slot D situated at vertical distance L from E, and below there is a photographic plate at vertical distance $\lambda$ from D. Then slot D is opened for one moment, during which it is passed by optical signals from endpoints F and G of the rods (where $FG=l'$), reaching points f and g on the plate.

Since the signals reaching D at the same time are sent from different locations, they were not emitted simultaneously. Assuming the signal from F was sent at time $t_{0}$ at position $x_{0}$ after which it traverses distance ${\sqrt {L^{2}+x_{0}^{2}}}$ to D, it follows that the signal from G was sent at an earlier time $t_{0}+\tau$ at position $x_{1}=x_{0}+l'+v\tau$ after which it traverses distance ${\sqrt {L^{2}+\left(x_{0}+l'+v\tau \right)^{2}}}$ to D. Thus the value of $\tau$ is given by

${\sqrt {L^{2}+\left(x_{0}+l'+v\tau \right)^{2}}}-{\sqrt {L^{2}+x_{0}^{2}}}=-c\tau$

or using angle $EDF=\phi _{0}$ and assuming very large L by which terms containing $l'^{2}$ and $\tau ^{2}$ can be neglected:

$\tau =-{\frac {\sin \phi _{0}}{c+v\sin \phi _{0}}}l'$, thus $x_{1}=x_{0}+{\frac {cl'}{c+v\sin \phi _{0}}}$.

After deriving the position of points f and g on the plate, and under consideration that the plate was moving with velocity $w$, he obtained the general formula representing the approximated visual length of rods:

$s={\frac {\lambda }{L}}{\frac {c+w\sin \phi _{0}}{c+v\sin \phi _{0}}}l'$.

The visual length $s_{1}$ of the resting rod follows by plugging in $v=0$ and $l'=l$, whereas visual length $s_{2}$ of the moving rod follows by plugging in $l'=l/a$ (where $a$ is the Lorentz factor), thus

${\frac {s_{1}}{s_{2}}}=\left(1+{\frac {v}{c}}\sin \phi _{0}\right)a$.

The case $\phi _{0}=0$ allows to recover the Lorentz contraction formula

${\frac {s_{1}}{s_{2}}}=a$,

which is possible because in this approximation neglecting terms containing $l'^{2}$, one can pretend as if the emission of the light signals at F and G happened simultaneously.

a) Setting $\phi _{0}=\pi /2$ (i.e. 90 degrees), Lorentz's formula produces

${\frac {s_{1}}{s_{2}}}=\left(1+{\frac {v}{c}}\right)a={\sqrt {\frac {1+v/c}{1-v/c}}}$

equivalent to the longitudinal visual length formula (2), independently derived by § Petzoldt (1914).

b) Lorentz's last case $\phi _{0}=0$ agrees with §§ Mangoldt (1910)​ and Grünbaum (1911) who argued that Lorentz contraction can be recovered from the data on photographs under certain approximations.

Lorentz (1922)

In 1922 (published 1927), w:Hendrik Lorentz simplified his earlier treatment of 1914 by writing:

Let there be two “equal” rods P and Q, placed along the line AB parallel to OZ, and let the first be at rest in x, y, z, t, the other in x’, y’, z’, t’. The screen CD coincides with the plane YOZ, the distance OK being $p$, and EF is the photographic plate, at a distance $OM=q$ behind the screen.

In A’s experiment the screen and the photographic plate are at rest in the system x, y, z, the pinhole being constantly at the origin O of those coordinates. We shall suppose it to be opened for an instant at time $t=0$, and we shall define the “effective” position of a moving point by the condition that light emitted by the point in that position shall reach the hole O just when it is opened; the time at which the effective position is reached may be called the effective time. We shall denote by $2l$ the length of the rod P in the system x, y, z, so that in the same system $2a/l$ [with $a$ as Lorentz factor] is the length of Q. For the sake of simplicity we shall suppose $l$ to be so small that terms with $l^{2}$ may be neglected.

Now A must be sure that he compares simultaneous positions of the ends of the moving rod Q. He can satisfy this condition by arranging the experiment in such a way that the middle point of Q passes through $K(z=0)$ at the time $-p/c$. We shall further assume that the middle point of P always has this position.

It is easily seen that in this first experiment the images of the two rods have the lengths $2{\tfrac {q}{p}}l$ and $2{\tfrac {q}{p}}{\tfrac {l}{a}}$, the second being $a$ times shorter than the first.

a) Lorentz's method, in which both endpoints of the rod have the same distance to hole O, can indeed accurately represent the different lengths of resting and moving 1D rods on a photograph.

b) As later shown by § Penrose (1958/59) and Terrell (1959), this method doesn't work anymore when 3D objects are considered.

Lampa (1924)

w:de:Anton Lampa devoted an entire paper to the question of the visual length of moving rods. He pointed out, that the relativistic length contraction formula doesn't say anything about the visual appearance of moving rods from the perspective of non-comoving observers. He started by defining a rod of rest length $l'$ with endpoint coordinates $x'_{A}=y'_{A}=0$ and $x'_{B}=l',\,y'_{B}=0$, letting a light signal be sent from B to the origin starting at time $t'_{B}=-l'/c$ and arriving at time $t'=0$. The Lorentz transformation into another frame K which is moving parallel to the x-axis gives:

${\begin{aligned}x_{B}&={\frac {x'_{B}+vt'_{B}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {l'-{\frac {v}{c}}l'}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=l'{\frac {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}{1+{\frac {v}{c}}}}\\t_{B}&={\frac {t'_{B}+{\frac {v}{c^{2}}}x'_{B}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {-{\frac {l'}{c}}+{\frac {v}{c^{2}}}l'}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=-{\frac {l'}{c}}{\frac {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}{1+{\frac {v}{c}}}}\end{aligned}}$

from which he derived the Lorentz contracted length as follows:

$l=x_{B}+v(0-t_{B})=x_{B}-vt_{B}=l'{\sqrt {1-v^{2}/c^{2}}}$

As in the case of § Mangoldt (1910), Lampa's position formula that incorporates the time-of-flight of light

$x_{B}=l'{\frac {\sqrt {1-v^{2}/c^{2}}}{1+v/c}}=l'{\sqrt {\frac {1-v/c}{1+v/c}}}$,

is equivalent to equation (2) by re-interpreting $x_{A}=0$ and $x_{B}$ as endpoints defining the visual length of the rod.

In order to explicitly derive the visual length of the rod, he discussed a more general case in which the endpoints A and B of the rod at rest in frame K' are given by coordinates $x'_{A}=0,\,y'_{A}=y'$ and $x'_{B}=l',\,y'_{B}=y'$. So when an observer resting at the origin sees the rod at time $t'=0$ , the rod's light must have been sent earlier at time $t'_{A}=-y'/c$ from A and at an even earlier time $t'_{B}=-{\sqrt {y^{\prime 2}+l^{\prime 2}}}/c$ from B. Applying the Lorentz transformation, he showed that the visual distance $\lambda$ between A and B seen by an observer at the origin of another frame K which is moving parallel to the x-axis, is given as follows:

$\lambda =x_{B}-x_{A}={\frac {l'-{\frac {v}{c}}\left({\sqrt {y^{\prime 2}+l^{\prime 2}}}-y'\right)}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

He concluded:

This distance is a function of $v/c$ at given $l'$ and $y'$. If $v=0$ or $v/c=0$, we have $x_{B}=l'$ and $x_{A}=0$, thus $\lambda =l'$ as it must be, because the coordinate systems always coincide at $v=0$. If we increase $v/c$ beginning with value zero, then $x_{B}$ decreases and finally becomes negative, while the absolute value of $x_{A}$ steadily increases; $x_{B}-x_{A}$ initially decreases, then reaches a minimum and then permanently increases. If $v/c=l'/{\sqrt {y^{\prime 2}+l^{\prime 2}}}$, then $x_{B}=0$ and $x_{A}=-l'$ and $\lambda$ becomes equal to $l'$. We see that the distance of the points from which the light rays emanate as measured in system K of the stationary observer, is not at all smaller than the length of the rod measured in the coordinate system attached to it; it even becomes infinitely large if the rod moves relative to the observer with the speed of light. Yet the visual angle under which the stationary observer sees points A and B of the rod, is always finite.

In frame K, he defined $\alpha$ as angle of the light ray emanating from endpoint A at time $t_{A}$ and $\beta$ as angle of the light ray emanating from endpoint B at time $t_{B}$, and then derived the relation between the w:visual angle $\varphi =\beta -\alpha$ observed in K and angle $\varphi '$ observed in K' by:

${\begin{matrix}\sin \varphi =\sin(\beta -\alpha )=\sin \beta \cos \alpha -\cos \beta \sin \alpha \\\sin \varphi '={\frac {l'}{\sqrt {y^{\prime 2}+l^{\prime 2}}}},\ \cos \varphi '={\frac {y'}{\sqrt {y^{\prime 2}+l^{\prime 2}}}}\\\Downarrow \\\sin \varphi ={\frac {\sin \varphi '-{\frac {v}{c}}(1-\cos \varphi ')}{1-{\frac {v}{c}}\sin \varphi '}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\end{matrix}}$

He finally showed how to recover the Lorentz contracted length $l$ from visual length $\lambda$:

${\begin{aligned}l&=\lambda +v\left(t_{A}-t_{B}\right)\\&=l'{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\end{aligned}}$

which he rewrote in terms of angles $\alpha$ and $\beta$ as:

${\begin{aligned}\lambda &=y'(\tan \beta -\tan \alpha );\quad t_{A}-t_{B}={\frac {y'}{c}}\left({\frac {1}{\cos \beta }}-{\frac {1}{\cos \alpha }}\right)\\\Rightarrow l&=y'\left[(\tan \beta -\tan \alpha )+{\frac {v}{c}}\left({\frac {1}{\cos \beta }}-{\frac {1}{\cos \alpha }}\right)\right]\end{aligned}}$

a) Lampa's formula for $\lambda$ is more general than equation (2), producing the latter in the special case $y'=0$:

$\lambda ={\frac {l'-{\frac {v}{c}}\left({\sqrt {y^{\prime 2}+l^{\prime 2}}}-y'\right)}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{\underset {y'=0}{=}}{\frac {l'-{\frac {v}{c}}l'}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=l'{\sqrt {\frac {1-v/c}{1+v/c}}}$

b) He was evidently unaware of previous work on visual positions and lengths by §§ Grünbaum (1911), Petzoldt (1914), and Lorentz (1914).

c) In a short review of Lampa's paper, w:Cornelius Lanczos (1925) described it as deriving the relation between the visual angles of a moving rod seen by resting and moving observers, as well as deriving the length of the rod from the data of its appearance.

Penrose (1958/59) and Terrell (1959)

§§ Petzoldt (1914), Lorentz (1914), and Lampa (1924) restricted their investigations to the visual appearance of 1D rods, whereas the only early description of the visual appearance of 3D objects was given by § Plummer (1910) in relation to the celestial sphere based on the conformal property of aberration. However, these early investigations were evidently completely forgotten, and it took decades before those questions were taken up again:

Using the conformal property of aberration, w:Roger Penrose (1958/59) described the visual appearance of spheres, and discovered a completely new effect: Due to the motion of the sphere, points that were in the background become visible due to the sphere's motion, leading to the effect that the sphere appears not to be Lorentz contracted but rather rotated. Independently, w:James Terrell (1959) noticed the same rotation effect in relation to 3D rods, leading him to famously declare the "invisibility of the Lorentz contraction". Independently, the special case of the visual length of 1D rods was discussed by Weinstein (1959/60). A large number of follow-up papers by many authors appeared, which demonstrate the visual appearance of 1D and 3D objects in detail, see w:Penrose-Terrell effect.