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Visual appearance of wireframe objects in special
relativity
Thomas M¨uller
Visualisierungsinstitut der Universit¨at Stuttgart (VISUS)
Allmandring 19, 70569 Stuttgart, Germany
E-mail: Thomas.Mueller@vis.uni-stuttgart.de
Sebastian Boblest
Visualisierungsinstitut der Universit¨at Stuttgart (VISUS)
Allmandring 19, 70569 Stuttgart, Germany
E-mail: Sebastian.Boblest@vis.uni-stuttgart.de
Abstract. The visual appearance of a moving object in special relativity can
be constructed in a straightforward manner when representing the surface of the
object, or at least a wire frame model of it, as a point cloud. The apparent
position of each individual point is then found by intersecting its worldline with
the observer’s backward light cone. In this paper, we present a complete derivation
of the apparent position of a point and some more complex geometric objects
for general parameter settings (configurations). We implemented our results in
python and asymptote and used these tools to generate scripts that create the
figures in this paper. These scripts are directly applicable in an undergraduate
course to special relativity and can also serve as the basis for student projects
with the aim to study more complex sceneries.
PACS numbers: 03.30.+p, 95.75.Pq
1. Introduction
The strange visual appearance of objects moving with a velocity close to the speed
of light relative to an observer is one of the puzzling predictions of Einstein’s special
theory of relativity [1] and was studied already by Lampa [2] in 1924. Unfortunately,
Lampa’s discussion of the apparent shape of a moving rod was not recognized for a
long time and even the famous physicist Gamow gave an incorrect conclusion about
the visual appearance of a moving wheel in his book “Mr. Tompkins in Wonderland”
(edition 1940) [3]. In 1959, Terrell [4] pointed out that the Lorentz contraction is not
visible to an observer, a direction that was similarly pursued by Weinstein [5], while
Penrose [6] proved that a relativistically moving sphere always has a circular outline,
a problem that was again considered by Boas [7]. However, the visual appearance of
relativistically moving objects is one of the consequences of special relativity, where
an intuitive understanding is hard to reach by performing calculations alone. The field
of relativistic visualization bridges this gap between mathematical results and human
imagination.
arXiv:1410.4583v1 [physics.comp-ph] 14 Oct 2014
Visual appearance of wireframe objects in special relativity 2
The rapid increase in computer power and the emergence of very powerful graphics
hardware made the development of several sophisticated techniques possible that are
capable of generating high quality imagery of special and general relativistic scenarios.
The most natural of these methods is relativistic ray tracing, where the physical
propagation of light is being reversed and the finite speed of light is taken into account,
see for example Hsiung and Dunn [8], Weiskopf [9], or M¨uller [10], amongst others.
As this rendering technique is generally very time consuming even on contemporary
computers, it is not used for interactive simulations showing the visual effects of
relativity. A popular alternative method is to transform the polygonal mesh of an
object into the observer’s rest frame [11]. But, this polygon rendering technique
leads to image artefacts because only the vertices are transformed and the connecting
edges are still straight lines. Ray tracing and polygon rendering can be combined to
circumvent the respective disadvantages, however. For that, it is necessary to restrict
oneself to triangular meshes and make use of the high parallelism of graphics processing
units (GPUs) and the free programmability of the graphics pipeline. Details of this
local ray tracing technique are described in M¨uller et al. [12]. A recent survey of
visualization methods for special relativity was given by Weiskopf [13]. The reader
interested in a comprehensive overview is referred to that paper.
Among the first visualization techniques employed in special relativity is to
consider the apparent shape of wireframe models as such images can be generated
also by hand. Various authors published work that uses this technique. Scott and
Viner [14] considered the appearance of plane grids and rectangular boxes, Scott and
van Driel [15] studied, among other things, the look of a sphere passing close to the
observer, however without giving a full description of the scenarios they looked at.
Hickey [16] considered the two-dimensional appearance of a relativistically moving
cube and Suffern [17] again discussed the outline of a relativistically moving sphere,
where he focused on a motion directed towards the observer. One of the first interactive
computer simulations showing the apparent distortion effect at relativistic velocities
is Visual Appearance by Taylor [18]. He also uses wireframe objects but does not give
any inside in how the visualization is accomplished and his program seems to suffer
from polygon rendering artefacts.
In this article, we as well concentrate on wireframes of objects. In contrast
to earlier work, we not only transform the complete edges in between the vertices
according to the Lorentz transformations and the finite speed of light so that we can
properly visualize how straight lines in general appear bent, but also include depth
information to emphasize the apparent shape. While it is clear that other methods
can easily create images of much higher quality, especially by using textures and
simple shading techniques, the wireframe method is still very powerful didactically.
On modern computers such visualizations can be created completely interactive and
students can create their own sceneries and study the effects of special relativity in
these cases. In this article we give a general derivation of the apparent view of lines
and spheres. We allow for a free positioning of these objects in their reference frame,
of the observer in his frame, and of the spatial separation and relative speed of the
two frames with the only restriction that we assume the axes of the two frames to be
aligned. This allows to construct complex scenes on the one hand and to study how
different observers perceive the same scenery on the other hand.
Our results are implemented in asymptote [19] and python scripts that we used to
create the figures in this paper but which, more importantly, may be used in courses
to special relativity or in student projects where other scenes could be constructed and
Visual appearance of wireframe objects in special relativity 3
studied. Our scripts can be downloaded from http://go.visus.uni-stuttgart.de/
srwireframe. With the python scripts, some scenes can also be animated.
The structure of this paper is as follows. In section 2 we recapitulate the Poincar´e
transformation that is the basis for all further calculations. In section 3 we give a
detailed mathematical derivation of the parametrized equations for the apparent view
of a single point, a rod, and a sphere. In section 4, we specialize to some descriptive
examples and compare our wireframe models with the corresponding rendered images
which follow from four-dimensional ray tracing. Appendix A gives some further
examples in forms of exercises.
2. Poincar´e Transformation
Consider two frames of reference Sand S0equipped with their individual coordinate
systems xµ= (x0, x1, x2, x3) = (ct, ~x) and x0µ= (x00, x01, x02, x03) = (ct0, ~x0),
respectively. The clocks of both frames are synchronized to x0=x00= 0 when the
origin of S0is located at ~a with respect to the origin of S, see figure 1. The coordinate
axis of both frames are aligned to each other and S0moves with constant velocity
~
βwith respect to S. We will refer to this setup as being the standard configuration
in special relativity without rotations. Our observer will be at rest in the system S,
while the system S0is the rest frame for our sceneries.
Figure 1. The frame of reference S0is moving with constant velocity ~
βwith
respect to S. Both systems are synchronized to x0=x00= 0 when S0is located
at ~a with respect to S; and their axes are aligned.
The Poincar´e transformation between both frames is defined by
xµ= Λµ
νx0ν+aµ,(1)
and the Lorentz matrix Λµνis given by
Λ0
0=γ, Λ0
i=γβi,Λi
0=γβi,Λi
j=δi
j+γ2
γ+ 1βiβj,(2)
where βiβi=βiβi=~
β·~
β < 1 and γ= 1/p1−βiβi, see e.g. Misner et al. [20],
and δi
jis the Kronecker-δ. We also use Einstein’s sum convention to sum over indices
that appear twice in the same term. The displacement four-vector reads aµ= (0,~a).
Visual appearance of wireframe objects in special relativity 4
Here and in the following, Greek indices run from 0 to 3, where the 0-th coordinate
represents time, and Latin indices go from 1 to 3.
The inverse of the Lorentz matrix ¯
Λ=Λ−1reads
¯
Λ0
0=γ, ¯
Λ0
i=−γβi,¯
Λi
0=−γβi,¯
Λi
j=δi
j+γ2
γ+ 1βiβj,(3)
which differs from the initial Lorentz matrix only by the sign of the velocity ~
β. The
corresponding Poincar´e transformation reads x0µ=¯
Λµν(xν−aν).
3. Apparent view of an object
The finite speed of light is responsible for the fact that we do not see a moving object
where it actually is, but where it was when it sent the light that we now observe.
3.1. Apparent position of a point
In the simplest case, the object is just a point Pand its apparent position can be
determined by intersecting the point’s worldline x0
p, ~xp=~xp(x0
p)with the backward
light cone of the observer, who is static with respect to S,
0 = −x0−x0
obs2+
3
X
i=1 xi−xi
obs2.(4)
(For the rest of this paper, we drop the index pof the point.)
If Pis at rest with respect to the moving frame S0,~x0= const = ~x0
p, we have
to transform its worldline into Sby means of the Poincar´e transformation (1). Then,
equation (4) yields
0 = −Λ0
νx0ν−x0
obs2+δij Λi
νx0ν+ai−xi
obsΛj
µx0µ+aj−xj
obs(5)
=−Λ0
0x00+ Λ0
nx0n−x0
obs2
+δij Λi
0x00+ Λi
nx0n+ai−xi
obsΛj
0x00+ Λj
mx0m+aj−xj
obs.(6)
Here, the only unknown is x00which is the time when light must be emitted by the
point in order to reach the observer at time x0
obs. Solving the quadratic equation (6)
for x00and using the abbreviations
ρ=γ~
β·~x0−x0
obs, ~η =~x0+γ2
γ+ 1 ~
β·~x0~
β+~a −~xobs,(7)
ω2
0=γ2(~
β·~η −ρ)2−ρ2+~η ·~η, (8)
yields
x00=γ~
β·~η −ρ−ω0,(9)
~x =~x0+γx00+γ2
γ+ 1 ~
β·~x0~
β+~a =γx00~
β+~η +~xobs .(10)
Note that in these expressions the scalar product as usual is an abbreviation for
the sum over all products of the vector components, like for example, ~
β·~η =
β1η1+β2η2+β3η3. However, it must not be interpreted with respect to either
one of the reference frames Sor S0, respectively. Hence, it has to be taken by care
how to interpret the situation when the scalar product vanishes, ~
β·~η = 0. In general,
Visual appearance of wireframe objects in special relativity 5
it cannot be interpreted as both vectors being “perpendicular”, because some of them
are a mixture of vectors measured with respect to Sor S0.
Equations (9) and (10) simplify considerably if the point Pand the observer are in
the origin of their respective reference frames, i.e. ~x0=~
0 = ~xobs, and the displacement
vector ~a =~
0. Then,
x00=γ(1 ±β)x0
obs =s1±β
1∓βx0
obs =: Dβx0
obs and ~x =γx00~
β. (11)
While x0
obs <0, the point approaches the observer and we have to use the upper signs
in the square root factor Dβ. After Phas passed the observer, we have to use the
lower signs, respectively. Dβis also called Doppler factor and is responsible for a blue-
or red-shift if the spectrum of the light would be taken into consideration.
We could also accomplish the light cone intersection within the frame S0, where
the observer’s current position at their observation time x0
obs follows from the inverse
Poincar´e transformation, x0µ
obs = (Λ−1)µν(xν
obs −aν), see figure 2.
Figure 2. The intersection between the worldline of the point Pand the
backward light cone of the observer determines the event ~x0
p(x00
p), where Phas
to emit light that is seen by the observer at time x00
obs.
Then, the intersection of the light cone
0 = −x00−x00
obs2+
3
X
i=1 x0i−x0i
obs2(12)
with the static point Pimmediately yields
x00=x00
obs −∆(~x0, ~x0
obs),∆(~x0, ~x0
obs) := v
u
u
t
3
X
i=1 x0i−x0i
obs2.(13)
Visual appearance of wireframe objects in special relativity 6
The apparent position ~x follows from the back transformation by means of
equation (1). This second approach appears to be more straight, but it needs two
Poincar´e transformations.
With the above transformations at hand, we could determine the virtual shape
of any relativistically moving object by means of representing its surface by a cloud
of points. The resulting apparent positions make up the photo-object, which is the set
of all points where light is emitted from the object’s surface that reaches the observer
at the same time. This photo-object is what the observer or its camera will see.
However, the observer’s perception might differ from what he sees and depends also
on the texture of an object; we will come to this point later in Sec. 4.4.
3.2. Apparent view of a line/rod
Instead of a single point, we now consider a straight line segment ~x0+s0~σ0,s0∈[s0
1, s0
2],
that is defined by a specific reference point ~x0and a direction ~σ0with k~σ0k= 1 as
measured in S0. Replacing ~x0in equation (10) by ~x0+s0~σ0yields
~x (s0, ~σ 0) = γx00(s0, ~σ 0)~
β+~η +~xobs +s0~σ0+γ2s0
γ+ 1 ~
β·~σ0~
β, (14)
x00(s0, ~σ0) = γ~
β·~η −ρ−ωline(s0),(15)
where ρand ~η are the same abbreviations as in (7), and
ωline(s0)2=ω2
0+ 2s0υ+s02,(16)
υ=~µ ·~σ0,with ~µ =~η +~
β−γρ +γ2
γ+ 1 ~
β·~η,(17)
with ω0from equation (8). Each point of a line has to emit light at a different time
such that it is being received by the observer at their observation time. Hence, the
apparent shape of the line will not be straight, in general. Details can be determined
using the Frenet-Serret frame along the line segment defined by the tangent ~e1(s0),
the main normal ~e2(s0), and the binormal ~e3(s0) = ~e1(s0)×~e2(s0), where
~e1(s0) = d~x(s0)/ds0
kd~x(s0)/ds0k(18)
with derivative
d~x(s0)
ds0=−γs0+υ
ωline(s0)~
β+~σ0+γ2
γ+ 1 ~
β·~σ0~
β(19)
and corresponding norm
d~x(s0)
ds0
2
=γ2(s0+υ)2
ωline(s0)2β2−2γ2s0+υ
ωline(s0)~
β·~σ0+1+γ2~
β·~σ02
:= f(s0)2.(20)
Note that ~e1, ~e2, ~e3are parametrized by s0but are given with respect to the frame S.
As the Frenet-Serret frame is only valid for a curve parametrized by its arc
length, the main normal cannot be determined directly from the second derivative
of equation (14), but has to be calculated from the derivative of the tangent ~e1(s0).
Thus, ~e2(s0) = (d~e1/ds0)/kd~e1/ds0kwith
d~e1
ds0=−ωline(s0)2−(s0+υ)2
f(s0)3ωline(s0)3~
β−γ2s0+υ
ωline(s0)(~
β·~σ0) + γ+γ3
γ+ 1(~
β·~σ0)2
+~σ0γ2β2s0+υ
ωline(s0)−(~
β·~σ0).(21)
Visual appearance of wireframe objects in special relativity 7
The absolute value of (21) not only yields the normalization factor for the main normal
but it also yields the curvature of the curve, κ(s0) = kd~e1/ds0k, which is given by
κ(s0)2=ωline(s0)2−(s0+υ)2
f(s0)3ωline(s0)32
γ2 s0+υ
ωline(s0)2
γ2β2hβ2−(~
β·~σ0)2i(22)
−2γ2(~
β·~σ0)s0+υ
ωline(s0)hβ2−(~
β·~σ0)2i+hβ2+ (γ2−2)(~
β·~σ0)2−γ2(~
β·~σ0)4i.
The exact form of the binormal ~e3(s0) is of no interest here.
In the special case ~σ0k~
β, or ~
β=±β~σ0, respectively, the straight line points
in the direction of motion, and the curvature in this parallel case κk(s0)≡0 for all
s0∈R. Hence, such lines only change their apparent length but do not appear to be
bent, see also the example in Sec. 4.2.
If ~
β·~σ0= 0, the curvature, equation (22), simplifies considerably. Then,
κ⊥(s0) = ωline (s0)2−(s0+υ)2
f(s0)2ωline(s0)3γβ (23)
with maximum given at s0
max =−υ=−~η ·~σ0. Thus, κ⊥
max =γβ/ωline(−υ). As to be
expected, the tangent and the main normal read ~e⊥
1(−υ) = ~σ0and ~e⊥
2(−υ) = −~
β/β,
respectively. The apparent line, equation (14), reduces to
~x⊥(s0, ~σ 0) = ~p −γωline(s0)~
β+s0~σ0(24)
with the reference point ~p =γ2[x0
obs +~
β·(~a−~xobs)]~
β+~η+~xobs. To show that ~x⊥(s0, ~σ0)
has the form of a hyperbola, the local coordinates (ζ, ξ) with respect to the coordinate
system spanned by ~
βand ~σ0are defined. With ξ=~σ0·~x⊥=~σ0·~p +s0=ξ0+s0and
ζ=~
β/β ·~x⊥=~
β/β ·~p −γβωline(s0), the ansatz (ζ−ζ0)2/a2−(ξ−c)2/b2= 1 yields
1 = (ζ−ζ0)2
γ4β4/(κ⊥)2−(ξ−ξ0+υ)2
γ2β2/(κ⊥)2,(25)
where ζ0=γ2β[x0
obs +~
β·(~a −~xobs )] + γ(~
β·~x0) + ~a ·~
βand ξ0= (~x0+~a)·~σ0.
The osculating circle at the point of maximum curvature has radius (κ⊥)−1and
is centred at ~m =~x⊥(−υ , ~σ)+(κ⊥)−1~e⊥
2(−υ). The corresponding local coordinates
read ζm=ζ0−γ2(κ⊥)−1and ξm=~xobs ·~σ0.
3.3. Apparent view of a sphere
The surface of a sphere within the reference frame S0can be defined by the central
point ~x0, the orthonormal basis vectors {~σ0
1, ~σ0
2, ~σ0
3}, the radius r0, and the spherical
coordinates ϑ0∈(0, π) and ϕ0∈[0,2π). An approach similar to the one for the line,
where ~x0is now replaced by ~x0+r0sin ϑ0cos ϕ0~σ0
1+r0sin ϑ0sin ϕ0~σ0
2+r0cos ϑ0~σ0
3=
~x0+P3
i=1 s0
i~σ0
iin equation (10), yields,
~x =γx00~
β+~η +~xobs +
3
X
i=1
s0
i~σ0
i+γ2
γ+ 1 3
X
i=1
s0
i~σ0
i·~
β!~
β, (26)
x00=γ~
β·~η −ρ−ωsph(s0
1, s0
2, s0
3),(27)
where
ωsph(s0
1, s0
2, s0
3)2=ω2
0+ 2~µ ·
3
X
i=1
s0
i~σ0
i+r02,(28)
Visual appearance of wireframe objects in special relativity 8
and ~µ is the same expression as in equation (17).
Equations (26) and (27) simplify considerably if the sphere’s center point ~x0, the
observer position ~xobs, and the system offset ~a vanish identically. Additionally, the
basis vectors ~σ0
1,~σ0
2, and ~σ0
3are equal to the standard basis vectors ~e0
1= (1,0,0)T,
~e0
2= (0,1,0)T, and ~e0
3= (0,0,1)T, and the velocity ~
β= (β, 0,0)Thas only a non-
vanishing component in the x1-direction. Then,
ω2
sph =γ2β2x0
obs2+ 2γx0
obsβr0sin ϑ0cos ϕ0+r02,(29)
x1=γγx0
obs −ωsphβ+γr0sin ϑ0cos ϕ0,(30)
x2=r0sin ϑ0sin ϕ0,(31)
x3=r0cos ϕ0.(32)
As expected, the x2- and x3-components are not influenced, because the sphere
only moves along the x1-direction and the other parameters are like in the standard
literature.
The silhouette of a sphere always appears to be circular irrespective of the sphere’s
motion, as shown already by others. We give a short sketch in Appendix B of how
this could be proven.
4. Examples
In the following, we will present some typical examples. All of them can be reproduced
by the accompanying asymptote and python scripts. We also compare the wireframe
representations with the corresponding images rendered using the four-dimensional
ray tracing code GeoViS [10]. The great advantage of the python scripts is the
possibility to animate the scenes without delay while ray tracing codes might take
several minutes to render an image sequence which has to be concatenated into a film
afterwards. Besides the script names mentioned in the figure captions, we use the
common script sr apparent that contains the calculation of the apparent positions
discussed in the previous sections. Note that script names without file ending are valid
for asymptote as well as python.
4.1. Eye or camera transformation
In section 3, we deduced the apparent position of a single point, a point on a line,
or a point on a sphere. This apparent position is the position in space where light
has to be emitted by the point in order to reach the observer at their observation
time. The next step is to map the apparent position of the point into the eye or the
camera of the observer which we both represent by a pinhole camera. For that, we first
transform the apparent position into the camera’s standard reference frame by means
of the View matrix. Then, the Projection matrix emulates the perspective projection
of the pinhole camera. The View and Projection matrices are defined in sr camera,
see also Appendix C. For further details, we refer the reader to the standard literature
of computer graphics like, e.g., Foley [21] or Shirley et al. [22].
4.2. Apparent view of a line/rod oriented along its direction of motion
The most fundamental object besides a point is a straight line or rod. If the rod’s
orientation is alongside its direction of motion, then equations (14) and (15) can be
Visual appearance of wireframe objects in special relativity 9
simplified. Thus, with ~
β=β~σ0,~a =~
0, ~x0=~
0 and s0∈[−l0/2, l0/2], we obtain
~x(s0, ~σ 0) = γx00(s0, ~σ0)β+s0~σ0,(33)
x00(s0, ~σ0) = γ−β~σ0·~xobs +x0
obs−ωline (s0),(34)
ωline(s0)2=γ2−β~σ0·~xobs +x0
obs2−x0
obs2+k~xobsk2+ 2s0υ+s02,(35)
υ=γβx0
obs −~σ ·~xobs .(36)
If additionally ~xobs =~
0, the direction ~σ0is insignificant, and the rod can only move
towards or away from the observer. Then, the apparent length l2
ap =k~x(l0/2, ~σ 0)−
~x(−l0/2, ~σ 0)k2of the rod is given by
l±
ap =γl0(1 ±β),(37)
where the upper (lower) sign represents the approaching (receding) rod. This is also
true for the slightly more general case ~xobs =ξ~σ. The Minkowski diagram, figure 3(a),
depicts this situation. At observation event O1, the apparent length of the approaching
rod is determined by the x1-coordinates of the events P1land P1r. Thus, for β= 0.5,
we obtain l+
ap ≈1.732 l0. At O2, the rod recedes from the observer and has an
apparent length l−
ap ≈0.577 l0. The length lm, measured by two observers who are
x1
x0
1
1
x′1
x′0
l′
W1W2
O1
O2
P1l
P1r
P2l
P2r
(a) Minkowski diagram, β= 0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
velocity β
length
l−
ap
l+
ap
lm
(b) Length depending on β
Figure 3. (a) A rod of length l0= 1 with respect to S0is oriented along its
direction of motion ~
β=β~σ0, where ~σ0points in the positive x1-direction of S. In
this Minkowski diagram, the projections of the green lines onto the x-axis yield
the apparent lengths of the rod. (b) Apparent length l±
ap of the approaching (red)
and the receding (blue) rod. The green line represents the measured length lm.
in synchronicity with respect to S, follows from the Lorentz-Fitzgerald contraction
equation: lm=l0/γ. Thus, a rod which approaches the observer always appears
longer than it actually is. A receding rod, however, appears to be even shorter than
its measured length with respect to S. Figure 3(b) shows the apparent lengths l±
ap and
the measured length lmboth as functions of the velocity β.
Strictly speaking, if a line (rod) is oriented alongside its direction of motion, only
a point (the tip) is visible. Even if the line or rod is slightly off-axis, the perspective
projection has to be taken into account which prevents the observer from seeing the
calculated apparent lengths.
Visual appearance of wireframe objects in special relativity 10
4.3. Apparent view of a die
Consider a row of 8 dice with edge length l= 0.5 at rest in the reference frame S.
The centre of the n-th die is located at ~xn= (0,−10 + n·2,−0.75)T,n= 0,...,7.
Another die of the same size is at rest in the centre of the reference frame S0, while
the frame S0itself moves with velocity ~
β= (0,0.9,0)T. An observer located at ~xobs
looking into the direction of the origin of Swill see the row of dice and the moving
die as shown in figure 4.
(a) Rendered, ~xobs = (14.422,0,2)T
34567
(b) Wireframe model, ~xobs = (14.422,0,2)T
(c) Rendered, ~xobs = (8,12,2)T
0123
4
5
6
7
(d) Wireframe model, ~xobs = (8,12,2)T
Figure 4. Apparent view of a die with edge length l0= 0.5 and velocity
~
β= (0,0.9,0)Tmoving above a row of static dice. The observer’s pinhole camera
has 32◦×16◦field of view. The front edges of all die are coloured red to
make their orientations easier to recognize. The observation times are x0
obs =
{14.012,15.1,16.12}. In the top row, the observer is at ~xobs = (14.422,0,2)T.
In the bottom row the observer has the same distance to the x3-axis but she is
located at ~xobs = (8,12,2)T, i.e. by an angle ϕ= 56.3◦shifted away from the
x1-axis. (Script: appDie, animDie.py).
Although the observation times for the moving die are equal in figure 4, the
apparent positions differ dependent on the position of the observer. If she looks
perpendicular to the row of dice, the light travel times are nearly the same. But that
is no longer true if the observer has a tilted view to the row. Light from the rearmost
position needs much more time than from a closer position. Hence, the distances
between the apparent positions are longer.
4.4. Apparent view of a circle/ball
Penrose [6] has already shown in 1959 that the apparent shape of a relativistically
moving sphere is again a sphere. However, the shape of the photo-object of the sphere
is more similar to an ellipsoid, see figure 5. Hence, the observer will “see” an ellipse but
the projection on his plane of sight leads to a circular outline and his brain therefore
interprets it as a rotated sphere.
Visual appearance of wireframe objects in special relativity 11
observer
x1
x2
x0
obs = 3.0x0
obs = 6.0
fov
x0
obs = 10.0x0
obs = 25.0
Figure 5. Apparent shapes (red) of a moving circle for an observer located at
~xobs = (0,−10,−1)T, an offset ~a =~
0, a velocity ~
β= (0.9,0,0)T, and observation
times x0
obs. The central point of the circle (black dot) is at ~x0= (0,0,0)Twith
respect to S0. The grey-dashed disks represent a row of static circles. (Script:
appCircle, animCircle.py)
In order to follow the apparent image of a moving circle/ball, the camera has
to point in the direction of the apparent position of the circle’s/ball’s centre ~x0=~
0.
In the standard configuration, ~a =~
0, ~xobs = (0,−yobs,0)T,~
β= (β, 0,0)T, we have
ρ=−x0
obs and ~η =−~xobs . Thus, the camera has to follow the apparent point ~xc,
~xc=γβx00,with x00=γ x0
obs −qγ2β2(x0
obs)2+y2
obs.(38)
Figure 6 shows a ball moving along the x1-axis in positive direction above a row
of static balls where the axes of all of them point in the same direction. The image
rendered using GeoViS demonstrates clearly that the moving ball still appears as a
ball but appears to be rotated only. If we visualize this situation with our standard
wireframe model, we lose the spatial impression because the lines on the front and
on the back of the sphere intersect on the plane of sight, resulting in a “cluttered”
impression. This of course is an intrinsic property of our wireframe models. However,
the transformation into the plane of view using the view and projection matrices
preserves depth information in the ˆp3-component of the projected point, see Appendix
C for a short discussion. We use this information to draw lines closer to the observer
thicker and with stronger colours than lines further away. In fact, this can even help us
to extract information that is not perceivable in the rendered images, namely that the
right pole of the moving sphere is the part closest to the observer as we can already see
in figure 5. This can nicely be seen in figure 6(b) while it is not visible in the rendered
image 6(a). However, the scripts that produce these figures are significantly more
complicated, as we have to subdivide the picture in small line segments, sort them
with respect to their depth value and draw them in depth-ascending order. Therefore,
we also include simpler scripts that do not use depth information but which might be
easier to read.
4.5. Apparent view for close encounters
In our previous examples the distortion effects due to the finite speed of light are
relatively small, because the distance of the observer to the objects is large in
Visual appearance of wireframe objects in special relativity 12
(a) Rendered (b) Wireframe model
Figure 6. A chequered ball of radius r0= 1 moves with β= 0.95 along the
x1-axis in positive direction. The lower balls are static and are positioned at
~xn= (−17.5 + n·2.5,0,−2.2)T,n={4,5,6}. The observer is located at
~xobs = (0,−10,−1)T. The camera’s field of view is 40◦×30◦. The inclusion
of depth information in the wireframe picture reveals that the region around the
moving sphere’s right pole is closest to the observer as can also be seen from
figure 5 but which is hardly visible in the rendered image. (Scripts: appSphere
and appSphereZ.asy with depth information)
comparison to their size. If the observer’s distance is comparable to the object’s scale,
the time of flight for light rays originating from different locations on the object’s
surface varies strongly. Hence, the observer sees different regions of the object at very
different times and therefore locations and so the object appears strongly distorted.
4.5.1. Apparent view of a line/rod oriented perpendicular to its direction of motion
We again start with the discussion of a moving rod, but contrary to section 4.2 we
now assume it to be aligned perpendicularly to its direction of motion. In this case,
the apparent view becomes more interesting. Let ~xobs =~a =~
0 = ~x0,~
β= (β, 0,0)T,
and ~σ0= (0,1,0)T. Then, ~
β·~σ0= 0, ρ=−x0
obs,~η =~
0, and υ= 0. Furthermore,
ωline(s0)2=γ2β2(x0
obs)2+s02.
As already discussed in section 3.2, the perpendicularly oriented line appears as
a hyperbola which can be described by the implicit equation
1 = (x1−γ2βx0
obs)2
γ4β4(x0
obs)2−(x2)2
γ2β2(x0
obs)2.(39)
The apex resides on the x1-axis with curvature κ⊥(s0= 0) = 1/x0
obs, see equation (23).
The centre of the osculating circle has coordinates x1=γ2(βx0
obs − |x0
obs|) and x2= 0,
see figure 7. At x0
obs = 0, the hyperbola degenerates to a corner. As light rays
originating from points close to the middle of the rod take much less time to reach the
observer than those from its outer parts, the observer sees the outer parts at earlier
times and, hence, at larger distances than the centre and the rod appears to be bent.
Figure 8 illustrates this situation for a rod moving with β= 0.75. At observation time
x0
obs =−0.5, were x0
obs = 0 is defined as the time when the rod reaches the observer, the
moving rod is already very close to the observer, but the light rays from its outer parts
left its surface as early as approximately x0
obs =−3.0 and therefore the observer gets
Visual appearance of wireframe objects in special relativity 13
x1
x2
x0
obs =−1.0−0.5−0.1 0.5 1.0 3.0
Figure 7. A rod of length l0= 4 is oriented perpendicularly to its direction of
motion, ~
β·~σ0= 0. Here, β= 0.75. The observer is located at ~xobs =~
0 and ~a =~
0.
The osculating circle has radius (κ⊥)−1=|x0
obs|. (Script: appRod, animRod.py)
x1
x2
x0
obs =−0.5
observer
Figure 8. Photo-object for a rod with length l0= 4 moving with β= 0.75
towards the observer. At observation time x0
obs =−0.5, the observer receives
light rays from the outer parts of the rod that started already at approximately
x0
obs =−3.0. (Script: appRodLight.asy, animRodLight.py)
the impression that the rod is still quite far away. Figure 9 shows a moving rod which is
described by a cuboid with lower left corner ~c0
ll = (−0.1,−1.0,−0.1)Tand upper right
corner ~c0
ur = (0.1,1.0,0.1)T. The longitudinal direction is oriented along the x2-axis
and the rod moves along the positive x1-direction. In figure 9(a) we show an example
of polygon rendering, where only the rod’s vertices are transformed to their apparent
positions. This technique obviously is insufficient to correctly visualize situations
where strong distortions appear, because the edges connecting the vertices remain
straight lines, for a more detailed discussion see [12]. Figures 9(b) and 9(c) compare
Visual appearance of wireframe objects in special relativity 14
the results of the four-dimensional ray tracing with GeoViS and of our wireframe
model asymptote script. As we transform the entire edges and not just the vertices,
their hyperbolic shape becomes apparent.
(a) Polygon rendering
(b) Rendered (c) Wireframe model
Figure 9. A rod moves with β= 0.9 along the positive x1-direction. The
observation time is given by x0
obs =−0.104 and the observer looks along the
negative x1-direction. The front edges of the rod are coloured red to make its
orientation easier to recognize. (Script: appRodView, animRodView.py)
4.5.2. Sphere and Cube in close fly by We conclude our examples with a comparison
of a cube and a sphere at rest with their moving counterparts closely passing the
observer, see figures 10 and 11.
In both cases the scene is chosen such that the apparent centre of the moving
object coincides with the centre of the static object. Contrary to the rod example,
these cases are not symmetric because the objects are not moving towards the observer.
In the cube example, the different appearances of lines oriented perpendicularly
or parallely to their direction of motion becomes quite apparent. The upper and lower
edges of the cube are oriented almost parallely to the direction of motion and hence
appear straight. On the other hand the edges of the back and the front are oriented
almost perpendicularly to the direction of motion and appear bent. This effect is
much stronger for the front of the cube than for its back, because these edges are close
to the observer and the flight times for the light rays from different points on these
edges differ more strongly.
The sphere example clearly shows that the sphere retains its circular shape while
its surface is strongly distorted, in accordance with the results by Penrose [6]. However,
it appears larger than the sphere at rest, see also figure 12. Please note again that
the centres of the moving sphere and the one at rest coincide so this is indeed a, well-
known, relativistic effect. This example also again impressively demonstrates, how we
can enhance the visual impression by including depth information.
Visual appearance of wireframe objects in special relativity 15
(a) Cube at rest (b) Cube with β= 0.9
(c) Cube at rest (d) Cube with β= 0.9
Figure 10. Apparent distortion of a cube with edge length l0= 0.5 in close fly
by. The camera has a field of view of 32◦×32◦. The observer is at position
~xobs = (0.5,2,0.05)T, i.e. at a distance d= 1.5 to the x3-axis, and the cube
moves along the positive x2-direction. (Script: appCube)
5. Summary
In this article we derived general equations that describe the apparent view of
relativistically moving points, lines, and spheres. We implemented our results in
asymptote and python scripts and generated some exemplary scenes of wireframe
objects and compared our results for these cases with images created with a four-
dimensional ray tracer. We showed that by taking into account the depth information,
our wireframe figures can provide a realistic impression of the special relativistic
Visual appearance of wireframe objects in special relativity 16
(a) Sphere at rest (b) Sphere with β= 0.9
(c) Sphere at rest (d) Sphere with β= 0.9
Figure 11. Apparent distortion of a sphere with radius r0= 0.5 in close fly
by. The camera has a field of view of 50◦×50◦. The observer is at position
~xobs = (0.5,2,0.05)T, i.e. at a distance d= 1.5 to the x3-axis, and the
sphere moves along the positive x2-direction. (Scripts: appSphereSingle and
appSphereSingleZ.asy with depth information)
distortion effects. The tools that we created are very flexible and may be used to
study other scenes, while the examples that we created can already serve as an aid in
teaching of the visual appearance of relativistically moving objects.
Visual appearance of wireframe objects in special relativity 17
x2
x1~xobs
~x
α′
α
Figure 12. Two-dimensional analog of figure 11. The angular size α0of the
apparent sphere, represented by the red photo-object, is greater than the angular
size αof the static sphere (gray disk) at the apparent position of the sphere’s
centre, here ~x =~
0.
Appendix A. Further examples
In the following, we will give some additional examples that could be used directly in
the classroom either for demonstration purposes or as exercises.
Exercise 1: Given a rod of length l0= 4 which moves perpendicularly to
its orientation towards an observer, see Figs. 7 and 8. Play around with
the velocity βand explain why the rod appears to be bent stronger the faster
it moves.
Result: When the velocity of the rod comes ever closer to the speed of
light, the light travel times from the different positions of the rod to the
observer become more and more diverse. Thus, light from the top of the
rod has to start ever earlier than light from the center of the rod in order to
reach the observer at the same time which results in an increasing bending
of the rod.
Configure file: demoRodLight.py
Exercise 2: Given a sphere of radius r0= 0.5moving along the positive
x1-direction with velocity β. The observer is located at ~xobs = (0,−100,0)T
and looks along the x2-axis, compare Fig. 5. At fixed observation time
x0
obs = 100, he will see that the sphere is apparently rotated in its direction
of motion. Determine the relation between the sphere’s velocity βand the
apparent rotation angle α.
Result: For β= 0, the axis/pole of the sphere points towards the observer.
With increasing velocity, the sphere appears to be rotated by an angle
α= arctan(βγ). (Terrell [4] uses the complementary angle.) The angle α
can be read from the image generated by the script. Given the distance d
of the pole to the center of the sphere and the radius r, both in relative
units or pixels, the angle reads α= arcsin(d/r).
Configure file: demoSphere.py
Visual appearance of wireframe objects in special relativity 18
Exercise 3: Analogous to Fig. 4(d), a die moves with β= 0.9above
a row of static dice. Here, we fix the observation times to x0
obs =
{13.422,14.422,15.422}and let the observer rotate around the point of
interest (0,0,2)Ton the circle ~xobs = (robs cos ϕ, robs sin ϕ, 2.0)Twith
robs = 14.422 and 0≤ϕ≤2π. Explain why the distances between the
apparent positions of the moving die for the different observation times
depend on the angle of observation ϕ. What happens if βis changed?
Result: The observation time x0
obs = 14.422 is chosen such that the
apparent position of the die keeps its position irrespective of the observation
angle ϕas long as robs = 14.422. If ϕ=π/2 or ϕ= 3π/2, the observer looks
along the row of dice towards the approaching or receding die, respectively.
Then, the finite speed of light has strong influence on where the moving
die appears. This can be most easily understood by means of a Minkowski
diagram, see Fig. A1 for a similar situation with only a point-like object. If
ϕ= 0 or ϕ=π, light travel times from the current positions of the moving
die to the observer are nearly the same. Hence, the distances between the
apparent positions approximately reflect the actual distances between the
current positions for the different observation times.
x1
x0
O1
O2
W
x0
c
x0
b
x0
a
A1c
A1b
A1a
A2c
A2b
A2a
Figure A1. A point-like object, e.g. the center of the die, moves with β= 0.75 in
the positive x1-direction, here indicated by the gray worldline Win the Minkowski
diagram. The observation times for the static observers O1and O2are the
same, x0
a< x0
b< x0
c. The dashed lines represent parts of the backward light
cones for the corresponding observers and observation times. The distances
between the apparent positions A1ifor the approaching point are bigger than
the distances of the apparent positions A2ifor the receding point; for example
|x1(A1a)−x1(A1b)|>|x1(A2a)−x1(A2b)|
Configure file: demoDie.py
Visual appearance of wireframe objects in special relativity 19
Appendix B. Circular silhouette of a moving sphere
As proven already by several authors, see for example Penrose [6] or Boas [7], the
silhouette of a relativistically moving sphere keeps circular irrespective of its velocity.
To show this circular silhouette for our general standard configuration, we could follow
two approaches.
Appendix B.1. Straightforward calculation
The straightforward approach works as follows. First, we determine the normal vector
~n at each apparent point ~x which is given by the cross product between the derivatives
of ~x with respect to ϑ0and ϕ0, respectively,
~n =∂~x
∂ϑ0×∂~x
∂ϕ0.(B.1)
For that, we need the derivatives
∂x00
∂ϑ0=−∂ωsph
∂ϑ0,∂x00
∂ϕ0=−∂ωsph
∂ϕ0,∂~µ
∂ϑ0=∂~µ
∂ϕ0=~
0,(B.2)
where
∂ωsph
∂ϑ0=1
ωsph
~µ ·
3
X
i=1
∂s0
i
∂ϑ0~σ0
iand ∂ωsph
∂ϕ0=1
ωsph
~µ ·
3
X
i=1
∂s0
i
∂ϕ0~σ0
i.(B.3)
Therefrom, we obtain
∂~x
∂ϑ0="3
X
i=1
∂s0
i
∂ϑ0~σ0
i·−γ
ωsph
~µ +γ2
γ+ 1 ~
β#~
β+
3
X
i=1
∂s0
i
∂ϑ0~σ0
i,(B.4)
∂~x
∂ϕ0="3
X
i=1
∂s0
i
∂ϕ0~σ0
i·−γ
ωsph
~µ +γ2
γ+ 1 ~
β#~
β+
3
X
i=1
∂s0
i
∂ϕ0~σ0
i.(B.5)
When building the cross product of (B.4) and (B.5), we can make use of ~
β×~
β=~
0
and the orthonormality of the basis vectors, ~σ0
i×~σ0
j=ijk ~σ0
kwith the totally anti-
symmetric Levi-Civita symbol ijk . Hence, we obtain
~n =
3
X
i=1
hi(ϑ0, ϕ0)~
β×~σ0
i+
3
X
i,j,k=1
∂s0
i
∂ϑ0
∂s0
j
∂ϕ0ij k~σ0
k(B.6)
with the abbreviation
hi(ϑ0, ϕ0) =
3
X
j=1 ∂s0
j
∂ϑ0
∂s0
i
∂ϕ0−∂s0
j
∂ϕ0
∂s0
i
∂ϑ0~σ0
j·−γ
ωsph
~µ +γ2
γ+ 1 ~
β(B.7)
By means of the normal vector, we can construct the equation for the tangent plane
~n ·(~x −~y) = 0, where the apparent point ~x is the reference point of the plane. The
arbitrary positional vector ~y has to be replaced by the observer position ~xobs. The
resulting implicit equation for ϑ0and ϕ0defines the silhouette of the photo-object
which has to lie on a right circular cone with apex at the observer.
Visual appearance of wireframe objects in special relativity 20
~x′
~x′
obs
x′1
x′2
~σ′
1
~σ′
3
α′
R′
r′
~
t′
1
Figure B1. Tangent cone (blue lines) with apex angle α0of a sphere with radius
r0and centre point ~x0. The basis vector ~σ0
2points into the plane of projection.
Appendix B.2. Boas strategy
Another possibility to prove the circular silhouette of a moving sphere starts from
within the moving frame S0where the sphere is at rest, see also Boas [7]. For that,
we first have to transform the observer via the inverse Poincar´e transformation from
Sinto S0,x0µ
obs =¯
Λµν(xν
obs −aν).
In S0, the parameters of the cone tangential to the sphere can be easily determined
(see figure B1 for a two-dimensional equivalent),
~
d0=~x0−~x0
obs,sin α0=r0
k~
d0k, R0=r0cos α0,(B.8)
where ~
d0is the cone axis, α0the apex angle, and R0is the radius of the contact ring.
As the orientation of the sphere has no influence, we can set the basis vectors ~σ0
1,~σ0
2,
and ~σ0
3as shown in figure B1. Then, the contact ring ~x0
c(ψ0) = ~x0
obs +~
k0(ψ0) can be
parametrized by the angle ψ0and
~
k0(ψ0) = r0cos α0
tan α0~σ0
1+R0cos ψ0~σ0
2+R0sin ψ0~σ0
3.(B.9)
Now, the time x00(ψ0) when the point ~x0(ψ0) has to emit light that reaches the observer
at x00
obs follows from equation (13),
x00(ψ0) = x00
obs −∆ (~x0(ψ0), ~x0
obs) = x00
obs − k~
d0kcos α0.(B.10)
From that, we can determine the contact ring ~xc(ψ0) with respect to Svia the Poincar´e
transformation (1),
~xc(ψ0) = γx00
obs − k~
d0kcos α0~
β+~x0
obs+~
k0(ψ0)+ γ2
γ+ 1 h~
β·~x0
obs +~
k0(ψ0)i~
β+~a.(B.11)
Again, ~xc(ψ0) has to lie on a right circular cone with apex at the observer.
Visual appearance of wireframe objects in special relativity 21
Appendix C. View- and perspective projection
The reference frame {~ex, ~ey, ~ez}of the pinhole camera is defined by the eye point,
which corresponds to the observer’s position ~xobs , the point-of-interest ~p, and a
preliminary up-vector ~u. Please note that the pinhole camera looks along the negative
~ezdirection, which is defined by ~ez=−(~p −~xobs)/k~p −~xobsk. The right-axis is given
by ~ex=~u ×~ez/k~u ×~ezk, and finally the corrected up-vector follows from ~ey=~ez×~ex.
Thus, the View matrix , which maps a point into the reference frame of the camera,
reads
=
e1
xe2
xe3
x−x1
obs
e1
ye2
ye3
y−x2
obs
e1
ze2
ze3
z−x3
obs
0 0 0 1
.(C.1)
The perspective projection emulating the view of a pinhole camera is described by the
Projection matrix
=
1
acot fovy
20 0 0
0 cot fovy
20 0
0 0 −f+n
f−n−2fn
f−n
0 0 −1 0
(C.2)
with aspect ratio a, near clipping plane n, far clipping plane f, and vertical field of
view fovy. Note that, for these matrices, we need homogeneous coordinates (x, y , z, w)
and the calculations are done in projective space. Then, for our purpose, mapping a
point p= (p1, p2, p3) from world space onto the camera’s view plane works as follows.
Append the homogeneous coordinate w= 1 to the point and determine the matrix-
matrix-vector multiplication
ˆp =pwith p= (p1, p2, p3,1)T(C.3)
resulting in the projected point ˆp. The perspective division ˆp 7→ (ˆp1,ˆp2,ˆp3)/ˆpwyields
the view plane coordinates vx= ˆp1/ˆpwand vy= ˆp2/ˆpwwith vx, vy∈(−1,1). The
coordinate vz= ˆp3/ˆpwincorporates depth information of the point.
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