Visualization of Tangent Developables
on a Volumetric Display
Ou Yamamoto and Masatoshi Kokubu
Tokyo Denki University
5 Senju-Asahi-cho, Adachi-ku, Tokyo 120-8551, JAPAN{ou,kokubu}@mail.dendai.ac.jp
http://www.dendai.ac.jp

Abstract. A tangent developable is a developable surface constructedby the union of the tangent lines of a space curve. These surfaces haveapplications not only in mathematics but also in engineering, such as fordesigning cars, ships, and apparel. However, since tangent developables
typically have complicated and twisted surfaces, it is diﬃcult to understand their structures from their images on a 2D screen. For ease in understanding such complicated structures, a more improved visualization
method is required. In this study, we propose and evaluate a softwaretool for visualizing tangent developables on a volumetric display thatdraws 3D images directly in real 3D space.
Keywords: Tangent developable, Surface visualization, Volumetric display, 3D display

1Introduction

A tangent developable is a developable surface that is constructed by the unionof the tangent lines of a space curve. As the surface can be developed ontoa plane without stretching, it has applications not only in mathematics but
also in engineering, such as for designing cars [1], or ships [1], and apparel [2].
Fig. 1 illustrates a tangent developable of a space curve deﬁned by the following
parametric equation:√c(t) = (cos3 t, sin3 t, 2 − cos3 t − sin3 t ), − 1.0 ≤ t < 1.0.
(1)
As shown in Fig. 1, we can visualize tangent developables on a 2D screen.
However, they typically have complicated and twisted surfaces, and therefore itis diﬃcult to understand their structures from 2D images. To ease understandingof their structures, an improved visualization method is required.
A volumetric display is a device that creates 3D images in real 3D space [3].
Most volumetric displays satisfy the physiological factors for stereoscopic viewing
while requiring no 3D goggles or head-mounted displays. To date, several typesof volumetric displays have been proposed and developed, including a variablefocal type [4], a volume-scanning type [5] [6], an up-conversion type [7] [8], and2Ou Yamamoto, Masatoshi Kokubu

Fig. 1. Example of a tangent developablea plasma-emission type [9]. Some of these are commercially available; however,they are expensive for researchers who are developing applications for them.
We have developed a volume-scanning type volumetric display that can beeasily constructed from commercially available components [10]. Our display
creates sequential cross-sectional images of 3D objects on a vacuum ﬂuorescentdisplay (VFD) moving in reciprocating motion; 3D images of the objects appear
because of the afterimage eﬀect. Furthermore, we have proposed an additional
hardware and a software tool for visualizing isometric deformations of minimal
surfaces [11]. As an application of our display, we propose and evaluate, in thisstudy, a software tool for visualizing tangent developables.

2

Tangent Developable

In this section, we quickly review some mathematical aspects of tangent developables. For details, please refer to standard textbooks on the diﬀerential
geometry of curves and surfaces, e.g., [12] and [13].
A surface in Euclidean 3-space R3 is said to be ruled if it is formed by a oneparameter family of straight lines. It is a classical fact that any ruled surface ofconstant Gauss curvature zero is locally congruent to either a cylindrical surface,a conical surface, or a tangent surface. By deﬁnition, a tangent surface is a ruled
surface formed by tangent lines of a space curve. Precisely, given a regular curvec: I(⊂ R) → R3 , a map f (t, u) = c(t) + uċ(t) draws a tangent surface. It is easily
veriﬁed that f (t, u) = c(t) + uċ(t) has zero Gaussian curvature on the regularregion. Note that a tangent surface f (t, u) has singularities on u = 0 or t = t0
where the curvature of c vanishes.
In classical terminology, ruled surfaces of constant Gauss curvature zero havebeen called developable surfaces because they can be isometrically developed toa plane. The fact introduced above asserts that a surface is developable if andonly if it is locally congruent to either a cylindrical surface, a conical surface, orVisualization Tool of Tangent Developable on Volumetric Display

3a tangent surface. For this reason, a tangent surface is usually called a tangent
developable surface or a tangent developable, in short.
One may easily imagine that a cylindrical or conical surface can be developed to a plane. Moreover, this is not hard to demonstrate using a sheet ofpaper. However, it would be rather diﬃcult to directly verify the developabilityof tangent developables. A typical example is the tangent developable of a helix,()ttbth(t) = a cos √, a sin √,√(2)
a2 + b2
a2 + b2
a2 + b2
where a(> 0) and b are constant real numbers. Note that t is an arc-lengthparameter. Its tangent developablef (t, u) = h(t) + uḣ(t)has the ﬁrst fundamental form()
a2 u2
I = 1+ 2
dt2 + 2dtdu + du2(a + b2 )2

(3)

(4)

so that the choice of a, b preserving a/(a2 + b2 ) induces an isometric family oftangent developables. For example, consider the tangent developable fθ (t, u) ofa helix (2) with the choicea = 1 + cos θ, b = sin θ,θ ∈ (0, π).

(5)Then {fθ (t, u) | θ ∈ (0, π)} gives a one-parameter family of isometric tangent
developables, which converges to a part of the plane as θ → 0 (or π). It implies
that, in mathematical terms, a tangent developable of a helix can be developed
to a plane as non-zero θ continuously goes to zero. As shown in Section 5,our volumetric display visualizes this as an animation in real 3D space without
spoiling isometry.
We are more interested in a situation in which the generating curve c(t) canbe stretched and contracted. How does the tangent surface of a stretchable curvec(t) behave under c(t) stretching? In such cases, the deformation of the tangent
developable is not necessarily isometric and is more complicated. The ease ofunderstanding such deformations is evaluated in Section 5.

3

Our Volumetric Display

Our display creates sequential cross-sectional images of 3D objects on a VFD
moving in reciprocating motion; the 3D images of the objects appear becauseof the afterimage eﬀect. Fig. 2 shows a schematic overview of our display, thereciprocating mechanism, and an example 3D image on our display.
The display has a small display area of (W × H × D) 4 × 2 × 4[cm]; however,as shown in Fig 2, it has suﬃcient resolution to display a single 3D object of(W ×H ×D) 128×64×128 voxels, and it has a wide view angle (approx. vertical:
170◦ and horizontal: 170◦ ).

4Ou Yamamoto, Masatoshi Kokubu

Fig. 2. (Above left) Schematic overview of our display, (Above right, below left)
Schematic and photo of the reciprocating mechanism. (Below right) Sample 3D images on the display from three diﬀerent viewpoints

4

Our Software Tool

Our application provides users with a software tool, an interface called TDsquare(an abbreviation for 3D viewer for tangent developable). After a user provides anequation for a space curve to the tool, it diﬀerentiates the equation symbolically,
and then generates 3D-voxel data for the union of tangent lines. The user cansend the data to our volumetric display via a USB interface and view a true 3D
image of the tangent developable. Furthermore, the user can view a 2D preview
image of the tangent developable on the PC screen. TDsquare is written in Javaand accepts such text input as the following,x = 2*cos(pi*sin(2*pi*t))y = cos(2*pi*t + pi/6)
z = 2*sin(2*pi*sin(2*pi*t))
t: 0.0 -> 1.0
The above input describes the following parametric equation, which deﬁnes aspace curve,c(t) = (2 cos(π sin(2πt)), cos(2πt+π/6), 2 sin(2π sin(2πt))), 0.0 ≤ t < 1.0. (6)As TDsquare is a console application, the user speciﬁes the input text ﬁle as acommand line argument.
TDsquare provides the following basic mathematical functions: sin, cos, tan,log, exp, pow (power), arcsin, arccos, and arctan. Additionally, the user can use
these functions in composition form. TDsquare analyzes the equation by LL(1)parsing and builds a parse tree. Then, the tree is transformed to a tree corresponding to the symbolically diﬀerentiated equation. Finally, the coordinates of
the tangent lines are calculated using the tree, then data are generated that are
to be sent to the display. Although various ready-made math formula parsersare available today, we implemented an original formula parser including a differentiator for ease in modiﬁcation and extension in the future. As TDsquare iswritten in Java, it runs on any PC that supports Java. However, the software

Visualization Tool of Tangent Developable on Volumetric Display

5

that sends generated data to the display runs only on Windows PCs under thepresent conditions.
Fig. 3 shows sample images of tangent developables on the display and theirpreviews on the software tool. In the ﬁgure, TD1 is a tangent developable generated from the above input, and TD2 is a tangent developable of the space
curvec(t) = (t2 , t3 , t5 ), − 1.0 ≤ t < 1.0.
(7)
The previews can be rotated both horizontally and vertically by dragging amouse, and views are shaded in real-time. As the images on our display are
drawn directly in real 3D space, the user can view them from any point within
the view angle.

Fig. 3. (a) and (b) 3D images of tangent developable (TD1) on our display (from two
diﬀerent viewpoints). (c) and (d) 2D previews of TD1 on a PC screen. (e) 3D image oftangent developable TD2 on our display and (f) 2D preview of TD2 on a PC screen.

Changes in the constants in the equation that deﬁnes a space curve can cause
complicated deformations of a tangent developable. Furthermore, our tool cangenerate data for animating such deformations on our display. As an example,letc(t) = (1.7 cos(4πt) sin(kπt), 0.5 sin(kπt), − 1.7 sin(4πt)), 0 ≤ t < 1,(8)be a parametric equation of a space curve. When the constant k continuously
varies from 0 to 1, its tangent developable (TD3) deforms as shown in Fig. 4and in the supplementary movie ﬁle Movie1. In Movie11 , most of the ﬂicker iscaused by the diﬀerence in scanning frequencies of the display and the camera.
The preview animation can be rotated and is shaded in real time. The following
is the text input that describes the above deformation.
The following is the text input that describes the above deformation.
x = 1.7*cos(4*pi*t)*sin(_k*pi*t)
y = 0.5*sin(_k*pi*t)z = -1.7*sin(4*pi*t)
t: 0.0 -> 1.0_k: 0.0 -> 1.01

http://www.epi.dendai.ac.jp/Yamamoto/MathUI2014/Movie.html

6Ou Yamamoto, Masatoshi Kokubu

The underscore is a preﬁx for parameters that cause deformations. The numberof animation frames is speciﬁed by a command line argument.

Fig. 4. (a)-(c) Frames from 3D animation of the deformation of TD3 on our display.
(d)-(f) 2D previews on a PC screen for the same values of the deformation parameterk as in (a)-(c).

5Evaluation

For examples TD1, TD2, and TD3, we evaluated the diﬀerences in ease of understanding the structures (TD1 and TD2), and the deformation (TD3) between
the 2D preview and 3D images. For each tangent developable, each of thirteen
participants observed the preview image for 1 min, and then each observed the3D image on our display for 1 min. During observations of the 2D preview images, participants could rotate the image freely by dragging the mouse. Afterobserving the 2D and 3D images of each tangent developable, participants wereasked to state whether the 2D or 3D image was more helpful in understanding
the structure of TD1 and TD2 and the deformation of TD3. All participants are
students at the faculty of electronic engineering of our university and have nobackground knowledge of tangent developables. For TD1, ten participants chosethe 3D image (p = 0.092, two-sided binomial test). For TD2, seven participantschose the 3D image (p = 1.0, two-sided binomial test). For TD3, twelve participants chose the 3D image (p = 0.0034, two-sided binomial test). Sample TD2
has a simpler shape than the other two, and in this case, the results from participants show no statistically signiﬁcant diﬀerence between the preview and thedisplay. However, the results for TD1 and TD3 imply an advantage in using ourdisplay to help understand the structures of complicated tangent developables
and their deformations.
Another advantage to our display is preservation of isometry. As described in
Section 2, some deformations of tangent developables are isometric. Fig. 5 andthe supplementary movie ﬁle Movie22 illustrate an isometric deformation of atangent developable for the helix deﬁned by (3).
In those illustrations, the following parameter θ was used for the deformation,a = 0.25 × (1 + cos θ), b = 0.25 × sin θ,2θ ∈ (0, π).

http://www.epi.dendai.ac.jp/Yamamoto/MathUI2014/Movie.html(9)Visualization Tool of Tangent Developable on Volumetric Display

7Fig. 5. (a)-(d) Isometric deformation in (9) of a tangent developable on our 3D display.
(e)-(h) 2D previews on a PC screen for the same values of the deformation parameterθ as in (a)-(d).

As images on 2D displays are mappings from 3D space to 2D space, 2D
images do not preserve isometry. However, the images on our display are drawn
in real 3D space, thereby preserving isometry.

6Conclusions

We have proposed and implemented a tool for visualizing tangent developables
on a volumetric display. We further evaluated the ease in understanding structures of tangent developables using our tool as compared to using images on a 2D
display. Future tests should include evaluations that compare visualizations on
our display with those produced on other 3D displays. Nevertheless, the present
results show some advantages of using our tool rather than using images on a 2D
display; this is particularly true for visualizing elastic deformations of tangentdevelopables.
Acknowledgments. The second author was supported by Grant-in-Aid for
Scientiﬁc Research (A) No. 22540100 from the Japan Society for the Promotionof Science.

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