Abstract In this paper, a model to realize an interactive manipulation of virtual liquid using a virtual vessel is described first. Then, this model is extended to represent the swinging liquid in a vessel. The liquid receives reactionary acceleration according to the acceleration of the vessel. This reactionary acceleration is considered to represent swinging liquid. Our system with this proposed model makes it possible to swing the liquid surface with swinging the vessel, then to spill the liquid with swinging it

Kenji Funahashi

Yuji Iwahori

CiteSeerX

ICAT 2001
 December 5-7, Tokyo, Japan
Representation of Swinging Liquid
on Virtual Liquid Manipulation
Kenji Funahashi, and Yuji Iwahori
Center for Information and Media Studies, Nagoya Institute of Technology
Gokiso-cho, Showa-ku, Nagoya 466-8555 Japan
kenji@center.nitech.ac.jp
Abstract
In this paper, a model to realize an interactive manipulation
of virtual liquid using a virtual vessel is described first. Then,
this model is extended to represent the swinging liquid in a
vessel. The liquid receives reactionary acceleration according to
the acceleration of the vessel. This reactionary acceleration is
considered to represent swinging liquid. Our system with this
proposed model makes it possible to swing the liquid surface
with swinging the vessel, then to spill the liquid with swinging it.
Key words: Virtual Reality, Interactive Manipulation, Virtual
Liquid, Liquid Manipulation, Particle and Volume Model
1. Introduction
In the field of virtual reality for the recent years, the limitation
is that users can manipulate only the solid objects [1], [2], [3].
While in the computer graphics technology, the scene image
of water current has been represented by real-time simulation
[4], [5], [6]. However, an interactive manipulation of the virtual
liquid is not considered in these papers including other studies,
and direct application to the interactive manipulation is still
difficult.
In this paper, a model to realize an interactive manipulation of
virtual liquid using a virtual vessel [7], [8] is described first.
A virtual vessel is introduced to detect the interference with
liquid. We consider only the interaction between the virtual
vessel and the liquid. Then, this model is extended to represent
the swinging liquid in a vessel. The main purpose of this
model is the realization of the virtual liquid manipulation, while
the generation of high quality computer graphics images nor
simulating the exact behavior of the liquid, are not the main.
Our system with this proposed model makes it possible to catch
the liquid using the virtual vessel, to swing liquid surface with
swinging the vessel, then to spill the liquid with swinging it.
Also the system realizes the manipulation to spill the liquid by
tilting it and skim the liquid from another liquid vessel.
2.Liquid Manipulation Using Vessel
2.1 Virtual Liquid in Virtual Space
In this paper, we consider the liquid under the following condi-
tions.
(1) free fall condition (such as flowing water from a faucet)
(2) stay condition (such as holding water in a cup)
The proposed model represents the liquid in the condition (1) as
particles. Each particle moves according to the gravity and the
inertia. It is assumed that the size of the particle is ignored and
each particle does not interfere with other particles.
It also represents the liquid in the condition (2) as volume. The
exchange rate between each condition is taken to be N [number
of particles / volume]. When user falls all liquid by tilting
the vessel of the volume V , the volume of liquid in the vessel
becomes 0 and the number of particles of liquid in free fall
condition increases by NV .
2.2 Virtual Vessel in Virtual Space
We consider a vessel with the convex shape and a sphere includ-
ing the vessel (Fig. 1). The center of the sphere is designated
as C and its radius is designated as r. Then, the position of
the vessel is represented by C. The direction of the vessel is
represented by θ,φ and ψ (Fig. 2), while the tilt parameters are
represented by θ and φ. First, a vessel is rotated by φon axis xc,
and rotated by θ on axis zc. Then it is rotated by ψ on axis yc.
When θ and φ of a vessel are set to be 0 respectively, the status
means that it is not tilted.
Fig. 3 shows a virtual vessel with the liquid. First, functions of
the vessel with the liquid are defined in the following. The over
flow point F is represented by the relative vector from the point
C.
F = f (θ,φ,ψ). (1)
When the volume of the liquid in a vessel is V (V ≥ 0), the
height level of the liquid surface H is represented by the relative
distance from the point C.
H = h(θ,φ,V ). (2)
r
C
virtual vessel
Fig. 1: Virtual vessel and a sphere including it
C
xc
yc
zc
θ
φ
d1
d2
d3
ψ
Fig. 2: Representation of tilting of virtual vessel
When the parameter V of the function h is over the maximum
volume Vm, H becomes Fy (Fy is defined as the vertical coordi-
nate of F). The above equation is translated to
V = v(θ,φ,H). (3)
When the parameter H of the function v is under the vessel,
V becomes 0. When H is above the point F , V is taken as
v(θ,φ,Fy) = Vm(θ,φ) (Fy is independent of ψ).
When the height level is H , define the volume under H consid-
ering the thickness of vessel (Fig. 4) as
V+ = v+(θ,φ,H). (4)
When the volume of liquid is Vm, V + becomes Vm+. When the
thickness of vessel is not considered, V + is regarded as V .
2.3 Interaction Model of Liquid and Vessel
First, the system gets C(t),θ(t),φ(t) and ψ(t); the position and
the direction of the vessel moved by the user in the virtual
space. (This can be done using the 3-D position sensor.) In
the following discussions, the value at time t is assumed when
the description of the parameter t is omitted. Each particle is
moved according to the gravity and the inertia. The particles
moved under the predefined threshold plane will be deleted for
the drawing.
height level of
 liquid surface
over flow
 point


C
H
V
F
highest level
volume of
 liquid
Fig. 3: Each parameter of virtual vessel

V
V+
liquid
vessel
Fig. 4: Volume considering thickness of virtual vessel
2.3.1 Liquid in Free Fall Condition and Vessel
When the falling liquid (each particle) moves through the mouth
of a vessel, the status of the liquid changes from the condition
(1) to the condition (2), and the volume of the liquid in it will be
increased according to the exchange rate N (Fig. 5).
V (t) = V (t −∆t)+n/N, (5)
where n (n ≥ 0) means the number of particles which move
through the mouth. When the liquid in the vessel is not
interfered with another vessel, the height level is updated simply
according to the equation (2).
2.3.2 Liquid in Stay Condition and Vessel
In this section, we describe the interaction model between a ves-
sel and the liquid in another vessel. In the following, let each
vessel be designated as vessel 1 and vessel 2 respectively, also
let the symbol of each vessel be designated as subscript 1 and 2
for the distinction. When the equation (6) holds, vessel 1 and the
liquid in vessel 2 are interfered each other (Fig. 6).
C1y − r1 < C2y +h2(θ2,φ2,V2), (6)
C2y +h2(θ2,φ2 ,V2 +Vm+1 ) < C1y +F1y, (7)
where Cjy is the vertical coordinate of the position Cj .
When the equation (7) also holds, the situation is that the part
of vessel 1 under the over flow point interferes with the liquid
surface of vessel 2. Then, let the volume that vessel 1 upholds
the height level of vessel 2 be U . When it does not hold,
fall
stay
particles fall through
 the mouth of the vessel
add the volume
 of liquid in the vessel



exchange rate: N
Fig. 5: Relation between two conditions of liquid (1)
vessel 1
ve
rt
ic
al
 a
xi
s
C2y
V2
U: upheld
     volume
C1y
vessel 2
liquid surface
 of vessel 2
r1
F1y

h2(V2)
Vm +1
Fig. 6: Interference between virtual vessel and liquid of
another vessel
the situation is that the liquid in vessel 2 flows into vessel 1,
the volume of the liquid in each vessel is changed with the
appropriate rate [8].
2.3.3 Spilling Liquid from Vessel
When next equation holds at time t by the above interference and
tilting of a vessel, the liquid spills from the vessel (Fig. 7).
V +U > Vm. (8)
The part of the liquid in the vessel, which is expressed using
the volume, spills at the over flow point F as N(V +U −Vm)
particles according to the gravity. The volume in the vessel at
time t is determined as
V (t) = Vm−U. (9)
While if the equation (8) does not hold, the liquid dose not spill.
2.4 Implementation of Functions
For the implementation of a system, each function of each
vessel expressed by equations (1) – (4) should be made based
on the shape of each vessel respectively. In this system, each
virtual vessel are expressed by a convex shape polyhedron. At
loading up the shape of vessels, the LUTs (look up table) for
each vessel are made to calculate the equation (3). This LUT
gives V from the parameters θ,φ and H ·100/r (Fig. 8). In this
fall
stay
flow liquid
 as particles
the volume is over
 the maximum volume
Uexchange rate: N
Fig. 7: Relation between two conditions of liquid (2)
θ -90 90...
...
-100
100
(degree)
(%)
[ V ]
φ: -1
80 .
.. 18
0 (d
egr
ee)
H.100/r
Fig. 8: Look up table for calculation volume
LUT, if the height level is under the vessel, the volume is set
to 0. If it is above the over flow point, the volume is set to the
maximum volume Vm(θ,φ).
The volume of liquid in a vessel can be calculated by the inter-
polation from the elements of the LUT. The height level with
volume parameter (equation (2)) can be obtained by searching
elements which are located nearby the value of the volume then
by interpolating the height level parameter of elements. The over
flow point of a vessel (equation (1)) can be obtained by the se-
quential search for the vertices of the polygon which expresses
the mouth of the vessel. The thickness of the vessels is ignored
in this system. To consider the thickness, it is necessary to make
another LUT for the equation (4).
3.Representation of Swinging Liquid
3.1 Absolute Stay Condition
The stay condition; the condition (2) mentioned above is the
absolute stay condition. The liquid surface is horizontal when
the vessel is swung. However, many trial people swung the
vessel vertically and horizontally at the demonstration hall of
IEEE-VR2001 [9].
C (t-2∆t) C (t-∆t) C (t)
D (t-∆t) D (t) =
 C (t) -C (t - ∆t)
D (t-∆t)/∆t D (t)/∆tvelocity 
   and
acceleration
   of vessel
a (t) = (D (t) -D (t-∆t))/∆t2
Fig. 9: Position, Velocity and Acceleration of Vessel


H
V
F r
θ = θr
φ = φr
ψ = ψc
g  : gravity
 acceleration
g - a  : relative
 gravity vector
-a  : reactionary
 acceleration
α
θ = θa
φ = φa
ψ = ψc



g - a
α
θ = θa
φ = φa
ψ = ψc
spill
F = F a
Fig. 10: Relative Stay Condition
3.2 Reactionary Acceleration and Relative Gravity Vectors
We extend the absolute stay condition to the relative stay condi-
tion for a vessel. When the position of a vessel is C(t) at time t,
the moving vector of its position is defined as D(t) (Fig. 9).
D(t) = C(t)−C(t−∆t). (10)
Then the velocity vector is
D(t)/∆t. (11)
The acceleration vector of the vessel; a(t) at time t is obtained
as
a(t) =
D(t)−D(t−∆t)
∆t2
. (12)
The liquid in the vessel receives the reactionary acceleration
(−a) according to the acceleration of the vessel. The sum vector
of the reactionary acceleration and the gravity acceleration is
calculated as g − a. This vector is used to realize the relative
gravity vector (See Fig. 10(left)).
3.3 Relative Stay Condition
When the direction of the vessel is
θ = θa,φ= φa,ψ = ψc, (13)
we consider that the vessel is rotated by α as the relative gravity
overlaps with the actual gravity. Then the relative direction is
determined as
θ = θr ,φ= φr ,ψ = ψc (14)
for the following calculations (See Fig. 10(middle)). The height
level of the liquid surface is calculated by equation (2) with θr
and φr, and the liquid surface is set up. Also the relative over
flow point Fr is determined by equation (1).
Then, the vessel is rotated and returned in the direction of ori-
gin. The liquid surface is rotated with the vessel for the drawing.
Also Fr is rotated to the point Fa (See Fig. 10(right)). When the
liquid spills from the vessel;
V +U > Vm(θr,φr), (15)
each liquid particle flows out from the point F(= Fa) according
to the relative gravity (g−a).
4.Experimental Results
Using the model mentioned above, we implemented a system
for the manipulation of the virtual liquid using virtual vessels
in C language on a graphics workstation SGI OCTANE2. The
exchange rate N is taken to be 10 [number of particles / cc]. In
the experiment, 1 cc of the falling liquid is generated per 1 CG
frame.
An example operation is shown in Fig. 11. This system gets
the position and the direction of the real vessel via the motion
sensor using a low-frequency magnetic field. As shown in
the Fig. 12(right), the user can catch the falling liquid using
the virtual vessel, hold the liquid in it, then spill the liquid.
Fig. 12(left) shows that the user can skim the liquid from
another vessel. As shown in Fig. 13, the user can swing liquid
surface with swinging the virtual vessel, then spill the liquid.
The present situation is that the surface is still flat and the
representation of a wave cannot be fully performed.
Through this virtual liquid manipulation system in our labora-
tory, we get some positive evaluations such that “I feel that I
really manipulate the liquid using a cup.” The real cup with sen-
sor is passed to users and only the fact that users can manipulate
the water (liquid) in the screen using a real cup was explained
to users. It is considered that this system can keep the essential
characteristics of real liquid manipulation. This system can re-
fresh its screen at 24 frames/sec as the average speed when there
are about 500 particles.
5.Conclusion
In this paper, we proposed a model for liquid manipulation
using virtual vessels and realized the implementation of an
experimental system as a virtual reality system. Using the
virtual vessel, the system can swing the liquid surface, and spill
the liquid at an interactive refresh rate. Also the system can
catch the falling liquid, skim the liquid from another vessel.
Users have an impression close to that in the real world.
The system reported in this paper is under the stage of the re-
alization of the basic functions, and the following subjects are
remained as the future subjects.
• Representation of the waving liquid such as the ocean sur-
face.
• Interaction with the liquid such as the water current in the
river.
• Manipulation of the liquid using the vessels with the general
concave shape.
The addition of these functions will be required. It is considered
that the wave and current condition is represented as both par-
ticles and volume. For the latter, it is considered that a concave
polyhedron vessel should be divided into some neighboring
convex polyhedrons.
Acknowledgments
We are grateful to our colleagues in our laboratory for the useful
discussions. This research was supported in part by the NIT
research promotion program.
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Fig. 11: Appearance of proposed system
Fig. 12: Manipulation of liquid using virtual vessel
Fig. 13: Swinging liquid surface and spilling liquid


Representation of Swinging Liquid on Virtual Liquid Manipulation

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Representation of Swinging Liquid on Virtual Liquid Manipulation

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