Transcript Animation of complex objects

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Course Name: Programming Mini-project 2
Theme:
Computer generated pictures of
comets
http://cis.k.hosei.ac.jp/~vsavchen/MiniPr_2/
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Introduction
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One of the challenging problems of computer graphics (CG) and
computer arts is the visualization of different natural phenomena and
simulated flow data.
Examples are clouds, space phenomena such as evaporation process
of comet nucleus, mirages, rainbows, and other atmospheric effects.
Visualization of
simulated data can be useful in different
application areas to display behavior of studied phenomena or to
improve CG images with more correct scientific factors.
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Introduction
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An example of evaporation process of comet nucleus
 One process which transfers water from the ground back to the atmosphere is
evaporation. Evaporation is when water passes from a liquid phase to a gas
phase.
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The head of Comet Halley, May 1910. Photographed at Helwan Observatory,
Egypt
 Introduction
Characteristics of a Comet
• Structure
 Nucleus
 Coma
 Dust tail
 Plasma tail
• Evaporation process
Generating coma and two tails
 When a comet approaches within a few AU of the
Sun, the surface of the nucleus begins to warm, and
volatiles evaporate. The evaporated molecules boil
off and carry small solid particles with them,
forming the comet's coma of gas and dust.
 Begins inside of Mar’s orbit
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Introduction
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Modeling and Visualization of a Cometary Coma
To get the optical picture of a cometary coma, we can use
simple particle-based simulation, however, based on the main
premises of the Whipple’s theory.
According to this theory the cometary nucleus is an icy
conglomerate of dust and meteor material.
The icy particles evaporate on the surface exposed to the sun
and carry off the dust particles accelerated by the solar
radiation.
For more references, see,
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Whipple F. L. (1950) A comet model I. The acceleration of Comet Encke. Astrophys. J., 111,
375–394.
Bschorr, O.; Jochim, E.F.; Freund, J. (1979) Computer-Generated Pictures of Comets'. :l.
IAF-Congress, Amsterdam, 30 Sep-5 Oct 1974
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Introduction
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optical picture of the coma and tail is
visualized by a Java Applet.
Introduction. Newtonian particle systems
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A particle is described by its mass, m, and its
trajectory, r(t), as illustrated in Figure
Newtonian particles are the most common
and are governed by Newton’s second low
 F = md2r(t)/dt2,
where r(t) = [x1,x2,x3]T is the position of a
particle at time t and a(t) = d2r(t)/dt2 is the
instantaneous acceleration of the particle.
Newton’s second law is converted into two
coupled first-order differential equations
where a point p  R6 in phase space is
denoted by its position r and velocity v =
dr(t)/dt
Step 1. Modeling and Visualization of the Cometary
Coma
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Overview of the Modeling and Visualization of Comets
Applet
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Comet\Cometaimage2.html
Structure of the Applet
Input parameters
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Step 1. Modeling and Visualization of the Cometary
Coma
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Coordinate systems
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The heliocentric system
The motion of the cometary nucleus
and particles is described in the
heliocentric Cartesian coordinate
system r(x,y,z) related to the orbital
plane of the comet.
The coordinate
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x is directed from the sun to the cometary
nucleus,
y is lying in the orbital plane,
z is perpendicular to the orbital plane.
Step 1. Modeling and Visualization of the Cometary
Coma
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Coordinate systems
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Image coordinates
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If the heliocentric coordinates rb of the observer and rk of the comet
are given, the parallel or central projection of a particle with the
heliocentric coordinates rp can be calculated.
The parallel projection of a particle with heliocentric position vector
rp has the image position vector
r*p = (rb – rk)[ (rb – rk)  (rp – rk)]/ rb – rk2
The central projection of a particle with heliocentric position vector
rp has the image position vector
r*p = (rb – rk)[ (rb – rk)  (rp – rk)]/ ((rb – rk)  (rp – rb))
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Step 1. Modeling and Visualization of the Cometary
Coma
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Coordinate systems
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Image coordinates
The parallel projection
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The central projection
-A mapping of a configuration into a
plane that associates with any point of
the configuration
Step 1. Modeling and Visualization of the Cometary
Coma
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World window – a rectangular region in the world
that is to be displayed
Define by
W_T
W_L, W_R, W_B, W_T
W_B
W_L
W_R
Step 1. Modeling and Visualization of the Cometary
Coma
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Viewport
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The rectangular region in the screen for displaying the graphical
objects defined in the world window
Defined in the screen coordinate system
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V_T
V_B
V_L
V_R
Step 1. Modeling and Visualization of the
Cometary Coma
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Develop functions of scalar dot() and vector
multiplications vec_m().
Develop functions for calculating the parallel and central
positions of the image position vector.
Develop Window to Viewport supporting class
Use a driver program shown in Listing1 (download zip
file) as an example of a future Applet.
Naturally, you can use your own driver program!
Step 2. Modeling and Visualization of the Cometary
Coma
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Coordinate systems
The comet centered system
We assume the nucleus as a sphere which is covered
by a meridian system (, ) .
The north pole of this system is normally directed to
the sun.
The latitude  is measured from the north pole
toward the equator.
The longitude is characterized by the angle .
There are kG intervals on the latitude circle and lG
intervals on the longitude circle.
Each of these intervals is numbered by k and l resp.
Step 2. Modeling and Visualization of the Cometary
Coma
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The cometary model
The evaporation process – regular evaporation
The cometary material accelerated from the sun and forming the
coma and tail is composed of different i classes of particles
The evaporation process is divided into time intervals t, which are
numbered by the index j: tj.
The surface of the nucleus is divided into elementary areas kl
kl ~sin k   .
The number of particles nijkl of class i evaporated from surface
element kl in the unit time is related to ~cos kkl and directly
proportional to the relative evaporation rate of the given particle
class.
Step 2. Modeling and Visualization of the Cometary
Coma
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The cometary model
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The evaporation process – regular evaporation
The number (particle lump) is calculated by
nijkl = Ai cos k kl tj/r2, if cos k < 0,
where r is the heliocentric distance of the nucleus and Ai is the
evaporation rate of particle class i.
The position vector ri of the particle lump is set equal to the
position vector rk of the nucleus in the moment of the beginning of
the evaporation process.
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Step 2. Modeling and Visualization of the Cometary
Coma
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The cometary model
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Influence due to rotation of the nucleus
The nucleus of a comet has a diameter of about 1 to 100 km and in
the general case has rotation that causes the surface temperature
deviation.
The rotational axis may be oriented in any direction. The
temperature deviation can be approximated by a shift of the
meridian system (, ) covering the nucleus such that its north pole
points away from the sun by the angle 0.
The latitude  is measured from the north pole toward the equator
and the north pole is directed to the sun,  is the longitude.
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Step 2. Modeling and Visualization of the Cometary
Coma
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The cometary model
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Influence due to rotation of the nucleus
The state vector of the particle lump nijkl at the end of expansion
phase (occurred after the evaporation) is ri and vi
ri = rk +  i ,
vi = vk + wi .
rk is the position vector of the nucleus, vk its velocity vector.
The vectors i and wi are normal to the surface.
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Step 2. Modeling and Visualization of the Cometary
Coma
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The cometary model
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Influence due to rotation of the nucleus
Let ekl the direction vector in our coordinate system. Then
i = i ekl, wi = wi ekl,
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where
i is the boundary of the zone of expansion for the particle class i
and wi is its individual final velocity.
Step 2. Modeling and Visualization of the
Cometary Coma
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Develop functions needed for simulation
regular evaporation.
Use a driver program shown in Listing1 as
an example of a future Applet.
Step 3. Modeling and Visualization of the Cometary
Coma
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Phase of free flying cometray particles (the particle bundle)
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The state vector of the particle bundle at the end of the expansion phase is
r0 , v0 .
If we know the boundary of the expansion zone , the final thermal velocity
w, and the repulsive factor f, after the expansion phase the motion of the
given particle bundle is calculated by the cubic approximation as follows:
r(t) = [1- f t2/2r03 +  ft3/2r05(r0v0)]r0 + (t -  f t3/6r03 )v0,
v(t) = [- f t/r03 + 3 f t2/2r05(r0v0)]r0 + (1 -  f t2/2r03 )v0.
This system of equations is implication of the difference equation
d2r/dt2 +  f r/r3 = 0
and Taylor’s expansion of r(t).
 - heliocentric gravitational constant.
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Step 3. Modeling and Visualization of the Cometary
Coma
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The motion of a comet and an observer (the Earth)
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Comets and planets necessarily obey the same physical laws as every other
object.
They move according to the basic laws of motion and of universal gravitation
discovered by Newton in the 17th century (ignoring very small relativistic
corrections). If one considers only two bodies -- either the Sun and a planet,
or the Sun and a comet -- the smaller body appears to follow an elliptical path
or orbit about the Sun, which is at one focus of the ellipse.
Nevertheless, comets are the perfect examples both of large perturbations and
their possible consequences. Comets expel dust and gas, usually from
localized regions, on the sunward side of the nucleus. This action causes a
reaction by the cometary nucleus, slightly speeding it up or slowing it down.
For simplicity, we define the comet and observer motion by numerical
integration of the difference equation
d2r/dt2 +  r/r3 = 0
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Step 3. Modeling and Visualization of the Cometary
Coma
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The Runge-Kutta algorithm
Consider the initial value problem y = f(x,y) with y(x0) = y0 over the interval a x  b.
The Runge-Kutta method iterates the x-values by simply adding a fixed step-size of h
at each iteration.
Here is a summary of the method:
xn+1 = xn + h
yn+1 = yn + (1/6)(k1 + 2k2+ 2k3 + k4)
where
k1 = h f(xn, yn)
k2 = h f(xn + h/2, yn + k1/2)
k3 = h f(xn + h/2, yn + k2/2)
k4 = h f(xn + h, yn + k3)
Step 3. Modeling and Visualization of the
Cometary Coma
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Develop a function needed for simulating free flying
cometray particles.
Develop a function for numerical integration of the
difference equation, for example, use the common
fourth-order Runge–Kutta method
Use a driver program shown in Listing1 as an example
of a future Applet.
Step 4. Modeling and Visualization of the Cometary
Coma
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Presentation of the images
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We know the position vector r for each particle lump nijkl of class
i originated at time tj, j = 1,2, 3,…, N. N is the number of time
intervals defined by the user.
The number nijkl is used as a measure of brightness Hijkl and is
directly proportional to the light emission Di of particles of the
class i.
The superposition of all brightness elements yields an image of
the cometary tail.
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Step 4. Modeling and Visualization of the
Cometary Coma
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Presentation of images
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For obtaining a well illuminated image we seek the brightest and
the darkest point in the image.
This range of brightness will be divided into 255 brightness steps
(levels).
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Figure (a) shows that the straightforward approach to draw a flow of tiny particles cannot
provide a realistic picture.
Figure (b) presents the picture with smoothed data approximated by an algorithm.
Simplest way is to use the mean filter - a simple sliding-window spatial filter that
replaces the center value in the window, for instance, 3x3 pixels with the average value
of its neighbors.
Step 4. Modeling and Visualization of the
Cometary Coma
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Develop functions which are necessary for
visualization.
Finish development of the Applet.
Step 5. Modeling and Visualization of the
Cometary Coma
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Applet’s presentation