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Post on 19-Dec-2015
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The Virtual World Space
In order to render coloured images of a virtual world, knowledge of geometric basis or set of rules needed The Cartesian system: a set of 3D axes where each
axis is orthogonal (90 o) to the other two
assume all Cartesian coordinate systems are right-handed, and the vertical axis is always the y-axis
Positioning the Virtual Observer(VO)
Procedure depends on the VO’s frame of reference within the VE which may involve Direction cosines XYZ fixed angles XYZ Euler angles or quaternions
Direction cosines
A unit 3D vector has three axial components which are also equal to the cosines of the angles formed between the vector and the three axes
the angles are known as direction cosines and can be computed by taking dot products of the vector and the axial unit vectors
the direction cosine enable any point P(x,y,z) in one frame of reference to be transformed into P’(x’,y’,z’)
Xw
Zw
Yw
v
αγ
β
Cont.
x’ r11 r12 r13 x
y’ = r21 r22 r23 y
z’ r31 r32 r33 z
where:
r11 r12 r13 are the direction cosines of the secondary x-axis
r21 r22 r23 are the direction cosines of the secondary y-axis
r31 r32 r33 are the direction cosines of the secondary z-axis
Example 1: shows that (x’,y’,z’)= (x,y,z)
x’ 1 0 0 xy’ = 0 1 0 yz’ 0 0 1 z
XwZw
Yw
P’(x’,y’,z’) = P(x,y,z)
XVO
YVO
ZVOAs the two axial system are coincident, any point P’= P
XYZ fixed angles
Another approach for specifying orientation
3 separate rotations about a fix frame of reference, angles referred to as
yaw: angle of rotation about the y-axis
pitch: angle of rotation about the x-axis
roll: angle of rotation about the z-axis Xw
Zw
Yw
yaw
rollpitch
Stereo perspective projection
Stereoscopic vision helps us estimate depth and works up to distances of approximately 30 m - beyond this disparity between the images in left and right eyes is significant
Projection plane
Left eye Right eye
Xvo
Yvo
Zvo
Se
To compute the perspective projection of the box for the right eye, the box’s x-coordinates are translated by Se/2 and for the left eye the box’s x-coordinate are translated by - Se/2
Cont. The viewing plane is
located d from the origin and the interocular distance is Se.. Any point on the object P(x,y,z) must now be translated - Se/2:
xpl = x- Se/2
d z+ d
left eye place at origin
Xvo
Zvo
- Se/2
zxpl
d
Left eye Right eyeVO
P(x,y,z)
A plan elevation of the geometry relating the VO’s left eye the projection plane and the box
Yvo
Zvo
z
ypl
d
Left eye
Projection plane
P(x,y,z)
y
ypl = yd z+ d
left eye place at origin
A side elevation of the geometry relating the VO’s left eye, the projection plane and the box
Similar equation for the right eye
xpr = x +Se/2
d z + d
ypr = yd z+ d
•In order to see an object in focus as a single fused image, we must focus and allow both eyes to fixate upon the object by adjusting their convergence angle (natural 3D view)
•however, computer model does not include any convergence angle
•it assumes the two eyes are gazing at infinity
•we will see two overlapping views of an object
•to simulate convergence overlap images in the HMD using diverging or converging optics
3 D clipping
There are many occasions when part of an object is visible and the rest is invisible which implies that every object must be trimmed or clipped against some visible viewing envelope or volume
3D clipping must be applied separately for the left and right eyes
Left eyeRight eye
Xvo
Yvo
Zvo
The two viewing volumes associated with left and right eyes
3 D clipping
Right eye
Near (hither) plane
Projection plane
Far (yon) plane
A single viewing frustum (a truncated pyramid) with near, projection and far plane
3 D clipping
Clipping algorithms
to establish as efficiently as possible whether an object requires clipping or not
example if every object has an associated rectangular bounding box that completely contains the object, and if every vertex of the box is visible than the object must be completely visible
2 popular methods Cohen-Sutherland: employs a 6 bit code to describe
whter the end of a line is visible or not Cyrus-Beck: clips lines against a 3D convex
polyhedron using a parametric definition of a 3D line.
Back-face removal Clipping is computationally expensive process Any way of reducing the number of polygons to be
clipped must be investigated. This can be done by the back-face removal technique.
A
B
na
θa nb
θb
Vb
va
YVO
ZVO
XVO
cosθa = na . va
| na | | va | If cosθa is positive then the surface is visible otherwise it is invisible. In diagram above, A is visible to the VO as θa is less than 90o, whereas surface B is invisible as θb is greater than 90o
Simple 3D modelling
Geometric considerations Euler’s rule: a polyhedron without holes have the number of
edges is always the sum of the faces and vertices minus two:Edges = faces + vertices -2
Surface normal: can be determined by the cross product of two edges
Surface planarity: forms the boundary of an object. If a polygon is defined as a chain of arbitrary edges, it will be very easy to construct twisted surfaces
Modelling tools databases containing surface elements, light points, texture,
colour and ect. Extruding and swept surfaces are examples of modelling
tools
Illumination models
Approaches to create a coloured view of a 3D scene 2 approaches
assign a fix colour to every surface simulate the interaction of light sources with coloured surfaces
Point light sources: radiates light energy equally in all direction
Directional light source: assumes to be located so far away that all of the incident light rays are parallel
Spot light source: a directed beam of light with its associated spot angle
Ambient light: illumination schema that allow the existence of some level of background light level
Shadows: still regarded as a luxury Transparency: to simulate such as the effects of glass and
other transparent media
How to increase realism?
Multiple light source Intensity fall-off with distance negative light X-ray sources
Reflection models
Describe the reflective behaviour of imaginary light Diffuse reflection such as carpets, textiles… can also
give rise to surface gloss effects Specular reflection: smooth or polished surfaces The complete reflection expression: ambient, diffuse
and specular
Shading algorithms
Involves: The frame store: store the image for display
purposes Mapping to the display device Gouraud shading: flat and smooth Phong shading: to compute specular refelctions
Radiosity
A global illumination model that attempas to simulate the multiple diffuse reflections that occur between surfaces
Hidden-surface removal
Problem in distinguishing two surfaces separated by a distance. Approaches taken: The painter’s algorithm: sorts surfaces within the
VO’s field of view in depth sequence Scan-line algorithm: renders images on a line-by-line
basis The z-buffer algorithm: introducing a depth buffer
that always maintains the z-depth for the nearest surface rendered into pixel.
Realism
Realism can be increased by texture mapping of the real world. Some issue has to be considered aliasingcan cause visual artefacts anti-aliasing to overcome the visual artefacts
problems bump mapping: modulating surface normal during
lighting calculation environment mapping: consideration for reflection of
objects to their surroundings
Stereographic images
Red and green glasses red for right eye and green for the left eye to view streographic images: an image for each eye
overlaid with a suitable horizontal overlap
Tranformation
System must be prepared to indicate an object is able to move given the right condition
Modelling transformations translate scale reflection rotation
Translate
The translate transformation enables an object to be positioned anywhere within the VE simply by specifying three offset values that are associated with every 3D vertex of the object
Scale
The scaling transformation alters the size of an object by scaling all of its coordinates
Reflection
Imagine seeing an object in the mirror the same concept for reflection modelling
an object can be computed by reversing the sign of either the x-, y- or z-coordinates
Rotation
Using direction cosines compund rotations: by subjecting an object to a
sequence of matrix operations XYZ fixed angles XYZ Euler angles rotation about an arbitary axis general rotation matrix: can be used to genarate all
of the above rotation matrices
Scaling the VE
VE can be scaled up or down making the VE larger, makes the VO becomes
smaller, can explore the world from the standpoint of a child, or a scene in the dinasour world…
making the VE smaller until a point so small, we can lost all the view of the VE
Animating Position
Some simple linear animation of objects linear tranlation non-linear translation linear angular rotation non-linear angular rotation
shape and object inbetweening process of deriving an object or characterfrom two key
images free-form deformation
works by surrounding an object with a 3D lattice of control points that can be used to deform the object
particle systems a powerful technique for modelling natural phenomena
such as water, rain, fire, grass...
Others to be considered
Collision detection Instances picking an object pyhsical simulation
object falling, rotating wheels, clocks,projectiles, pendulum, springs, flight dynamics