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Introduction to 3D Computer
Graphics and Virtual Reality
McConnell text
Vectors
• Vectors have direction and magnitude –
generally given in terms of three
coordinates and hence the representation
is an arrow from the origin to that point
(3,1,1)
• Vectors are important for viewpoint,
orientation, scaling, rotating and other
transformations
Vectors (con’t)
•
•
•
•
Length: ||v||= x2 y2z2
Addition, scalar multiplication
Dot product: vw||v|||| w|| cos
Cross-product: vector that is perpendicular
to both
Courtesy Wikipedia for symbols
Camera
• Point of view of a camera; viewpoint
• Clipping window is the part of the scene
that is visible
• View direction
• What is up?
Coordinate Systems
• 3D coordinate system – right or left handed (curl
fingers from pos X-axis to pos Y-axis: thumb points pos
Z)– Virtools is left-handed, Processing is left-handed, but
the y axes (and hence the z axes) point in opposite
directions
Y
X
Virtools
Z
Processing
Z
Y
X
• 2D screen coordinate system:
X
Y
Coordinate System: 3D
Environments
• Most 3D environments have at least two
coordinate systems: a world coordinate
system and a local coordinate system for
each object (sometimes parts of objects)
• The world coordinate system does not
change
• The local coordinate system is generally
located in the “middle” or in a corner of the
object and is set in the 3D modeling
program.
Coordinate Systems (con’t)
• Clipping window is the visible area of the
3D scene (it is 2D)- window through which
you look
• Viewport is where on the screen (also 2D)
the visible scene appears; uses the
coordinate system of the screen
• The viewport and the clipping window may
be different sizes, in which case there is
stretching or squishing
• Aspect ratio= width/height- easier if both
clipping window and viewport have same
Interplay of 2D and 3D systems
• There are often 2D objects (buttons,
interfaces, screen text) – these are in the
2D system
• 3D objects must actually be rendered on
the screen so they ultimately end up with
2D coordinates
• This projection onto the screen must take
into account the z-position of objects, as
well as perspective
Orientation
• Objects in a 3D world have spatial location
(position) and orientation
• Orientation is given in many different
forms: pitch, roll, yaw or Euler angles
(around each axis); quaternions
• In Processing can rotate in 2D or in 3D
around the axes
• In the Virtools setup Euler angles are used
for orientation
Object Representation
• 3D objects are represented with meshes; points
that are joined together in convex, planar
polygons (faces); typically these polygons are
triangles because then there is assurance that
they will be planar
• The set of points forms the mesh
• Each face of the mesh may have a material
associated with it; these materials can be
textures and/or colors
• Details (and realism) increase with the numbers
of polygons
Point Representation
• These representations allow for algorithms
for calculation of intersections, collisions,
positioning
• Also have algorithms to find which objects
are in front, partial view, occlusion
• For speed, objects can be surrounded by
a bounding box – allows quick calculation
of intersections
Graphics Pipeline
• Model the individual objects (color,
transformations, realism, where located)
and together they constitute a scene
• Render the scene (lights, shading,
camera, etc.) in an image
• Display the image as output
• If in a virtual environment have real-time,
interaction and navigation