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

• Vectors are important for viewpoint, orientation, scaling, rotating and other transformations

(3,1,1)

Vectors (con’t)

• Length: ||v||=

• Addition, scalar multiplication

• Dot product:

• Cross-product: vector that is perpendicular to both

2 2 2x y z

|| || || || cosv w v w

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

X

Virtools Processing

Z Y

• 2D screen coordinate system:

X

Z

Y

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