Computer Graphics (CS 4300) 2010S: Lecture 2

Today

• HW1 due next class meeting!
• coordinate frames
• segments, rays, and intersections 2D

Coordinate Frames

• components of an -dimensional vector define a point in an -dimensional coordinate frame as a linear combination of the -dimensional basis vectors defining the frame
• linear combination is computed as
• same as product of a matrix and a vector:
• (general multiplication and its uses in graphics will be covered later in the course)
• technically, any set of -dimensional vectors can form a basis for an -dimensional frame as long as they are linearly independent
• orthonormal bases are of particular interest: all and for all
• an orthonormal basis defines a Cartesian frame
• “standard” basis is
• , so
• note this is right-handed; if we had defined instead, it would have been left handed
• there are always exactly two options for the “handedness” (chirality) of a 3D orthonormal basis
• right-handed coordinate frames are typical in graphics, but not totally universal
• it is often useful to define other local coordinate frames with arbitrary pose relative to the standard global frame
• pose of an object is its position plus its orientation
• we will cover this more formally later, but basically, the pose of a local frame is defined by the location of its origin and an orthonormal basis
  • all of these are vectors in the global frame
  • the matrix defines the orientation of the local frame
  • to transform a vector from the local to the global frame, compute
  • note this does not change the magnitude of the vector, only its direction
    • this is because the columns of are orthonormal
  • since a vector is technically only a magnitude and a direction, with no specific “place in space”, no need for in this operation
  • to transform a point or location from the local to the global frame, compute
  • going the other way: to transform a vector from the global to the local frame, compute
    • this uses the fact that the inverse of an orthogonal matrix is its transpose (show easy derivation from above eqn for )
    • note this is the same as computing , as in the text
  • transforming a point or location from the global to the local frame: solve above eqn for
    • 
    • note this is not the same as
• there are algorithms to “square” up a set of vectors to ensure they are orthonormal
• also sometimes required to construct a basis given only two or one vector
  • if given two non-parallel vectors, use cross product to produce third, and square up if necessary
  • if given only one vector, first find some other non-parallel vector, then proceed as above

Rays and Segments

• points have already been discussed: an -dimensional vector can be used to represent a point in dimensions by giving a displacement relative to the origin in some coordinate frame
• a ray is a continuum of points starting at some location and continuing in a straight line to infinity
  • can be represented as a start point plus a unit vector giving the direction of the ray
  • def works in any dimension
• a line segment, or just a segment, is the continuum of points along a straight line between two given locations
  • usually represented as the two endpoints, typically identified as start and end
  • can also represent as start and a displacement from there to end
    • easy to convert:
  • either def works in any dimension
  • can define the “left” and “right” sides in 2D: stand at and face along
  • on what side of the segment does a given point fall?
    • compute component of , extending each to 3D with :
    • (note that and components of cross product of any two vectors each with will be 0 anyway)
    • if positive, is on left of segment
    • if zero, is coincident with line through segment
    • if negative, is on right of segment
    • this can be used e.g. to pick triangles in the plane
  • unit normal can also be used to answer this question (we will see this when studying line equations and the signed distance from a point to a line)
  • can parametrize points along the segment as where
    • works in any dimension
  • intersection of two 2D segments:
    • 
    • two linear equations in two unknowns
    • 
    • 
    • rearranging into standard form,
    • 
    • 
    • 
    • can solve for and (carefully):
      • e.g. use Cramer’s rule, watch out for “no solution”
      • same result can be derived using law of sines and cross product
        
      • let
      • law of sines:
      • treat , , and as 3D vectors with
      • thus, the magnitude any cross product will be just the z component
      • let
      • let
      • let
      • 
      • 
      • 
      • similarly,
      • 
      • 
      • if segments are collinear
      • segments intersect iff
  • same math works for segment/ray and ray/ray intersection, except the condition for the ray(s) is that

Next Time

• HW1 due!
• lines in 2D
• “triangle asteroids” example
• output devices
• reading on website

this page subject to change  |  last updated Mon Apr 19 13:06:50 EDT 2010  |  course homepage