Meeting 3: Projection, Duality,
Inversion
6.838/4.214: Interactive Geometric
Data Structures and Computation
Presenters: Marc, Prof.
T
-> Presentation
<-
Resources:
To present:
-
Representation of 2D Cartesian
coordinates (naive approach)
-
Show expressions for line from
two points, point from two lines
-
Show that only "local" (non-infinite)
points are representable
-
Show that representing a direction
requires a "tag bit"
to distinguish x,y points
from x,y directions
-
Representation of homogeneous
coordinates
-
Show addition of line at infinity;
allows representation of directions
-
Operationally, extend (x,y)
to (x,y,w) representation
-
one point is excluded from
the representation
-
Show operators: cross
product of points is line; etc.
-
Show again the result of "degenerate"
cases from above
-
Show perspective transformation
(example of a collineation;
a transformation that takes
lines to lines, preserves incidence)
-
Initial look at predicates
(incidence, sidedness, etc.)
-
three points collinear <=>
dual statement?
-
three points a, b, c make "left
turn" <=> dual statement?
-
(note: ordering removes sign
ambiguities!)
-
what should be relation of
leftof (a,b,c), leftof (b,a,c), etc.?
-
exhibit a,b,c with inconsistent
predicates as floats
-
what predicates are preserved
under perspective?
-
Introduce notion of duality
between spaces
-
Show single duality (polar,
d -> 1/d), w/help of idual
-
What is the benefit of duality:
-
Detect degenerate configurations
(many copunctual lines)
-
Introduce convex hull (nails
in plane, wrapped with string, etc.)
-
Show point inside/outside convex
hull of other points
<==>
halfspace redundant/non-redundant
-
Intersection of convex polygons
-
Finding a line that intersects
set of line segments, etc.
-
Intersection of set of circles,
etc.
-
Show several other dualities,
from relabeling to more complex
-
For each, show equation, static
example, dynamic example
-
Generalizations to 3D
-
point (3D euclidean, 3D homogeneous),
plane
-
3 points <=> plane, 3 planes
<=> point; show
-
point and plane are dual, analogous
to 2D case
-
show relabeling dual; show
polar dual
-
what happens to 3D lines under
this duality? why?
To demonstrate:
-
in geometer:
-
in idual:
-
in sphere:
To discuss:
-
Primitive representation, data
structures
-
Fundamental operations on points,
lines in 2D
-
Extensions to higher dimensions
Meeting 4 (Monday, XXX [YYY]):
Previous Meeting .... Next
Meeting ... Course Page
Last modified: Feb 1998
Seth Teller, MIT Computer Graphics
Group, seth@graphics.lcs.mit.edu