Three-dimensional projections
Sometimes, projecting a complex function graph down to only two dimensions
results in images which are hard to interpret. For instance, functions
which are not injective appear to "map over themselves" in a
two-dimensional mapping window. In situations like this, we can instead
choose to project our four dimensions of data into three-space, taking
advantage of our visual capacity to interpret depth information. The extra
dimension allows us to spread out the information, hopefully making it
easier to understand.
Let (x,y) represent the real and imaginary domain-coordinates, and (u,v)
represent the real and imaginary image-coordinates. Orthogonal projection
from four-dimensions down to three means simply dropping one of the four
coordinates. There are four such choices, resulting in four simple
projections:
- (x,y,u,v) -> (x,y,u) .... The real part of the function
- (x,y,u,v) -> (x,y,v) .... The imaginary part of the function
- (x,y,u,v) -> (x,u,v) .... The real part of the inverse function
- (x,y,u,v) -> (y,u,v) .... The imaginary part of the inverse function
When you switch a mapping window to 3-d mode, the default projection is
the real part of the function (x,y,u). If you don't want to stick with the
default projection, you can use the "Projection Options" window to change
the projection.