1. Laura Brown
Design Computation
Spring 2012

3. Context
How are three dimensional forms represented on two-dimensional
surfaces?
Camera Obscura using light and mirror to project image onto drawing surface

4. Brunelleschi method of using string to literally form lines from a single point
through an object and onto a plane

5. Masolino, Saint Peter Heals a Cripple,
1424-1425
Paolo Uccello, perspective
drawing of a chalice, 1450

7. Newest addition to this cannon—3D modeling
Like its predecessors, it is a system to describe objects in space
Like, say, one-point perspective, it has its own language and inaccuracies

8. If computer rendering of 3D objects is another way of
projecting three dimensions onto two, what characterizes
this strategy and the images it produces?
With the computer’s 3D modeling we no longer strive to
represent an object on paper but on the screen.
Already flattened representations evade the physicality of
objects. Models in the computer exist outside the
physicality even of paper or canvas.
As in the other examples listed, a revolution in how
space is represented is a revolution in how it is created
and perceived. The invention of perspective systems had
huge affects on architecture and by proxy on culture in
general. The invention of modeling software has likewise
affected the collective visual vocabulary through the
trends in architecture it has enabled.

9. Objects as the computer treats them—as pure math—a
sort of Platonic ideal? The perfect square that can never
exist? The infinitely thin surface?
What if the natural biases of computer modeling are
exploited?
A few things come to mind for me:
The idea of projection of an object onto the planar
computer screen
The possibility of objects that could never exist
outside this specific context
No longer are planar representations static—they can
be manipulated and re-built as pixels on our screens in
real time.

10. The goal:
Create a system resulting in a set of objects
which highlight/exploit the peculiarities
specific to the computer’s treatment and
representation of three-dimensional space.
Compile them in some form that goes
beyond a folder on my desktop of object
files. Looking towards distributable media.

11. Systems for projection
Human depth perception is dependent upon
stereoscopic vision.
In the computer, the view is from a single
camera’s point of view—depth perception is
dependent upon illusions created by shading
or forshortening (as in a drawing or painting)
PLUS the added ability to move things
around with your mouse

12. A sphere floating over a
plane? Or a series of both
curved lines and straight
lines on a plane?
Here the same model was rotated a bit—
the lines that looked like a tilted sphere
have lost their depth. While the spherical
isocurves of a ball retain their 3D
information when rendered to our 2D
screen, here they are projected
programmatically onto a plane in Rhino.
Only when the plane of our viewport is
parallel to the planar surface object in
Rhino does the projected image take on
a spherical appearance

13. Projecting onto non-planar surfaces, taking advantage of the computer’s ability to
handle 3D forms that don’t make sense in real space:
Two views of the same scene in Rhino. In the computer’s conception of the space,
the topmost form is spherical (really curve objects which mimic isocurves), the
middle form consists of curves which all lie on the same plane, and the
bottommost is the result of projecting the sphere not onto a planar surface but a
warped one. From above, all three appear the same. We come out with curves that
occupy all three of Rhino’s virtual dimensions but it is not at all what we started
with.

14. Process so far
1. Create isocurves
1. Pick a surface/object to create isocurves
2. Depending on the object type, the script treats it differently to try to
avoid error messages—for example weird things happen with
extrusions
3. Evaluate a given surface at regular intervals in its U and V domains
4. Save the resulting coordinates in a nested list so that index values
correspond to the points position on the surface
5. Connect the points—for each row of points, connect all the points
6. Do the same for columns.

15. Top and perspective views of setup
2. Project those curves onto another surface
Script is built so that projection direction is parallel to the z axis [0,0,-1]. This m
1. Pick the curves to project
2. Pick the surface to project onto
3. Again some if:then statements to try to avoid error messages and account
4. Voila your curves have been projected

16. Curves have been projected onto the Move the original curves and sphere away to see the new
surface ones
Top view—still looks like the top of a sphere

17. More process
3. Make them occupy space
Right now these curves are infinitely thin and don’t display very well
anywhere outside of Rhino. Curves are replaced by narrow tubular forms,
technically circles extruded along the original curve.
1. Select the curves to make into tubes
2. For each curve, find the first endpoint
3. Also find the vector tangent to the curve at this point
4. Find the plane normal to this vector
5. Add a circle at this endpoint and orient it according to this normal plane,
giving it an arbitrary radius. The default made sense for the scale I was
working at but it is by no means written in stone.
6. Extrude the circle along the curve

18. Examples

19. Using facets instead of isocurves
In this variation, the same points are evaluated on the surface of an object,
but now instead of connecting them and projecting the resulting curve, we
project the points themselves and then use them to create surfaces. As the
surface is evaluated, for each point a class is instantiated. Each point’s
coordinates are stored as well as three of its neighbors. Its X Y and Z values
are translated into R G B values. The points are then projected onto a
surfaces and the new coordinates are stored. The four stored points for each
instance of the class are used to make a surface, which is then colored
according to the color that was calculated earlier.
A cube projected onto an irregular surface using this method

20. Here, there is no projection at all, only random displacement along the z-axis
of the surfaces. Because the individual surfaces are moved only up and down,
not side to side, from above they appear to form a sphere. In reality they do
not form a flush surface.

21. Shortcomings
So many types of objects and surfaces, hard to make a script that will handle
them all nicely
Do I live with this limitation and just set up my objects in a way that the script
can handle, or do I spend time fixing the script but perhaps coming out with
fewer satisfying results—horizontal vs vertical investigation?
Assigning colors to the objects to create the illusion of depth/lighting rather
than relying on scene lighting. The computer’s lighting sometimes gives away
the fact that surfaces are not what they seem. Also, getting object color to
export has not worked for me yet.