| Andrew Cooke | Contents | Latest | RSS | Twitter | Previous | Next


Welcome to my blog, which was once a mailing list of the same name and is still generated by mail. Please reply via the "comment" links.

Always interested in offers/projects/new ideas. Eclectic experience in fields like: numerical computing; Python web; Java enterprise; functional languages; GPGPU; SQL databases; etc. Based in Santiago, Chile; telecommute worldwide. CV; email.

Personal Projects

Lepl parser for Python.

Colorless Green.

Photography around Santiago.

SVG experiment.

Professional Portfolio

Calibration of seismometers.

Data access via web services.

Cache rewrite.

Extending OpenSSH.

C-ORM: docs, API.

Last 100 entries

Re: Quick message - This link is broken; Adding Reverb To The Echo Chamber; Sox Audio Tools; Would This Have Been OK?; Honesty only important economically before institutions develop; Stegangraphy via PS4; OpenCL Mess; More Book Recommendations; Good Explanation of Difference Between Majority + Minority; Musical Chairs - Who's The Privileged White Guy; I can see straight men watching this conversation and laffing; When it's Actually a Source of Indignation and Disgust; Meta Thread Defending POC Causes POC To Close Account; Indigenous People Of Chile; Curry Recipe; Interesting Link On Marginality; A Nuclear Launch Ordered, 1962; More Book Recs (Better Person); It's Nuanced, And I Tried, So Back Off; Marx; The Negative Of Positive; Jenny Holzer Rocks; Huge Article on Cultural Evolution and More; "Ignoring language theory"; Negative Finger Counting; Week 12; Communication Via Telecomm Bids; Finding Suspects Via Relatives' DNA From Non-Crime Databases; Statistics and Information Theory; Ice OK in USA; On The Other Hand; (Current Understanding Of) Chilean Taxes / Contributions; M John Harrison; Playing Games on a Cloud GPU; China Gamifies Real Life; Can't Help Thinking It's Thoughtcrime; Mefi Quotes; Spray Painting Bike Frame; Weeks 10 + 11; Change: No Longer Possible To Merge Metadata; Books on Old Age; Health Tree Maps; MRA - Men's Rights Activists; Writing Good C++14; Risk Assessment - Fukushima; The Future of Advertising and Surveillance; Travelling With Betaferon; I think I know what I dislike so much about Metafilter; Weeks 8 + 9; More; Pastamore - Bad Italian in Vitacura; History Books; Iraq + The (UK) Governing Elite; Answering Some Hard Questions; Pinochet: The Dictator's Shadow; An Outsider's Guide To Julia Packages; Nobody gives a shit; Lepton Decay Irregularity; An Easier Way; Julia's BinDeps (aka How To Install Cairo); Good Example Of Good Police Work (And Anonymity Being Hard); Best Santiago Burgers; Also; Michael Emmerich (Vibrator Translator) Interview (Japanese Books); Clarice Lispector (Brazillian Writer); Books On Evolution; Looks like Ara (Modular Phone) is dead; Index - Translations From Chile; More Emotion in Chilean Wines; Week 7; Aeon Magazine (Science-ish); QM, Deutsch, Constructor Theory; Interesting Talk Transcripts; Interesting Suggestion Of Election Fraud; "Hard" Books; Articles or Papers on depolarizing the US; Textbook for "QM as complex probabilities"; SFO Get Libor Trader (14 years); Why Are There Still So Many Jobs?; Navier Stokes Incomplete; More on Benford; FBI Claimed Vandalism; Architectural Tessellation; Also: Go, Blake's 7; Delusions of Gender (book); Crypto AG DID work with NSA / GCHQ; UNUMS (Universal Number Format); MOOCs (Massive Open Online Courses); Interesting Looking Game; Euler's Theorem for Polynomials; Weeks 3-6; Reddit Comment; Differential Cryptanalysis For Dummies; Japanese Graphic Design; Books To Be Re-Read; And Today I Learned Bugs Need Clear Examples; Factoring a 67 bit prime in your head; Islamic Geometric Art; Useful Julia Backtraces from Tasks; Nothing, however, is lost with less discomfort than that which, when lost, cannot be missed; Article on Didion

© 2006-2015 Andrew Cooke (site) / post authors (content).

Cube - Series of Images for Laser Printer

From: "andrew cooke" <andrew@...>

Date: Fri, 26 Dec 2008 17:29:04 -0300 (CLST)

I've been messing around with Python (using NumPy for the calculations and
PyScript for the final Postscript rendering), generating a series of
images based on a cube that is "wrapped" by one or more lines.

Initially points on the surface of a cube are generated in 3D - this is
implemented using a Python generator.  Then they are filtered by a
"pipeline" of transforms that filter, project to 2D, and scale/rotate.

The initial generator takes as parameters further generators, one for each
line).  So, for example, the length of each line might be given by
"constant(10)", or "repeated_uniform(1, 5)" (repeated sampling from a
uniform distribution from 1 to 5), or [1, 2, 3] (lengths for 3 lines).

The transforms are not necessarily generators - they are functions that
are composed.  This is because some transforms require the entire set of
points to function correctly.  But in practice most, those that operate on
individual points, are implemented as generators.


I'm wondering about printing some out on "nice" paper and seeing if I can
get a gallery to stock them.  Is that crazy?  I have no idea...



From: "andrew cooke" <andrew@...>

Date: Sat, 27 Dec 2008 08:43:40 -0300 (CLST)

The conversion from 3D (spatial model) to 2D points (paper plot) is worth
recording.  This is a summary of what I learnt from googling around - it
may not be correct (the info on the web about projective geometry isn't
completely consistent).

There are two common kinds of projections - orthographic and perspective.

Orthographic projections are calculated using an affine transformation - 
a linear calculation using a matrix multiplication (with a shift that can
be implemented using an additional vector addition, or by using an extra
dimension with "1" extending the vectors).

Orthographic projections can be further divided into three groups.  First,
those perpendicular to "principal" planes (like plans), which are not very
interesting here.  Second, axonometric projections which show a "skew"
view.  These are equivalent to a perspective projection "from infinity"
(so parallel lines remain parallel in projection) and the most common is
probably the isometric view.  Third, oblique projections where one plane
is shown "face on" while others are skewed.  The most common here is
"cavalier perspective" with an angle of 45 degrees and no scaling of

My code for the orthographic projections is:

  def _orthographic(transform, offset):
    t = np.asarray(transform)
    o = np.asarray(offset)
    return lambda p: np.dot(t, p) + o

  def cavalier(offset=(0,0)):
    rsq2 = 1 / sqrt(2)
    return per_point(
	transform=((-1,rsq2,0), (0,-rsq2,1)),

  def isometric(offset=(0,0)):
    s32 = sqrt(3) / 2
    return per_point(

where "per_point" is the function to convert from a pointwise transform to
one for the whole image (image is a list of paths; paths are lists of

I couldn't find a simple description  of how to calculate a perspective
projection, but it's simple to derive by hand if you construct the
projection by taking the figure generated by rays from eye to object that
intersect a plane (the "image plane").

I imagine the image plane as a sheet hanging in space, through which I can
vaguely see the object that is to be projected; the perspective projection
is just tracing the outline seen.

For an "undistorted" view the image plane is perpendicular to the line of
sight to the "centre" of the object (ie the point on the object that will
be at the centre of the image).  Also, vertical lines in the object will
be vertical in the image.  This is sufficient to define the geometry if we
place the eye at E (capital letters for vectors) and the centre of the
image at C (so the image plane contains C and is perpendicular to EC).

I will use, for example, EC as shorthand for C-E - the vector from E to C
(the reversal of order between "EC" and "C-E" is a real pain).

There are two coordinate systems in use here, confusingly: Z is the
spatial vertical direction, while X and Y are the horizontal and vertical
directions in the image plane.  From the discussion above
("..undistorted...") we want Y to appear (from E) parallel to Z.  We can
use this to define X and Y in terms of C, E and Z.

First, let us define unit(V) = V / |V| (where |V| is the length of V, an
arbitrary vector) and N = unit(CE) (the normal to the image plane,
pointing towards E).

Then X = unit(Z^N) since it is perpendicular to N (in the image plane) and
perpendicular to the vertical.  I am using "^" for the "cross" or "vector"

Similarly, Y = N^X

Now we can do the projection.  The eye looks towards a point P on the
object.  The line EP intersects the image plane at Q (there is no need for
Q to be between E and P - it may lie on the extended line, in which case
we get weird "fisheye" effects).

So the projection consists in finding Q for each P on the object and then
resolving Q along X and Y.

How to find Q?

We know that Q lies on EP, so for some parameter t, Q = E + t EP (a simple
parametrised line).  We also know that Q is in the image line and so is
perpendicular to N: 0 = N.Q (where "." is the "dot" or "scalar" product).

We can therefore solve for t:
Q = E + t EP
0 = N.Q
0 = N.(E + t EP)
-(N.E) = t (N.EP)
t = -(N.E) / (N.EP)

So Q = E - EP (N.E) / (N.EP)

The final projection to 2D coords is then just the dot product "along" X
and Y: (x, y) = (Q.X, Q.Y).

My code is a bit ugly, but seems to work:

  def perspective(eye, screen):
    def unit(v):
      return v / sqrt(np.dot(v, v))
    eye = np.asarray(eye)
    screen = np.asarray(screen)
    normal = eye - screen
    normal2 = np.dot(normal, normal)
    n = normal / sqrt(normal2)
    # screen axes
    x = unit(np.cross(Z, n))
    y = np.cross(n, x)
    def process(p1):
      los = p1 - eye
      t = -normal2 / np.dot(los, normal)
      p2 = eye + t * los # point in the plane
      return (np.dot(p2, x), np.dot(p2, y))
    return per_point(process)


Comment on this post