projection matrix (and others) synthesis?

projection matrix (and others) synthesis?

Post by fleks » Fri, 11 Jul 2003 18:42:33



could someone pass a link to a paper from the times them scientist dudes
were making first steps toward 3d as we know it ?
i would like to see and understand how they synthesized the matrix not just
copy their values to 3x3 array.

thanks

 
 
 

projection matrix (and others) synthesis?

Post by Jerzy Karczmarczu » Fri, 11 Jul 2003 19:02:50



> could someone pass a link to a paper from the times them scientist dudes
> were making first steps toward 3d as we know it ?

Oh, you mean people like Descartes, or a bit later Monge?

Well, links to their Web sites are unfortunately dead. But - perhaps you
might think it over - there are some University libraries in this best of
worlds.

Jerzy Karczmarczuk

 
 
 

projection matrix (and others) synthesis?

Post by j » Fri, 11 Jul 2003 19:30:43


you probably mean how the 4x4 transformation
matrix we use in OpenGL and such was derived ?
well, there's this one book anyone must have read:

http://www.amazon.com/exec/obidos/tg/detail/-/0201848406/002-9460732-...
?vi=glance

they dont go into history ( much ) but youll get a good
feel on how things were derived.

J
--
-------------------------------------------------------------
  please tear the sticker off my eddress before use
-------------------------------------------------------------


> > could someone pass a link to a paper from the times them scientist dudes
> > were making first steps toward 3d as we know it ?

> Oh, you mean people like Descartes, or a bit later Monge?

> Well, links to their Web sites are unfortunately dead. But - perhaps you
> might think it over - there are some University libraries in this best of
> worlds.

> Jerzy Karczmarczuk

 
 
 

projection matrix (and others) synthesis?

Post by Just d' FAQ » Sat, 12 Jul 2003 01:24:24




Quote:>could someone pass a link to a paper from the times them scientist dudes
>were making first steps toward 3d as we know it ?
>i would like to see and understand how they synthesized the matrix not just
>copy their values to 3x3 array.

I applaud your intent; but you seek an anachronism! Any such papers
predate the web, so are unlikely to be online. For modern computer
graphics probably the most important early paper is

  Roberts, L.G., Homogeneous Matrix Representation and Manipulation of
  N-dimensional Constructs. MIT Lincoln Laboratory, MS 1405, May 1965.

The idea of using a matrix had to wait for the matrix to be invented,
which did not happen until 1858.

<http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_dete...>

Yet the eventual entries of the matrices had been known for quite some
time, merely expressed in other forms.

Using mathematics you already understand, like cartesian coordinates,
trigonometry, and so on, you can probably recreate the matrix forms
for yourself. Start in 2D, which is much simpler.

 
 
 

projection matrix (and others) synthesis?

Post by fleks » Sat, 12 Jul 2003 16:54:50






> >could someone pass a link to a paper from the times them scientist dudes
> >were making first steps toward 3d as we know it ?
> >i would like to see and understand how they synthesized the matrix not
just
> >copy their values to 3x3 array.

> I applaud your intent; but you seek an anachronism! Any such papers
> predate the web, so are unlikely to be online. For modern computer
> graphics probably the most important early paper is

>   Roberts, L.G., Homogeneous Matrix Representation and Manipulation of
>   N-dimensional Constructs. MIT Lincoln Laboratory, MS 1405, May 1965.

> The idea of using a matrix had to wait for the matrix to be invented,
> which did not happen until 1858.

<http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Matrices_and_dete...
ts.html>

Quote:

> Yet the eventual entries of the matrices had been known for quite some
> time, merely expressed in other forms.

> Using mathematics you already understand, like cartesian coordinates,
> trigonometry, and so on, you can probably recreate the matrix forms
> for yourself. Start in 2D, which is much simpler.

but what about projection matrix, that didn't? have any appliance in the 19
century    -did it?
 
 
 

projection matrix (and others) synthesis?

Post by Jerzy Karczmarczu » Sat, 12 Jul 2003 17:23:03



> but what about projection matrix, that didn't? have any appliance in the 19
> century    -did it?

Projections have been elaborated in a 'modern' setting by Monge. It was end of
18th century, beginning of 19th. Mind you, the descriptive geometry was and
still is, differently but is, used by architects.

I would - if you really want to learn some basic ideas - postpone matrices just
a little bit. *BEFORE* learning the rotation matrices try to understand well the
Rodrigues formula.

Try to grasp the geometric meaning of some objects, take into account that they
remain as they are, independently of the coordinate system used for their
analytic description, while matrices representing vectors and transforms
obviously change. Concrete mathematics is a wonderful tool, I would die without,
but I strongly believe that intuition should come first.

Jerzy Karczmarczuk

 
 
 

projection matrix (and others) synthesis?

Post by Gernot Hoffma » Sat, 12 Jul 2003 18:03:33





> >could someone pass a link to a paper from the times them scientist dudes
> >were making first steps toward 3d as we know it ?
> >i would like to see and understand how they synthesized the matrix not just
> >copy their values to 3x3 array.

> I applaud your intent; but you seek an anachronism! Any such papers
> predate the web, so are unlikely to be online.

  ....

Some key words for a Google search:

Filippo Brunelleschi discovered the geometrical laws of perspective
projections (1401 demonstrated on the market place in Firenze/Florence).
1855 or 1858: Cayley invented the matrices.

Best regards --Gernot Hoffmann

 
 
 

projection matrix (and others) synthesis?

Post by Just d' FAQ » Sat, 12 Jul 2003 21:01:33




Quote:>but what about projection matrix, that didn't? have any appliance in the 19
>century    -did it?

Again we must distinguish between forms. First artists began to paint
perspective. Then mathematicians developed the theory of "projective
geometry". Later, homogeneous coordinates gave that theory an elegant
representation. And eventually matrices gave homogeneous transforms a
happy home. Only recently did computer graphics enter the picture. ;-)

  <http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Art.html>

Consider Leonardo da Vinci's portrait of Mona Lisa (La Gioconda).

  <http://www.ibiblio.org/wm/paint/auth/vinci/joconde/joconde.jpg>
  <http://www.louvre.fr/anglais/visite/peint/grands/31n111.htm>

Behind the lady herself is some lovely scenery, small in the distance.
So even though the artist knew nothing of projection matrices, still
he could render perspective around the year 1504.

Computer graphics is so obsessed with coordinates it is easy to forget
that Euclid and most geometers worked without them, using figures,
axioms, and proofs. Also there is a subtle distinction between talking
about perspective in Euclidean geometry and formalizing self-contained
projective geometry.

  <http://www.geometryalgorithms.com/history.htm>

 
 
 

projection matrix (and others) synthesis?

Post by Larry Deac » Sun, 13 Jul 2003 00:27:17


"Just d' FAQs"

Quote:> Again we must distinguish between forms. First artists began to paint
> perspective. Then mathematicians developed the theory of "projective
> geometry".

Thanks for the links and I like the this quote from that great math history
site:

" Descriptive geometry originated with Drer in this work although it was
only put on a sound mathematical basis in later work of Monge. One of the
methods of overcoming the problems of projection, and describing the
movement of bodies in space, is descriptive geometry. Drer's remarkable
achievement was through applying mathematics to art, he developed such
fundamentally new and important ideas within mathematics itself."

 
 
 

projection matrix (and others) synthesis?

Post by Gernot Hoffma » Wed, 16 Jul 2003 03:31:07



> Thanks for the links and I like the this quote from that great math history
> site:

> " Descriptive geometry originated with Drer in this work although it was
> only put on a sound mathematical basis in later work of Monge. One of the
> methods of overcoming the problems of projection, and describing the
> movement of bodies in space, is descriptive geometry. Drer's remarkable
> achievement was through applying mathematics to art, he developed such
> fundamentally new and important ideas within mathematics itself."

Before and after asking several experts I thought this might be wrong.
Drer had never applied mathematics (in our understanding, or as Descartes
would have done it).
He had described four apparatus' for the construction of perspective
images from a real scenery. Not even his own inventions - import from
Italy (which was an achievement as well).
Nothing new about the geometry - applications of Brunelleschi's great
discovery: he placed the IMAGE PLANE between the eye point and the scene.
This step was misssing in Euclids knowledge, though he knew already the
principle of the center of projection.

Sorry, I cant follow further discussions - travelling for several weeks
to Web-free regions.

Best regards  --Gernot Hoffmann

 
 
 

projection matrix (and others) synthesis?

Post by Larry Deac » Thu, 17 Jul 2003 02:41:25


"Gernot Hoffmann"

Quote:> Before and after asking several experts I thought this might be wrong.

Oh, my! How sad. I wonder what else this might be suspect on this site?
Thanks for saying pointing this out.