Projective Geometry: Applications & Uses in the Real World

[mathmathmath]

hi everyone, I'm a very frustrated grade 11 student trying to figure out projective geometry.

i have been reading excerpts of several books, as well as some websites on the internet, but i am currently unable to synthesize the subject as a whole.

my teacher assigned this topic, and I am expected to present the topic to the class in a few days. this is not a math class, so really the math of projective geometry is not really crucial (other than basics), but it's more explaning what it means to us.

i guess the question i am trying to ask is, why do we study projective geometry? other than another intellectual curiousity, how can we use it? what applications are there in the real world that we can apply projective geometry to?

i have read a wikipedia article that briefly mentioned a man by the name of rudolph steiner who said projective geometry could be used to describe the real world (which seems to be perfect for my class). but sadly, it seems to be a very small topic in the topic of proj. goemetry. am i wrong? other studnets in the class have been assigned topics such as chaos theory(and my classmates would no doubt talk about the deterministic universe) and golden ratio(applications to aesthetics, seemingly ubiquitous ratio, etc.). perhaps steiner's topic is the topic the teacher expects me to talk about, and i feel it is a small topic because most books about projective geometry are about the maths(which i am not delving deeply into).

any opinions?

thanks
mathmathmath

 

[matt grime]

Opinions

1. Chaos theory is misrepresented, you'll hear complete rubbish about it in class. If the words topological and transitivity appear I will eat my hat.

2. There is no demonstrable relationship between aesthetics and phi (see the AMA article by keith devlin. use google to find it).

3. What I think by projective geometry probably isn't the stuff you need to know about (P^1, homogeneous coordinate rings etc), but projective things in general are useful exactly because they allow points at infinity. Sorry.

Bit of a non-post but I couldn't stand by and let someone apparently believe that chaos theory and aesthetics were more exciting than projective geometry.

The reference you give to Steiner appears to be supported by the first hit on google, but it also appears to be complete gibberish.

 

[Hurkyl]

I don't know if it will help for your purposes, but if I recall correctly, 3-D graphics are done entirely with projective geometry.

 

[shmoe]

A typical introduction involves the painter and his canvas, which I think lies at the roots of projective geometry. It gives a way of describing a 3-d space on a 2-d space (the canvas, or computer screen say). Probably lots to talk about around this.

There's also a relation between finite projective spaces (a finite numebr of points) and some combinatorial designs, I think mutually orthogonal latin squares is a common topic. There's also some kind of duality between the points and lines in projective space, if you swap the words 'lines' and 'points' everything essentially worked out the same. I'm pretty hazy on this, it's been a few years, but you might try looking in some combinatorics books for this sort of thing. You might have a hard time if you aren't planning on trying to understand the math yourself.

It seems like it might be worthwhile to compare the axioms of a projective space with other more familiar geometries.

I think I looked at the same site on Steiner as matt, a cursory view makes this look crackpottish. in the short biography of Steiner it claims,

"He developed spiritual science by applying the scientific method to his remarkable powers of clairvoyant perception.. When observing subtler aspects of existence he could change his consciousness so that instead of experiencing the world from a central point of view his consciousness moved to the cosmic periphery."

http://www.anth.org.uk/NCT/people.htm#steiner

I have a hard time taking that seriously.

 

[robphy]

Possibly useful to the OP:
http://www.math.utah.edu/~treiberg/Perspect/Perspect.htm
http://plus.maths.org/issue23/features/criminisi/
http://www.handprint.com/HP/WCL/tech10.html
http://www.unc.edu/courses/2004fall/comp/290/089/
http://www.cs.unc.edu/~marc/tutorial.pdf
http://mathforum.org/library/topics/projective_g/
http://www.du.edu/~jcalvert/optics/mirror.htm
http://www.math.toronto.edu/mathnet/questionCorner/projective.html

 

[mathmathmath]

thank you very much for those replies, especially the list of links.

i may be back soon as i try to wrap together by presentation though :)

 

[Sir_Deenicus]

On http://www.science.psu.edu/alert/Math10-2005.htm" . In one of the pictures you see a 2D projection (shadows) of a 3D projection of a 4D object.