Totally OT math question about projections

Heiko Wundram (Beenic) wundram at
Tue Nov 6 12:14:34 PST 2007

Hi all!

I can't think straight anymore (it's a little too late), that's why I decided 
to post here, and maybe someone knows the answer before I'll dig my way 
through my uni maths books tomorrow. Just think of it as a brainteaser if you 
feel compelled to answer. ;-)

Anyway, here we go:

I have a photography of an object, which I need to process to calculate 
the "relative" width of an object based on the projection on the photographic 
2D surface.

I decided to go with the "Zentralprojektion" model (sorry, I don't know the 
english name, most probably that's the "vanishing point projection", but I'm 
not sure), and arrived at the following sum to get an (increasingly better 
with increasing n) upper bound on the (relative) width of the projected range 
0 <= xs <= xe (both taken from the left side of the image), when the 
vanishing point is projected at xv > xe  from the left of the image:

d = ( xe - xs ) / n
relwidth = sum(i=0,n)[ d / ( 1 - ( xs + i * d ) / xv ) ]

Relative width meaning that for xs and xe close to 0, the relative width is 
close to xe - xs, whereas moving right in the direction of xv it rapidly 
increases (probably exponentially, but I didn't check yet).

Just to make a small (ascii) picture of the variables involved:

+  \ |     +
+   \|     +
+    \     +
+    |\    +
+    | \   +
+  |*| /|  +
+  |*|/ |  +
+  | /  |  +
+  |/   |  +
+  /    |  +
+ /|    |  +
0  xs   xv

* being the object to "measure".

What I'm now looking for is the limit with n -> infinity of that sum, not 
because I couldn't live with an upper bound, but rather because I have to 
implement this (for the biggest part) in integer math, which is pretty close 
to impossible with the sum given above.

Anyway, if anybody can nudge me in the right direction where to look for the 
limit of this specific type of sum, I'll be immensely grateful!

Thanks in advance!

Heiko Wundram
Product & Application Development

More information about the freebsd-questions mailing list