# Linear mapping of sliced hyperbox

Paul Heckbert ph+ at cs.cmu.edu
Thu Oct 7 20:56:04 PDT 2004

```Isn't it true that p(hypercube ^ halfspace) = p(hypercube) ^ p(halfspace) ?
where p(x) -> y projects an n-dimensional volume x into a 2-dimensional
region y

If so, you could compute the zonogon the usual way (project the edge vectors
and sort them by angle, and then string them together in sorted order to
create the zonogon) and after that slice with the projected halfspace.

----- Original Message -----
From: "Yaron Berman" <yaronber at cs.huji.ac.il>
To: <compgeom-discuss at research.bell-labs.com>
Sent: Tuesday, October 05, 2004 3:43 AM
Subject: Linear mapping of sliced hyperbox

> Hello,
>
> It is well known that a linear mapping of an n-dimensional hyperbox is a
> zonotope, and when the mapping is to R^2, it is simply a convex
> centrally symmetric polygon with at most 2n vertices, which is easy to
> compute (in O(nlogn) time).
> My question is about the linear map of a hyperbox intersected with a
> halfspace whose boundary contains the origin (which is also the center
> of the box), that is a box sliced through its center.
> Is there some efficient way to compute the map (still a convex polygon),
> other than enumerating all the 2^n vertices of the sliced box and
> computing the convex hull of their map? If there is an efficient method,
> can it be extended to the case of a box intersected with a cone?
>
> Thanks very much for the assistance,
> Yaron
>
>
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