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Conor Joseph Mccoid
UNIGE
Commits
9e3d588f
Commit
9e3d588f
authored
Sep 10, 2021
by
conmccoid
Browse files
SND 2021: fixed notes to be printable when showing them
parent
17009563
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Presentations/SND - Lausanne 2021/TALK_SND_2021_robust.tex
View file @
9e3d588f
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@@ -208,11 +208,11 @@ There are 3 types of points that make up the intersection:
\note
{
Pick an edge of Y; this edge divides the plane in two, one side containing Y the reference triangle and one side without Y.
We can reparametrize the coordinates of the vertices of X to be in line with this edge:
p
_
i is the distance the i-th vertex of X is away from the edge, with some other coordinate q
_
i.
If p
_
i is positive then the vertex of X lies on the same side of the edge as Y.
If we repeat this for each of the edges of Y and p
_
i is positive for all of the edges, then the vertex of X must lie inside Y.
$
p
_
i
$
is the distance the i-th vertex of X is away from the edge, with some other coordinate
$
q
_
i
$
.
If
$
p
_
i
$
is positive then the vertex of X lies on the same side of the edge as Y.
If we repeat this for each of the edges of Y and
$
p
_
i
$
is positive for all of the edges, then the vertex of X must lie inside Y.
If some p
_
i are positive and some negative then there are intersections between this edge of Y and those of X.
If some
$
p
_
i
$
are positive and some negative then there are intersections between this edge of Y and those of X.
Therefore, by discovering which vertices of X lie inside Y we have determined which edges of X intersect which edges of Y.
}
\begin{figure}
...
...
@@ -434,7 +434,7 @@ Now the sign of the numerator defines which side of the vertex of Y the face of
\note
{
How can we apply this to higher dimensions?
Suppose we now have the n-simplex with unit edges that align with the coordinate axes, which denote by vectors e eta.
Suppose we intersect this with another n-simplex with vertices at the points x
_
i for n+1 values of i.
Suppose we intersect this with another n-simplex with vertices at the points
$
x
_
i
$
for n+1 values of i.
At a given step in the algorithm we need to know the intersection between an m-dimensional `face' of X and m hyperplanes of Y.
This intersection can be written as this quotient.
The numerator of this quotient is shared over m of these intersections, keeping the given m-face of X constant but changing the m hyperplanes.
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