It is often necessary to identify whether a particular point in the
N-dimensional space is within the Delaunay tessellation of a set of
points in this N-dimensional space, and if so which N-simplex contains
the point and which point in the tessellation is closest to the desired
point. The functions tsearch and dsearch perform this
function in a triangulation, and tsearchn and dsearchn in
an N-dimensional tessellation.
To identify whether a particular point represented by a vector p
falls within one of the simplices of an N-simplex, we can write the
Cartesian coordinates of the point in a parametric form with respect to
the N-simplex. This parametric form is called the Barycentric
Coordinates of the point. If the points defining the N-simplex are given
by N + 1 vectors t(i,:), then the Barycentric
coordinates defining the point p are given by
p = beta * t
where beta contains N + 1 values that together as a vector represent the Barycentric coordinates of the point p. To ensure a unique solution for the values of beta an additional criteria of
sum (beta) == 1
is imposed, and we can therefore write the above as
p - t(end, :) = beta(1:end-1) * (t(1:end-1, :)
- ones (N, 1) * t(end, :)
Solving for beta we can then write
beta(1:end-1) = (p - t(end, :)) /
(t(1:end-1, :) - ones (N, 1) * t(end, :))
beta(end) = sum (beta(1:end-1))
which gives the formula for the conversion of the Cartesian coordinates of the point p to the Barycentric coordinates beta. An important property of the Barycentric coordinates is that for all points in the N-simplex
0 <= beta(i) <= 1
Therefore, the test in tsearch and tsearchn essentially
only needs to express each point in terms of the Barycentric coordinates
of each of the simplices of the N-simplex and test the values of
beta. This is exactly the implementation used in
tsearchn. tsearch is optimized for 2-dimensions and the
Barycentric coordinates are not explicitly formed.
idx = tsearch (x, y, t, xi, yi) ¶Search for the enclosing Delaunay convex hull.
For t = delaunay (x, y), finds the index in t
containing the points (xi, yi). For points outside the
convex hull, idx is NaN.
Programming Note: The algorithm is O(M*N) for locating
M points in N triangles. Performance is typically much faster if
the points to be located are in a single continuous path; a point is first
checked against the region its predecessor was found in, speeding up lookups
for points along a continuous path.
idx = tsearchn (x, t, xi) ¶[idx, p] = tsearchn (x, t, xi) ¶Find the simplexes enclosing the given points.
tsearchn is typically used with delaunayn:
t = delaunayn (x) returns a set of simplexes t,
then tsearchn returns the row index of t containing each point
of xi. For points outside the convex hull, idx is NaN.
If requested, tsearchn also returns the barycentric coordinates
p of the enclosing simplexes.
An example of the use of tsearch can be seen with the simple
triangulation
x = [-1; -1; 1; 1]; y = [-1; 1; -1; 1]; tri = [1, 2, 3; 2, 3, 4];
consisting of two triangles defined by tri. We can then identify which triangle a point falls in like
tsearch (x, y, tri, -0.5, -0.5) ⇒ 1 tsearch (x, y, tri, 0.5, 0.5) ⇒ 2
and we can confirm that a point doesn’t lie within one of the triangles like
tsearch (x, y, tri, 2, 2) ⇒ NaN
The dsearch and dsearchn find the closest point in a
tessellation to the desired point. The desired point does not
necessarily have to be in the tessellation, and even if it the returned
point of the tessellation does not have to be one of the vertices of the
N-simplex within which the desired point is found.
idx = dsearch (x, y, tri, xi, yi) ¶idx = dsearch (x, y, tri, xi, yi, s) ¶Return the index idx of the closest point in x, y
to the elements [xi(:), yi(:)].
The variable s is accepted for compatibility but is ignored.
idx = dsearchn (x, tri, xi) ¶idx = dsearchn (x, tri, xi, outval) ¶idx = dsearchn (x, xi) ¶[idx, d] = dsearchn (…) ¶Return the index idx of the closest point in x to the elements xi.
If outval is supplied, then the values of xi that are not
contained within one of the simplices tri are set to outval.
Generally, tri is returned from delaunayn (x).
The optional output d contains a column vector of distances between the query points xi and the nearest simplex points x.
An example of the use of dsearch, using the above values of
x, y and tri is
dsearch (x, y, tri, -2, -2) ⇒ 1
If you wish the points that are outside the tessellation to be flagged,
then dsearchn can be used as
dsearchn ([x, y], tri, [-2, -2], NaN) ⇒ NaN dsearchn ([x, y], tri, [-0.5, -0.5], NaN) ⇒ 1
where the point outside the tessellation are then flagged with NaN.