Previous: Plotting the Triangulation, Up: Delaunay Triangulation [Contents][Index]

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

, then the Barycentric
coordinates defining the point `t`(`i`,:)`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

, finds the index in`t`= delaunay (`x`,`y`)`t`containing the points`(`

. For points outside the convex hull,`xi`,`yi`)`idx`is NaN.

`idx`=**tsearchn***(*`x`,`t`,`xi`)*[*`idx`,`p`] =**tsearchn***(*`x`,`t`,`xi`)Search for the enclosing Delaunay convex hull.

For

, finds the index in`t`= delaunayn (`x`)`t`containing the points`xi`. For points outside the convex hull,`idx`is NaN.If requested

`tsearchn`

also returns the Barycentric coordinates`p`of the enclosing triangles.

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

to the elements`x`,`y``[`

.`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`)

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`

.

Previous: Plotting the Triangulation, Up: Delaunay Triangulation [Contents][Index]