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RADI: Computation of the Radius and of the Diameter of a spatial set,
plus various geometric descriptors.
E.g. put molecules in spheres, rectangular boxes, cylinders, ...
References:
M. Petitjean, Applications of the Radius-Diameter Diagram to
the Classification of Topological and Geometrical Shapes of
Chemical Compounds, J.Chem.Inf.Comput.Sci. 1992,32[4],331-337
M. Petitjean, About the Algebraic Solutions of Smallest Enclosing
Cylinders Problems. Appl. Alg. Eng. Comm. Comp., 2012, 23[3-4], 151-164
DOI 10.1007/s00200-012-0171-y, arXiv 1008.5259
Author email: petitjean.chiral@gmail.com
RADI reads the cartesian coordinates of the molecule. It extracts
the convex hull of the atomic positions, then computes the diameter
and the radius of the hull (smallest enclosing sphere).
Several shape coefficients are computed.
In addition to the quantities above, RADI computes the smallest size
of the set, i.e. it finds the two closest enclosing parallel planes.
The minimal height enclosing cylinder, the smallest enclosing box,
and the minimal radius enclosing cylinder are computed, too.
Input data and parameters:
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INPUT FORMAT:
BIO : Biosym (MSI) files
CAS : Reserved for internal purposes
HIN : Hyperchem-type files
ISU : Reserved for internal purposes
MDL : Cambridge Crystallographic Model files
ML2 : SYBYL Mol2 files
PDB : Protein Data Bank or Nucleic Acid Data Bank files
(only HEADER, ATOM, ENDMDL and END records are recognized)
SDF : Symyx Mol/SDF files
(data between 'M END' and '$$$$' are treated as comments)
XYZ : n+2 lines. Line 1: n; line 2: free comment,
Next n lines: label or atomic symbol, x, y, z
(separator: spaces; no tabulation allowed).
INPUT MOLEC FILE NAME: name of the input file containing the molecule
EPSTAB:
Generate randomly perturbated cartesian coordinates.
The coordinates are not modified when EPSTAB is negative or null.
Independant random 3-tuples (x,y,z) are added to the spatial atomic positions.
Each random 3-tuple follow an isotropic normal law of std.dev equal to EPSTAB.
with radius equal to EPSTAB and centered on the atomic position.
*** WE DO RECOMMEND TO USE THIS OPTION IN ORDER ***
*** TO AVOID POTENTIAL NUMERICAL INSTABILITIES ***
Most time, EPSTAB=1.D-7 is effective.
CYL:
When 'C' or 'c' is entered, the minimal radius enclosing cylinder is computed.
Enter a blank space to skip this calculation, which is time consuming.
Output results:
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EIGENVALUES of the inertia matrix, of the covariance matrix,
and percentages of inertia,
and direct orthonormal basis of eigenvectors
3D CONVEX HULL: the K atoms of the hull, i.e. the K vertices of the
smallest convex polyhedron containing the set of the
N atomic positions, and the corresponding list of
the oriented triangular faces (such that the
direct normal is the exterior one)
3D MOL. MEAN POINT: arithmetic mean of the N atomic positions
3D HULL MEAN POINT: arithmetic mean of the K atoms of the hull
The SURFACE and VOLUME of the full 3D convex hull (no cavity is considered)
DIAMETER: the distance D of the longest atom-pair of the molecule
The two D-extremal nodes are exhibited.
RADIUS: the radius R of the smallest sphere containing the N atoms
The center and the R-extremal nodes are exhibited.
MIN SPHERE CENTER: the center of this minimal sphere
GEOMETRICAL CONVEX-SHAPE COEFFICIENT : the ratio (D-R)/R
SURFACIC CONVEX-SHAPE COEFFICIENT:
The ratio: surface of the hull, divided by the surface of the minimal sphere,
followed by the square root of this ratio.
VOLUMIC CONVEX-SHAPE COEFFICIENT:
The ratio: volume of the hull, divided by the volume of the minimal sphere,
followed by the cubic root of this ratio.
SMALLEST SIZE:
The distance separating the two closest parallel planes enclosing
the convex hull.
In general one of these planes contains one node of the hull,
and the other plane contains three nodes, defining a triangular
face of the hull.
In some particular situations, each plane contains two nodes,
both defining an edge of the hull.
In both cases, these four nodes are exhibited and an unit vector
vector perpendicular to the parallel planes is output.
MINIMAL HEIGHT ENCLOSING CYLINDER:
(not to be confused with the minimal radius enclosing cylinder)
The height is identical to the SMALLEST SIZE above.
The height, the radius, and the contact points are given.
MINIMAL HEIGHT ENCLOSING BOX:
The height is identical to the SMALLEST SIZE above.
The second smallest size is minimal in the space orthogonal
to the unit vector associated to the first SMALLEST SIZE,
and the box is closed such that the third smallest is minimized, too.
All angles of the parallelepipedic box are right.
MID BOX: center of the box above.
ROTATION TO SET X,Y,Z PARALLEL TO BOX:
Applying this rotation (with or without prior translation)
to the molecule put it so that the largest size of the box
is parallel to the x axis and the smallest size of the box
is parallel to the z axis.
MINIMAL RADIUS ENCLOSING CYLINDER:
(not to be confused with the minimal height enclosing cylinder)
The radius, the height, and the contact points are given.
Remarks:
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The number of atoms is currently limited to 50000 for each molecule.
The source has to be recompiled to read larger molecules.
Perfectly planar molecules are not handled. When there are at least
four atoms in a planar molecule, it is suggested to slightly perturbate
the atomic coordinates to get a spatial set (EPSTAB should be postive).
More generally, if numerical instabilities occur, it is adviced to
perturbate the coordinates with the help of EPSTAB: a suggested value
is 1.d-7, although most conformers will not generate numerical
instabilities, so that EPSTAB=0 remains acceptable.
The computing time is worst-case bounded by O(K^5), K being the number
of vertices of the hull, and K is at worse equal to N.
The worst situation is not expected in chemistry, even for proteins.
When the minimal radius enclosing cylinder is computed,
the computing time is worst-case bounded by O(K^6).
For some molecules, this cylinder is not ensured to be the one
with the smallest radius, although the result may be exact,
and in any case it is indeed an enclosing cylinder.
It is due to numerical instabilities (lack of convergence),
which cannot be overcome, even by setting EPSTAB sligthly positive.
In this situation, a warning is printed.
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