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# 15. `geomtools` — Basic geometrical operations.

This module defines some basic operations on simple geometrical entities such as lines, triangles, circles, planes.

`Classes defined in module geomtools`

`Functions defined in module geomtools`

`geomtools.``areaNormals`(x)

  Compute the area and normal vectors of a collection of triangles.

  x is an (ntri,3,3) array with the coordinates of the vertices of ntri triangles.

  Returns a tuple (areas,normals) with the areas and the normals of the triangles. The area is always positive. The normal vectors are normalized.

`geomtools.``degenerate`(area, normals)

  Return a list of the degenerate faces according to area and normals.

  area,normals are equal sized arrays with the areas and normals of a list of faces, such as the output of the `areaNormals()` function.

  A face is degenerate if its area is less or equal to zero or the normal has a nan (not-a-number) value.

  Returns a list of the degenerate element numbers as a sorted array.

`geomtools.``levelVolumes`(x)

  Compute the level volumes of a collection of elements.

  x is an (nelems,nplex,3) array with the coordinates of the nplex vertices of nelems elements, with nplex equal to 2, 3 or 4.

  If nplex == 2, returns the lengths of the straight line segments. If nplex == 3, returns the areas of the triangle elements. If nplex == 4, returns the signed volumes of the tetraeder elements. Positive values result if vertex 3 is at the positive side of the plane defined by the vertices (0,1,2). Negative volumes are reported for tetraeders having reversed vertex order.

  For any other value of nplex, raises an error. If succesful, returns an (nelems,) shaped float array.

`geomtools.``smallestDirection`(x, method='inertia', return_size=False)

  Return the direction of the smallest dimension of a Coords

  - x: a Coords-like array
  - method: one of ‘inertia’ or ‘random’
  - return_size: if True and method is ‘inertia’, a tuple of a direction vector and the size along that direction and the cross directions; else, only return the direction vector.

`geomtools.``distance`(X, Y)

  Returns the distance of all points of X to those of Y.

  Parameters:

  - X: (nX,3) shaped array of points.
  - Y: (nY,3) shaped array of points.

  Returns an (nX,nT) shaped array with the distances between all points of X and Y.

`geomtools.``closest`(X, Y, return_dist=True)

  Find the point of Y closest to points of X.

  Parameters:

  - X: (nX,3) shaped array of points
  - Y: (nY,3) shaped array of points
  - return_dist: bool. If False, only the index of the closest point is returned. If True (default), the distance are also returned.

  Returns:

  - ind: (nX,) int array with the index of the closest point in Y to the points of X
  - dist: (nX,) float array with the distance of the closest point. This is equal to length(X-Y[ind]). It is only returned if return_dist is True.

`geomtools.``closestPair`(X, Y)

  Find the closest pair of points from X and Y.

  Parameters:

  - X: (nX,3) shaped array of points
  - Y: (nY,3) shaped array of points

  Returns a tuple (i,j,d) where i,j are the indices in X,Y identifying the closest points, and d is the distance between them.

`geomtools.``projectedArea`(x, dir)

  Compute projected area inside a polygon.

  Parameters:

  - x: (npoints,3) Coords with the ordered vertices of a (possibly nonplanar) polygonal contour.
  - dir: either a global axis number (0, 1 or 2) or a direction vector consisting of 3 floats, specifying the projection direction.

  Returns a single float value with the area inside the polygon projected in the specified direction.

  Note that if the polygon is planar and the specified direction is that of the normal on its plane, the returned area is that of the planar figure inside the polygon. If the polygon is nonplanar however, the area inside the polygon is not defined. The projected area in a specified direction is, since the projected polygon is a planar one.

`geomtools.``polygonNormals`(x)

  Compute normals in all points of polygons in x.

  x is an (nel,nplex,3) coordinate array representing nel (possibly nonplanar) polygons.

  The return value is an (nel,nplex,3) array with the unit normals on the two edges ending in each point.

`geomtools.``averageNormals`(coords, elems, atNodes=False, treshold=None)

  Compute average normals at all points of elems.

  coords is a (ncoords,3) array of nodal coordinates. elems is an (nel,nplex) array of element connectivity.

  The default return value is an (nel,nplex,3) array with the averaged unit normals in all points of all elements. If atNodes == True, a more compact array with the unique averages at the nodes is returned.

`geomtools.``triangleInCircle`(x)

  Compute the incircles of the triangles x

  The incircle of a triangle is the largest circle that can be inscribed in the triangle.

  x is a Coords array with shape (ntri,3,3) representing ntri triangles.

  Returns a tuple r,C,n with the radii, Center and unit normals of the incircles.

`geomtools.``triangleCircumCircle`(x, bounding=False)

  Compute the circumcircles of the triangles x

  x is a Coords array with shape (ntri,3,3) representing ntri triangles.

  Returns a tuple r,C,n with the radii, Center and unit normals of the circles going through the vertices of each triangle.

  If bounding=True, this returns the triangle bounding circle.

`geomtools.``triangleBoundingCircle`(x)

  Compute the bounding circles of the triangles x

  The bounding circle is the smallest circle in the plane of the triangle such that all vertices of the triangle are on or inside the circle. If the triangle is acute, this is equivalent to the triangle’s circumcircle. It the triangle is obtuse, the longest edge is the diameter of the bounding circle.

  x is a Coords array with shape (ntri,3,3) representing ntri triangles.

  Returns a tuple r,C,n with the radii, Center and unit normals of the bounding circles.

`geomtools.``triangleObtuse`(x)

  Checks for obtuse triangles

  x is a Coords array with shape (ntri,3,3) representing ntri triangles.

  Returns an (ntri) array of True/False values indicating whether the triangles are obtuse.

`geomtools.``lineIntersection`(P1, D1, P2, D2)

  Finds the intersection of 2 coplanar lines.

  The lines (P1,D1) and (P2,D2) are defined by a point and a direction vector. Let a and b be unit vectors along the lines, and c = P2-P1, let ld and d be the length and the unit vector of the cross product a*b, the intersection point X is then given by X = 0.5(P1+P2+sa*a+sb*b) where sa = det([c,b,d])/ld and sb = det([c,a,d])/ld

`geomtools.``displaceLines`(A, N, C, d)

  Move all lines (A,N) over a distance a in the direction of point C.

  A,N are arrays with points and directions defining the lines. C is a point. d is a scalar or a list of scalars. All line elements of F are translated in the plane (line,C) over a distance d in the direction of the point C. Returns a new set of lines (A,N).

`geomtools.``segmentOrientation`(vertices, vertices2=None, point=None)

  Determine the orientation of a set of line segments.

  vertices and vertices2 are matching sets of points. point is a single point. All arguments are Coords objects.

  Line segments run between corresponding points of vertices and vertices2. If vertices2 is None, it is obtained by rolling the vertices one position foreward, thus corresponding to a closed polygon through the vertices). If point is None, it is taken as the center of vertices.

  The orientation algorithm checks whether the line segments turn positively around the point.

  Returns an array with +1/-1 for positive/negative oriented segments.

`geomtools.``rotationAngle`(A, B, m=None, angle_spec=0.017453292519943295)

  Return rotation angles and vectors for rotations of A to B.

  A and B are (n,3) shaped arrays where each line represents a vector. This function computes the rotation from each vector of A to the corresponding vector of B. If m is None, the return value is a tuple of an (n,) shaped array with rotation angles (by default in degrees) and an (n,3) shaped array with unit vectors along the rotation axis. If m is a (n,3) shaped array with vectors along the rotation axis, the return value is a (n,) shaped array with rotation angles. Specify angle_spec=RAD to get the angles in radians.

`geomtools.``anyPerpendicularVector`(A)

  Return arbitrary vectors perpendicular to vectors of A.

  A is a (n,3) shaped array of vectors. The return value is a (n,3) shaped array of perpendicular vectors.

  The returned vector is always a vector in the x,y plane. If the original is the z-axis, the result is the x-axis.

`geomtools.``perpendicularVector`(A, B)

  Return vectors perpendicular on both A and B.

`geomtools.``projectionVOV`(A, B)

  Return the projection of vector of A on vector of B.

`geomtools.``projectionVOP`(A, n)

  Return the projection of vector of A on plane of B.

`geomtools.``pointsAtLines`(q, m, t)

  Return the points of lines (q,m) at parameter values t.

  Parameters:

  - q,`m`: (...,3) shaped arrays of points and vectors, defining a single line or a set of lines.
  - t: array of parameter values, broadcast compatible with q and m.

  Returns an array with the points at parameter values t.

`geomtools.``pointsAtSegments`(S, t)

  Return the points of line segments S at parameter values t.

  Parameters:

  - S: (...,2,3) shaped array, defining a single line segment or a set of line segments.
  - t: array of parameter values, broadcast compatible with S.

  Returns an array with the points at parameter values t.

`geomtools.``intersectionTimesLWL`(q1, m1, q2, m2, mode='all')

  Find the common perpendicular of lines (q1,m1) and lines (q2,m2)

  For non-intersecting lines, the base points of the common perpendicular are found. For intersecting lines, the common point of intersection is found.

  Lines are defined by a point (q) and a vector (m).

  Parameters:

  - qi,`mi` (i=1...2): (nqi,3) shaped arrays of points and vectors (mode=all) or broadcast compatible arrays (mode=pair), defining a single line or a set of lines.
  - mode: all to calculate the intersection of each line (q1,m1) with all lines (q2,m2) or pair for pairwise intersections.

  Returns a tuple of (nq1,nq2) shaped (mode=all) arrays of parameter values t1 and t2, such that the intersection points are given by `q1+t1*m1` and `q2+t2*m2`.

`geomtools.``intersectionPointsLWL`(q1, m1, q2, m2, mode='all')

  Return the intersection points of lines (q1,m1) and lines (q2,m2)

  with the perpendiculars between them.

  This is like intersectionTimesLWL but returns a tuple of (nq1,nq2,3) shaped (mode=all) arrays of intersection points instead of the parameter values.

`geomtools.``intersectionTimesLWP`(q, m, p, n, mode='all')

  Return the intersection of lines (q,m) with planes (p,n).

  Parameters:

  - q,`m`: (nq,3) shaped arrays of points and vectors (mode=all) or broadcast compatible arrays (mode=pair), defining a single line or a set of lines.
  - p,`n`: (np,3) shaped arrays of points and normals (mode=all) or broadcast compatible arrays (mode=pair), defining a single plane or a set of planes.
  - mode: all to calculate the intersection of each line (q,m) with all planes (p,n) or pair for pairwise intersections.

  Returns a (nq,np) shaped (mode=all) array of parameter values t, such that the intersection points are given by q+t*m.

  Notice that the result will contain an INF value for lines that are parallel to the plane.

`geomtools.``intersectionPointsLWP`(q, m, p, n, mode='all')

  Return the intersection points of lines (q,m) with planes (p,n).

  This is like intersectionTimesLWP but returns a (nq,np,3) shaped (mode=all) array of intersection points instead of the parameter values.

`geomtools.``intersectionTimesSWP`(S, p, n, mode='all')

  Return the intersection of line segments S with planes (p,n).

  Parameters:

  - S: (nS,2,3) shaped array (mode=all) or broadcast compatible array (mode=pair), defining a single line segment or a set of line segments.
  - p,`n`: (np,3) shaped arrays of points and normals (mode=all) or broadcast compatible arrays (mode=pair), defining a single plane or a set of planes.
  - mode: all to calculate the intersection of each line segment S with all planes (p,n) or pair for pairwise intersections.

  Returns a (nS,np) shaped (mode=all) array of parameter values t, such that the intersection points are given by (1-t)*S[...,0,:] + t*S[...,1,:].

  This function is comparable to intersectionTimesLWP, but ensures that parameter values 0<=t<=1 are points inside the line segments.

`geomtools.``intersectionSWP`(S, p, n, mode='all', return_all=False, atol=0.0)

  Return the intersection points of line segments S with planes (p,n).

  Parameters:

  - S: (nS,2,3) shaped array, defining a single line segment or a set of line segments.
  - p,`n`: (np,3) shaped arrays of points and normals, defining a single plane or a set of planes.
  - mode: all to calculate the intersection of each line segment S with all planes (p,n) or pair for pairwise intersections.
  - return_all: if True, all intersection points of the lines along the segments are returned. Default is to return only the points that lie on the segments.
  - atol: float tolerance of the points inside the line segments.

  Return values if return_all==True:

  - t: (nS,NP) parametric values of the intersection points along the line segments.
  - x: the intersection points themselves (nS,nP,3).

  Return values if return_all==False:

  - t: (n,) parametric values of the intersection points along the line segments (n <= nS*nP)
  - x: the intersection points themselves (n,3).
  - wl: (n,) line indices corresponding with the returned intersections.
  - wp: (n,) plane indices corresponding with the returned intersections

`geomtools.``intersectionPointsSWP`(S, p, n, mode='all', return_all=False, atol=0.0)

  Return the intersection points of line segments S with planes (p,n) within tolerance atol.

  This is like `intersectionSWP()` but does not return the parameter values. It is equivalent to:

      intersectionSWP(S,p,n,mode,return_all)[1:]

`geomtools.``intersectionTimesLWT`(q, m, F, mode='all')

  Return the intersection of lines (q,m) with triangles F.

  Parameters:

  - q,`m`: (nq,3) shaped arrays of points and vectors (mode=all) or broadcast compatible arrays (mode=pair), defining a single line or a set of lines.
  - F: (nF,3,3) shaped array (mode=all) or broadcast compatible array (mode=pair), defining a single triangle or a set of triangles.
  - mode: all to calculate the intersection of each line (q,m) with all triangles F or pair for pairwise intersections.
  Returns a (nq,nF) shaped (mode=all) array of parameter values t,
    such that the intersection points are given q+tm.

`geomtools.``intersectionPointsLWT`(q, m, F, mode='all', return_all=False)

  Return the intersection points of lines (q,m) with triangles F.

  Parameters:

  - q,`m`: (nq,3) shaped arrays of points and vectors, defining a single line or a set of lines.
  - F: (nF,3,3) shaped array, defining a single triangle or a set of triangles.
  - mode: all to calculate the intersection points of each line (q,m) with all triangles F or pair for pairwise intersections.
  - return_all: if True, all intersection points are returned. Default is to return only the points that lie inside the triangles.

  Returns:

  > If return_all==True, a (nq,nF,3) shaped (mode=all) array of intersection points, else, a tuple of intersection points with shape (n,3) and line and plane indices with shape (n), where n <= nq*nF.

`geomtools.``intersectionTimesSWT`(S, F, mode='all')

  Return the intersection of lines segments S with triangles F.

  Parameters:

  - S: (nS,2,3) shaped array (mode=all) or broadcast compatible array (mode=pair), defining a single line segment or a set of line segments.
  - F: (nF,3,3) shaped array (mode=all) or broadcast compatible array (mode=pair), defining a single triangle or a set of triangles.
  - mode: all to calculate the intersection of each line segment S with all triangles F or pair for pairwise intersections.

  Returns a (nS,nF) shaped (mode=all) array of parameter values t, such that the intersection points are given by (1-t)*S[...,0,:] + t*S[...,1,:].

`geomtools.``intersectionPointsSWT`(S, F, mode='all', return_all=False)

  Return the intersection points of lines segments S with triangles F.

  Parameters:

  - S: (nS,2,3) shaped array, defining a single line segment or a set of line segments.
  - F: (nF,3,3) shaped array, defining a single triangle or a set of triangles.
  - mode: all to calculate the intersection points of each line segment S with all triangles F or pair for pairwise intersections.
  - return_all: if True, all intersection points are returned. Default is to return only the points that lie on the segments and inside the triangles.

  Returns:

  > If return_all==True, a (nS,nF,3) shaped (mode=all) array of intersection points, else, a tuple of intersection points with shape (n,3) and line and plane indices with shape (n), where n <= nS*nF.

`geomtools.``intersectionPointsPWP`(p1, n1, p2, n2, p3, n3, mode='all')

  Return the intersection points of planes (p1,n1), (p2,n2) and (p3,n3).

  Parameters:

  - pi,`ni` (i=1...3): (npi,3) shaped arrays of points and normals (mode=all) or broadcast compatible arrays (mode=pair), defining a single plane or a set of planes.
  - mode: all to calculate the intersection of each plane (p1,n1) with all planes (p2,n2) and (p3,n3) or pair for pairwise intersections.

  Returns a (np1,np2,np3,3) shaped (mode=all) array of intersection points.

`geomtools.``intersectionLinesPWP`(p1, n1, p2, n2, mode='all')

  Return the intersection lines of planes (p1,n1) and (p2,n2).

  Parameters:

  - pi,`ni` (i=1...2): (npi,3) shaped arrays of points and normals (mode=all) or broadcast compatible arrays (mode=pair), defining a single plane or a set of planes.
  - mode: all to calculate the intersection of each plane (p1,n1) with all planes (p2,n2) or pair for pairwise intersections.

  Returns a tuple of (np1,np2,3) shaped (mode=all) arrays of intersection points q and vectors m, such that the intersection lines are given by `q+t*m`.

`geomtools.``intersectionTimesPOP`(X, p, n, mode='all')

  Return the intersection of perpendiculars from points X on planes (p,n).

  Parameters:

  - X: a (nX,3) shaped array of points (mode=all) or broadcast compatible array (mode=pair).
  - p,`n`: (np,3) shaped arrays of points and normals (mode=all) or broadcast compatible arrays (mode=pair), defining a single plane or a set of planes.
  - mode: all to calculate the intersection for each point X with all planes (p,n) or pair for pairwise intersections.

  Returns a (nX,np) shaped (mode=all) array of parameter values t, such that the intersection points are given by X+t*n.

`geomtools.``intersectionPointsPOP`(X, p, n, mode='all')

  Return the intersection points of perpendiculars from points X on planes (p,n).

  This is like intersectionTimesPOP but returns a (nX,np,3) shaped (mode=all) array of intersection points instead of the parameter values.

`geomtools.``intersectionTimesPOL`(X, q, m, mode='all')

  Return the intersection of perpendiculars from points X on lines (q,m).

  Parameters:

  - X: a (nX,3) shaped array of points (mode=all) or broadcast compatible array (mode=pair).
  - q,`m`: (nq,3) shaped arrays of points and vectors (mode=all) or broadcast compatible arrays (mode=pair), defining a single line or a set of lines.
  - mode: all to calculate the intersection for each point X with all lines (q,m) or pair for pairwise intersections.
  Returns a (nX,nq) shaped (mode=all) array of parameter values t,
    such that the intersection points are given by q+t*m.

`geomtools.``intersectionPointsPOL`(X, q, m, mode='all')

  Return the intersection points of perpendiculars from points X on lines (q,m).

  This is like intersectionTimesPOL but returns a (nX,nq,3) shaped (mode=all) array of intersection points instead of the parameter values.

`geomtools.``intersectionSphereSphere`(R, r, d)

  Intersection of two spheres (or two circles in the x,y plane).

  Computes the intersection of two spheres with radii R, resp. r, having their centres at distance d <= R+r. The intersection is a circle with its center on the segment connecting the two sphere centers at a distance x from the first sphere, and having a radius y. The return value is a tuple x,y.

`geomtools.``distancesPFL`(X, q, m, mode='all')

  Return the distances of points X from lines (q,m).

  Parameters:

  - X: a (nX,3) shaped array of points (mode=all) or broadcast compatible array (mode=pair).
  - q,`m`: (nq,3) shaped arrays of points and vectors (mode=all) or broadcast compatible arrays (mode=pair), defining a single line or a set of lines.
  - mode: all to calculate the distance of each point X from all lines (q,m) or pair for pairwise distances.

  Returns a (nX,nq) shaped (mode=all) array of distances.

`geomtools.``distancesPFS`(X, S, mode='all')

  Return the distances of points X from line segments S.

  Parameters:

  - X: a (nX,3) shaped array of points (mode=all) or broadcast compatible array (mode=pair).
  - S: (nS,2,3) shaped array of line segments (mode=all) or broadcast compatible array (mode=pair), defining a single line segment or a set of line segments.
  - mode: all to calculate the distance of each point X from all line segments S or pair for pairwise distances.

  Returns a (nX,nS) shaped (mode=all) array of distances.

`geomtools.``insideTriangle`(x, P, method='bary')

  Checks whether the points P are inside triangles x.

  x is a Coords array with shape (ntri,3,3) representing ntri triangles. P is a Coords array with shape (npts,ntri,3) representing npts points in each of the ntri planes of the triangles. This function checks whether the points of P fall inside the corresponding triangles.

  Returns an array with (npts,ntri) bool values.

`geomtools.``faceDistance`(X, Fp, return_points=False)

  Compute the closest perpendicular distance to a set of triangles.

  X is a (nX,3) shaped array of points. Fp is a (nF,3,3) shaped array of triangles.

  Note that some points may not have a normal with footpoint inside any of the facets.

  The return value is a tuple OKpid,OKdist,OKpoints where:

  - OKpid is an array with the point numbers having a normal distance;
  - OKdist is an array with the shortest distances for these points;
  - OKpoints is an array with the closest footpoints for these points and is only returned if return_points = True.

`geomtools.``edgeDistance`(X, Ep, return_points=False)

  Compute the closest perpendicular distance of points X to a set of edges.

  X is a (nX,3) shaped array of points. Ep is a (nE,2,3) shaped array of edge vertices.

  Note that some points may not have a normal with footpoint inside any of the edges.

  The return value is a tuple OKpid,OKdist,OKpoints where:

  - OKpid is an array with the point numbers having a normal distance;
  - OKdist is an array with the shortest distances for these points;
  - OKpoints is an array with the closest footpoints for these points and is only returned if return_points = True.

`geomtools.``vertexDistance`(X, Vp, return_points=False)

  Compute the closest distance of points X to a set of vertices.

  X is a (nX,3) shaped array of points. Vp is a (nV,3) shaped array of vertices.

  The return value is a tuple OKdist,OKpoints where:

  - OKdist is an array with the shortest distances for the points;
  - OKpoints is an array with the closest vertices for the points and is only returned if return_points = True.

`geomtools.``baryCoords`(S, P)

  Compute the barycentric coordinates of points P wrt. simplexes S.

  S is a (nel,nplex,3) shaped array of n-simplexes (n=nplex-1): - 1-simplex: line segment - 2-simplex: triangle - 3-simplex: tetrahedron P is a (npts,3), (npts,nel,3) or (npts,1,3) shaped array of points.

  The return value is a (nplex,npts,nel) shaped array of barycentric coordinates.

`geomtools.``insideSimplex`(BC, bound=True)

  Check if points are in simplexes.

  BC is an array of barycentric coordinates (along the first axis), which sum up to one. If bound = True, a point lying on the boundary is considered to be inside the simplex.

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