Geometric Functions

Geometric functions accept and return sequences that define the canonical form of geometrical entities. The canonical forms currently in use are:

• Point: {x,y,z} coordinates of the point

• Vector: {x,y,z} components of the vector; also called i,j,k components

• Line: {x,y,z,i,j,k} point on line and direction vector of the infinite line

• Plane: {d,i,j,k} distance from origin and normal vector

• Cylinder: {x,y,z,i,j,k,r} point on axis, axis vector and radius of the cylinder

It is important to note that geometric functions work with sequences and not arrays. The input arguments to geometric functions can be constructed using the $FSEQ function or the { } sequence operators. For example, the following are equivalent:

P1=$FSEQ(1,2,3)
P1={1,2,3}

Geometric functions will return a value of $NULL if the geometric entity cannot be created.

$FGCYPT3(point1, point2, point3)

Cylinder constructed from 3 points Returns: Cylinder sequence

Returns an {x,y,z,i,j,k,r} sequence defining a cylinder constructed through 3 points. The cylinder axis is perpendicular to a plane constructed through the 3 points, while also having the 3 points along its circumference. $NULL is returned if the points all lie on the same line.

$FGLNXPL(line, plane)

Intersection point of line and plane Returns: Point Sequence

Returns an {x,y,z} sequence defining the point at the intersection of an infinite line and plane. A value of $NULL is returned if the line does not intersect the plane.

$FGLSXCY(point1, point2, cylinder)	Intersection point(s) of line segment and cylinder Returns: Sequence
$FGLSXCY(point1, point2, point3, vector, radius)	Intersection point(s) of line segment and constructed cylinder Returns: Sequence

Returns a sequence defining the point(s) of intersection between a line segment and a cylinder. The line segment is defined by the start and ending points: point1 and point2. The cylinder is defined either by a single cylinder sequence or by 3 separate sequences defining its center point3, axis vector and radius. The return sequence has a length of 2, 5 or 8, with values as follows:

  (1) Number of intersection points: 0, 1 or 2
  (2) Segment location with respect to cylinder:
      –1: Segment is entirely inside the cylinder
       0: Segment intersects the cylinder
       1: Segment is entirely outside the cylinder
(3:5) First intersection point
(6:8) Second intersection point (furthest from start of line segment)

$FGLSXSP(point1, point2, point3, radius)

Intersection point(s) of line segment and sphere Returns: Sequence

Returns a sequence defining the point(s) of intersection between a line segment and sphere. The line segment is defined by the start and ending points: point1 and point2. The sphere is defined by its center point3 and radius. The return sequence has a length of 2, 5 or 8, with values as follows:

  (1) Number of intersection points: 0, 1 or 2
  (2) Segment location with respect to sphere:
      –1: Segment is entirely inside the sphere
       0: Segment intersects the sphere
       1: Segment is entirely outside the sphere
(3:5) First intersection point
(6:8) Second intersection point (furthest from start of line segment)

$FGPLPT3(point1, point2, point3)

Plane constructed from 3 points Returns: Plane Sequence

Returns a {d,i,j,k} sequence defining a plane constructed from 3 points. A value of $NULL is returned if 2 or more of the points are coincident.

$FGPLPTN(point, vector)

Plane constructed from point and normal vector Returns: Plane Sequence

Returns a {d,i,j,k} sequence defining a plane passing through a point and normal to a vector.

$FGPTRPL(point1, plane)	Projection of a point onto a plane Returns: Point Sequence
$FGPTRPL(point1, point2, vector)	Projection of a point onto a constructed plane Returns: Point Sequence

Returns an {x,y,z} sequence defining a point created as the projection of point1 onto a plane. The plane is defined either by a single plane sequence or as plane defined as passing through point2 and perpendicular to a vector.