For geometric specifications and geometric algebra, omdl adopts the following type specifications and conventions.

name	description
point	a list of numbers to identify a location in space
vector	a direction and magnitude in space
line	a start and end point in space (line wiki)
normal	a vector that is perpendicular to a given object
pnorm	a vector that is perpendicular to a plane
plane	a flat 2d infinite surface (plane wiki)
coords	a list of points in space
matrix	a rectangular array of values

When a particular dimension is expected, the dimensional expectation is appended to the end of the name after a '-' dash as in the following table.

name	description
point-Nd	a point in an 'N' dimensional space
vector-Nd	a vector in an 'N' dimensional space
line-Nd	a line in an 'N' dimensional space
coords-Nd	a coordinate list in an 'N' dimensional space
matrix-MxN	a 'M' by 'N' matrix of values

Lines and vectors

A vector has a direction and magnitude in space. A line, too, has direction and magnitude, but also has location, as it starts at one point in space and ends at another. Although a line can be specified in one dimension, most library functions operate on two and/or three dimensional lines. Operators in omdl make use of a common convention for specifying Euclidean vectors and straight lines as summarized in the following table:

Given two points 'p1' and 'p2', in space:

no.	form	description
1	p2	a vector from the origin to 'p2'
2	[p2]	a vector from the origin to 'p2'
3	[p1, p2]	a line from 'p1' to 'p2'

The functions is_point(), is_vector(), is_line(), line_dim(), line_tp(), line_ip(), vol_to_point(), and vol_to_origin(), are available for type identification and convertion.

Example

// points
p1 = [a,b,c]
p2 = [d,e,f]
 
// vectors
v1 = p2       = [d,e,f]
v2 = [p2]     = [[d,e,f]]
 
// lines
v3 = [p1, p2] = [[a,b,c], [d,e,f]]
 
v1 == v2
v1 == v2 == v3, iff p1 == origin3d

Planes

Operators in omdl use a common convention for specifying planes. A plane is identified by a point on its surface together with its normal vector specified by pnorm, which is discussed in the following section. A list with a point and normal together specify the plane as follows:

name	form
plane	[point, pnorm]

Planes' normal

The data type pnorm refers to a convention for specifying a direction vector that is perpendicular to a plane. Given three points 'p1', 'p2', 'p3', and three vectors 'v1', 'v2', 'vn', the planes' normal can be specified in any of the following forms:

no.	form	description
1	vn	the predetermined normal vector to the plane
2	[vn]	the predetermined normal vector to the plane
3	[v1, v2]	two distinct but intersecting vectors
4	[p1, p2, p3]	three (or more) non-collinear coplanar points

The functions is_plane() and plane_to_normal() are available for type identification and convertion.

Example

// points
p1 = [a,b,c];
p2 = [d,e,f];
p3 = [g,h,i];
 
// lines and vectors
v1 = [p1, p2] = [[a,b,c], [d,e,f]]
v2 = [p1, p3] = [[a,b,c], [g,h,i]]
vn = cross_ll(v1, v2)
 
// planes' normal
n1 = vn           = cross_ll(v1, v2)
n2 = [vn]         = cross_ll(v1, v2)
n3 = [v1, v2]     = [[[a,b,c],[d,e,f]], [[a,b,c],[g,h,i]]]
n4 = [p1, p2, p3] = [[a,b,c], [d,e,f], [g,h,i]]
 
n1 || n2 || n3 || n4
 
// planes
pn1 = [p1, n1]
pn2 = [p2, n2]
pn3 = [p3, n3]
pn4 = [n4[0], n4]
pn5 = [mean(n4), n4]
 
pn1 == pn4