# Unique representation of line in 3D space

In a 3D Euclidean Cartesian space, a plane can be represented by the vector from origin to the nearest point in plane. This vector can be expressed in spherical coordinates $$(r,\theta,\phi)$$, with the condition that vector $$(1,\theta,\phi)$$ must be always orthogonal to the plane (this additional condition is necessary for the case when origin is included in the plane, r=0). Moreover, $$r \ge 0$$, $$0 \le \theta \lt 2\pi$$ and $$0 \le \phi \le \pi$$, $$\theta=0$$ if $$\phi=0$$ or $$=\pi$$, as usual for uniqueness of a point in spherical coordinates, and $$\theta<\pi$$ if $$r=0$$ as @Berci comment explains.

This representation is specially useful for my objectives because every plane has one and only one representation (for a plane, it is the only representation I known that fulfills this condition).

I'm looking now for an equivalent in the case of a line in 3D space. A starting point can be the vector from origin to nearest point in line, given also in spherical coordinates, $$(r,\theta,\phi)$$. In addition, a unary vector to define the line direction $$(1,\theta',\phi')$$.

However, this representation is not unique, it has one degree of freedom. The origin of this indeterminate in the representation is that vectors $$(r,\theta,\phi)$$ and $$(1,\theta',\phi')$$ must be orthogonal, but this condition is not implicit in (used to define) the definition of the line representation. $$tan(\phi)tan(\phi')cos(\theta-\theta')=-1$$.

Kwons someone a 3D line representation, valid for any line and unique for each one ? Or it doesn't exists ?

• Your assignment $(r, \theta, \phi)$ indeed describes a plane almost always uniquely, but for $r=0, we have *two* orthogonal directions, so$(0,\theta,\phi)$and$(0,\theta+\pi,\phi)\$ correspond to the same plane.. – Berci Mar 22 at 14:17
• @Berci, fixed in the question, thanks a lot. – pasaba por aqui Mar 22 at 14:44
• There are plenty of ways to uniquely specify lines and planes. All you have to do is take an arbitrary means of representation, then restrict the allowed values until each line has only one representation left. You've actually done this restriction with your spherical coordinates representation. So what really matters is what other properties you want your representation to have. For example, although you didn't explicitly say so, your post sounds like you want a representation based on spherical coordinates. (FYI - if you don't mind wildly discontinuous representations, you do it in 1D.) – Paul Sinclair Mar 23 at 0:36
• If you are interested in defining lines, planes and other affine sub-spaces, would it not be more natural to work in a system of Cartesian co-ordinates ? – gandalf61 Apr 3 at 11:43
• @gandalf61: using cartesians, even in the case of a plane, I do not see how to define them with only 3 values, in a complete and unique way. – pasaba por aqui Apr 3 at 12:00