Spatial geometry 7

Mapping

The last topic covered here is the concept of mapping between “local” and “world” coordinates. This is the foundation of all linear rigid transformations. Let’s build this step by step from 1D to 2D and finally 3D.

Local to World

We expressed a point “P” on a line using $P = O + \hat{u} * t$ where $[O, \hat{u}]$ are the origin and unit direction vector. The same concept in 2D expresses points on a plane using $P = O + \hat{u}*s + \hat{v}*t$ , where $[o, \hat{u},\hat{v}]$ are the plane’s origin and basis vectors. In 3D we used $P = O + \hat{u}*r+ \hat{v}*s + \hat{n}*t$, where $[o, \hat{u},\hat{v},\hat{n}]$ are the coordinate system’s origin and basis vectors.

We can think of the “r”, “s” and “t” values as “local” or “dependent” coordinates with respect to the underlying support geometry. The point expressions for “p” map local coordinates into space in the sense of the “world” coordinate system: $[ t ] → [ x, y, z]$ in 1D, $[ s,t ] → [ x, y, z]$ in 2D and $[ r,s,t ] → [ x, y, z]$.

World to Local

We used projections i.e. dot products, to recover the “t” value of a point “P” along a line $t = (P - O)· û$. In the same sense, as long as the basis vectors are orthogonal and unit length we can use the same expression to recover the “r”, “s”, and “t” values for the plane and coordinate system cases, using the appropriate basis vectors.

We thus developed the tools to map from the “world” space to “local” coordinates using the inverse mapping: $[ x, y, z ] → [ t ]$ in 1D, $[ x, y, z ] → [s, t ]$ in 2D and $[ x, y, z ] → [ r,s,t ]$ in 3D, and the relevant support geometries.

Transformations

This indirection allow us for the same “local” coordinates to replace their supporting geometries easily e.g. keep $[r,s,t]$ but replace $[ o, û, v̂, n̂ ]$ with $[O, X, Y, Z]$. We just have transformed from one frame of reference to another but retained the same geometric content. In addition, because linear maps do not affect lengths and angles we also call them rigid transforms.

Conclusions

We conclude here after having reviewed the arithmetic of points and vectors; topics generally already covered prior to university education in both math and physics. We focused here on a geometric interpretation of algebraic concepts to build intuition and familiarity with the mechanics. You may practice on constructive geometry using the problems within in section. Additional problems are also available [>].

You will be formally introduced to general linear algebra in the future terms by math and sciences courses. The concepts presented here will be generalized to even higher dimensions. We hope the spatial interpretations developed here will be helpful in the following terms as the scope and depth of understanding is expanded.