Homogenous Coordinates

Posted on Posted in Computer Science, Geometry, Math

Homogeneous coordinates introduce by August Ferdinand Mobius in 1827. Homogeneous coordinates also called as projective coordinates, partially call 4D coordinate. This 4D coordinate is extra dimension that usually defined with W variable, when work with some 3D object we think on 3D space euclidian geometry (X,Y,Z), and when adding W variable to be (X,Y,Z,W) the object will called projective geometry, the 4th space dimension also called as “projective space”and coordinate on projective space called homogeneous coordinate. So homogeneous coordinates is coordinates system that used in projective geometry.

Projective geometry is a topic of mathematics. It is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called “points at infinity“) to Euclidean points, and vice versa.

Why projective geometry?

• The camera always show projection image from captured object in the real world.
• Euclidian geometry is not optimum in central projection description. Why central projection?, this thing is important to reconstruction image that projected.
• In euclidian geometry, math to be more difficult.
• Projective geometry become alternate of representing object geometry or transformation with algebra.

Advantages of projective geometry

• Math to be more simple.
• The geometry relation doesn’t change.
• In euclidian, computation could be done but too difficult. (that might affected in time consumption and complexity of algorithm), With projective geometry, compute will be more easy.

Ok, back to the homogenous coordinates

Homogeneous coordinates has characteristics…

• A formula that use homogeneous coordinates usually more simple than cartesian.
• Able to represent infinity point to be finite coordinate.
• Single matrix could represent affine and projective transformation so that it will able to transform until 8 degree of freedom.

Notation

• Point $\chi$ in homogeneous x, in euclidian $x$
• Line $l$ in homogeneous l
• Plane $A$ in homogeneous A
• 2D vs 3D space, lowercase for 2D, capitalized for 3D

Definition

The representation of a geometric object is homogeneous if and $\lambda$x represent the same object for $\lambda \neq 0$

where the lambda is transformation value, for example

$\texttt{x}= \lambda \texttt{x}$

homogeneous

$x \neq \lambda x$

euclidian

when geometry object multiply with transformation value, the geometry object can’t be same again with the result of multiplied in euclidian world but not in homogeneous coordinates, the geometry object doesn’t change itself. So what make the object doesn’t change itself?, in homogeneous coordinate use n+1 dimensional vector to represent the same (n-dimensional) point. for example $\mathbb{R}^{2}$

$x = \begin{bmatrix} x\\ y \end{bmatrix} \rightarrow \texttt{x} = \begin{bmatrix} x\\ y\\ 1 \end{bmatrix}$

$\texttt{x} = \begin{bmatrix} u\\ v\\ w \end{bmatrix} = \begin{bmatrix} u/w\\ v/w\\ 1 \end{bmatrix} = \begin{bmatrix} x\\ y\\ 1 \end{bmatrix} \rightarrow x = \begin{bmatrix} x\\ y \end{bmatrix}$

so homogeneous coordinates of a point $\chi$ in the plane $\mathbb{R}^{2}$ is  a 3-dim. vector. where $\mathrm{w} \neq 0$

As you can see at the above picture, there have some point that i want to explain.

1. the coordinate between homogeneous and euclidian have a distance, this distance that put as n+1 dimension for the example the distance has a value 1, in other cases that may have different values.
2. the origin point of homogeneous coordinate is at the centre of green plane (0,0,0)
3. the real world captured object represented as <X,Y,Z>, the point on blue plane through red line from captured object is object represented in euclidian that correspondent with captured object as same object.

So far i had explained in this article, i just explain in 2D point object that represented in homogeneous coordinate and how correspond in euclidian coordinate and its how to convert back. This article have relation with Camera Model Part. 2, may that article can complete that not get in this page for the reader. Next article i will write how point at infinity, line and 3D object represented in homogeneous coordinate.

Summary

• Homogeneous coordinates used to simplify math model so that computation be more easy.
• Homogeneous coordinates used to not change object itself by transformation value.
• The key how transformed real object can recover in homogeneous coordinates to euclidian coordinate is n+1 dimension value. Usually have value 1, or α = 1, α ≠ 0

Reference

1. http://www.tomdalling.com/blog/modern-opengl/explaining-homogenous-coordinates-and-projective-geometry/