What is homogeneous transform?

우선 homogeneous coordinate이 무엇인 지 생각해보자.

일례로, homogeneous coordinate에서는 (x, y)라는 하나의 포인트를 (x, y, 1)이라는 포인트로 표현한다.

이것은 projective plane 상의 점으로 생각하는 것이다. 조금 더 일반적으로 표현하자면 (wx, wy, w)로 표현할 수 있다. (하나의 표현으로 딱 정해진 것은 아님)

아래 페이지들을 보면 정말 설명이 잘 되어있기 때문에 나중에 다시 정리하도록 하자..

* Ref

http://blog.daum.net/shksjy/229

http://darkpgmr.tistory.com/78

Pose is described by just 3 numbers:

• Offset in the x-direction
• Offset in the y-direction
• A rotation angle

The homogeneous transformation is inefficient in this respect, requiring three times as many numbers as actually required. Some of those 9 numbers have fixed values like 0 or 1, and others repeated or repeated with a different sign. Its advantage is the ability to compound poses by matrix-matrix multiplication, or transform vectors by matrix-vector multiplication. These operations are very convenient to express in an environment like MATLAB, and with modern computers the extra computation required is almost insignificant.

In homogeneous coordinates a point and a line, on a plane, can both be represented by a 3-element vector.

An advantage of using the homogenous form is that it’s very easy to represent a line that is vertical where in Cartesian coordinates, that means that the grading of the line is equal to infinity. In homogeneous form, we treat that situation very simply and conveniently. We don’t need to introduce any infinities.

We also might be interested in how to describe a line that joins two points. So here we have two points and here’s a line that passes through those two points. Then the homogenous representation of the line which remember is the three vector is given simply by the cross-product of the two points that lie on the line. So a very, very simple way of finding the equation of a line that joins two points much, much more conveniently than it is for Cartesian coordinates.

* Ref :

[1] https://moocs.qut.edu.au/courses/814/learn

[2] https://moocs.qut.edu.au/courses/791/learn