![]() ![]() Well, before diving into derivations and mathematical complexity, let’s just start by looking at some very simple examples of homogeneous co-ordinates, and then we can maybe start to work out why this representation is actually beneficial afterwards. Okay… So that sounds interesting (especially the bit about projective transformations being easily represented by a matrix!) but that was all quite confusing, and we still don’t actually know what they are. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. In mathematics, homogeneous coordinates or projective coordinates are a system of coordinates used in projective geometry. ![]() Let’s see what our trusted Wikipedia has to say: So, you may ask, what exactly are homogeneous co-ordinates? One of the most important ideas to grasp when dealing with multiple view geometry in computer vision is the concept of projective geometry, and the associated homogeneous co-ordinates. For a softer entry into the topic, though, I would strongly recommend starting with this blog post! If you would like a more formal description on any of the topics subsequently covered, this online book could therefore serve as a helpful complimentary resource. ![]() NORMAL VECTOR 2D GEOMETRY FULLA rather “homogeneous” crowd, full of John Malkovich’sīefore diving in, it is worth emphasizing that everything covered in this post is derived from chapter 2.2 of the well known Hartley Zisserman book, in particular pages 26–29. ![]()
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