Everything about Homogeneous Coordinates totally explained
In
mathematics,
homogeneous coordinates, introduced by
August Ferdinand Möbius in his 1827 work
Der barycentrische Calcul, allow
affine transformations to be easily represented by a matrix. Also they make calculations possible in
projective space just as
Cartesian coordinates do in
Euclidean space. The homogeneous coordinates of a point of projective space of dimension
n are usually written as (
x :
y :
z : ... :
w), a row vector of length
n + 1, other than (0 : 0 : 0 : ... : 0). Two sets of coordinates that are proportional denote the same point of projective space: for any non-zero scalar
c from the underlying field
K, (
cx :
cy :
cz : ... :
cw) denotes the same point. Therefore this system of coordinates can be explained as follows: if the projective space is constructed from a vector space
V of dimension
n + 1, introduce coordinates in
V by choosing a basis, and use these in
P(
V), the equivalence classes of proportional non-zero vectors in
V.
Taking the example of projective space of dimension three, there will be homogeneous coordinates (
x :
y :
z :
w). The
plane at infinity is usually identified with the set of points with
w = 0. Away from this plane we can use (
x/w,
y/w,
z/w) as an ordinary Cartesian system; therefore the affine space complementary to the plane at infinity is coordinatised in a familiar way, with a basis corresponding to (1 : 0 : 0 : 1), (0 : 1 : 0 : 1), (0 : 0 : 1 : 1).
If we try to intersect the two planes defined by equations
x =
w and
x = 2
w then we clearly will derive first
w = 0 and then
x = 0. That tells us that the intersection is contained in the plane at infinity, and consists of all points with coordinates (0 :
y :
z : 0). It is a line, and in fact the line joining (0 : 1 : 0 : 0) and (0 : 0 : 1 : 0). The line is given by the equation
»
where
μ is a scaling factor. The scaling factor can be adjusted to
normalize the coordinates (0 :
y :
z : 0), thereby eliminating one of the two
degrees of freedom. The result is a set of points with only one degree of freedom, as is expected for a line.
Brackets versus parentheses
Consider projective 2-space: points in the projective plane are projections of points in 3-space ("3-D points"). Let the notation
»
refer to one of these 3-D points. Let
»
refer to another 3-D point. Then
»
Furthermore all
projective transformations can be represented by other matrices. This representation simplifies calculation in computer graphics as all necessary transformations can be performed by matrix multiplications. As a result, a series of affine transformations can be combined simply by multiplying successive matrices together. This is at the heart of real-time graphics systems such as
OpenGL and
DirectX which can use modern
graphics cards to perform operations with homogeneous coordinates.
Further Information
Get more info on 'Homogeneous Coordinates'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://homogeneous_coordinates.totallyexplained.com">Homogeneous coordinates Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
