| Homogeneous Coordinates |
Article Index for Homogeneous |
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| CATEGORIES ABOUT HOMOGENEOUS COORDINATES | |
| linear algebra | |
| projective geometry | |
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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 . 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 : On the other hand, let the notation : refer to the projection of 3-D point (''x'' : ''y'' : ''z'') onto the projective plane. The point {Link without Title} can be considered to be equal to an Equivalence Class of 3-D points which belong to the 3-D line passing through the points (''x'' : ''y'' : ''z'') and (0 : 0 : 0). If : is another projective point, then : Two 3-D points are Equivalent if their projections onto the projective plane are equal: : Thus, : ADDITION OF HOMOGENEOUS COORDINATES This distinction between brackets and parentheses means that addition of points in homogeneous coordinates will be defined in two different ways, depending on whether the coordinates are enclosed with brackets or parentheses. Consider once again the case of the projective plane. Addition of a pair of 3-D points is the same as for ordinary coordinates: : On the other hand, addition of a pair of projected points can be defined thus: : For projective 3-space, similar considerations apply. Addition of a pair of unprojected points is : whereas addition of a pair of projected points is : SCALAR MULTIPLICATION OF HOMOGENEOUS COORDINATES There are two kinds of scalar multiplication: one for unprojected points and another one for projected points. Consider a scalar ''a'' and an unprojected 3-D point (''x'' : ''y'' : ''z''). Then : Notice that : even though : Now consider the scalar ''a'' and a projected point : ''y'' : ''z'' . Then : so that : Notice however a special case - if , the above formula gives {Link without Title} as result, which as we know does not represent any point. Indeed is undefined, so this is not a flaw in the definition. LINEAR COMBINATIONS OF POINTS DESCRIBED WITH HOMOGENEOUS COORDINATES Let there be a pair of points A and '''B''' in projective 3-space, whose homogeneous coordinates are : : It is desired to find their Linear Combination where ''a'' and ''b'' are coefficients which can be adjusted at will, with the condition that , or (more exactly) that , to avoid degenerate points. There are three cases to consider:
The ''X'', ''Y'', and ''Z'' coordinates can be considered as Numerator s, whereas the ''W'' coordinate can be considered as a Denominator . To add homogeneous coordinates it is necessary that the denominator be common. Otherwise it is necessary to rescale the coordinates until all the denominators are common. Homogeneous coordinates are equivalent Up To any uniform rescaling. Both points are affine If both points are in affine 3-space, then and . Their linear combination is : :: :: :: Both points are at infinity If both points are on the plane at infinity, then ''W''''A'' = 0 and ''W''''B'' = 0. Their linear combination is : :::::: One point is affine and the other at infinity Let the first point be affine, so that . Then : :: :: which means that the point at infinity is "dominant". General case The calculation can also be carried over without distinguishing between cases, similarly to the addition of two points: : :: Starting from this, you can re-obtain the formulas for above cases. In particular, applying this formula in the degenerate cases gives us that summing with anything else produces again. USE IN COMPUTER GRAPHICS Homogeneous coordinates are frequently used in Computer Graphics as they allow all Affine Transformation to be represented by a matrix operation. A Translation in R^2:(''x'',''y'') -> (''x''+''a'',''y''+''b'') can be represented as : where column vectors are the homogeneous coordinates of the two point. All the Linear Transformation such as Rotation and Reflection can also be represented, by matrices of the form : Furthermore all Projective Transformation can be represented by other matrices. This representation simplifies calculation in computer graphics as all necessary transformations can be performed by matrix multiplications. SEE ALSO |
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