Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül,^[1]^[2] are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of a point, even those at infinity, can be represented using finite coordinates. Often formulas involving homogeneous coordinates are simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix.

If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. An additional condition must be added on the coordinates to ensure that only one set of coordinates corresponds to a given point, so the number of coordinates required is, in general, one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point on the projective plane.

Introduction

The projective plane can be thought of as the Euclidean plane with additional points, so called points at infinity, added. There is a point at infinity for each direction, informally defined as the limit of a point that moves in that direction away from a fixed point. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. A given point (x, y) on the Euclidean plane is identified with two ratios (X/Z, Y/Z), so the point (x, y) corresponds to the triple (X, Y, Z) = (xZ, yZ, Z) where Z ≠ 0. Such a triple is a set of homogeneous coordinates for the point (x, y). Note that, since ratios are used, multiplying the three homogeneous coordinates by a common, non-zero factor does not change the point represented. So, unlike Cartesian coordinates, a single point is represented by many sets of homogeneous coordinates.

The equation of a line through the point (a, b) may be written l(x − a) + m(y − b) = 0 where l and m are not both 0. In parametric form this can be written x = a + mt, y = b − lt. Let Z=1/t, so the coordinates of a point on the line may be written (a + m/Z, b − l/Z)=((aZ + m)/Z, (bZ − l)/Z). In homogeneous coordinates this becomes (aZ + m, bZ − l, Z). In the limit as t approaches infinity, in other words as the point moves away from (a, b), Z becomes 0 and the homogeneous coordinates of the point become (m, −l, 0). So (m, −l, 0) are defined as homogeneous coordinates of the point at infinity corresponding to the direction of the line l(x − a) + m(y − b) = 0.

To summarize:

Any point in the projective plane is represented by a triple (X, Y, Z), called the homogeneous coordinates of the point, where X, Y and Z are not all 0.
The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor.
Conversely, two sets of homogeneous coordinates represent the same point only if one is obtained from the other by multiplying by a common factor.
When Z is not 0 the point represented is the point (X/Z, Y/Z) in the Euclidean plane.
When Z is 0 the point represented is a point at infinity.

Note that the triple (0, 0, 0) is omitted and does not represent any point. The origin is represented by (0, 0, 1).^[3]

Notation

Some authors use different notations for homogeneous coordinates which help distinguish them from Cartesian coordinates. The use of colons instead of commas, for example (x:y:z) instead of (x, y, z), emphasizes that the coordinates are to be considered ratios.^[4] Brackets, as in [x, y, z] emphasize that multiple sets of coordinates are associated with a single point. Some authors use a combination of colons and brackets, as in [x:y:z].^[5]

Homogeneity

Homogeneous coordinates are not uniquely determined by a point, so a function defined on the coordinates, say f(x, y, z), does not determine a function defined on points as with Cartesian coordinates. But a condition f(x, y, z) = 0 defined on the coordinates, as might be used to describe a curve, determines a condition on points if the function is homogeneous. Specifically, suppose there is k so that

f(\lambda x,\lambda y,\lambda z)=\lambda ^{k}f(x,y,z).\,

If a set of coordinates represent the same point as (x, y, z) then it can be written (λx, λy, λz) for some non-zero value of λ. Then

f(x,y,z)=0{\text{ if and only if }}f(\lambda x,\lambda y,\lambda z)=\lambda ^{k}f(x,y,z)=0.\,

A polynomial g(x, y) of degree k can be turned into a homogeneous polynomial by replacing x with x/z, y with y/z and multiplying by z^k, in other words by defining

f(x,y,z)=z^{k}g(x/z,y/z).\,

The resulting function f is a polynomial so it makes sense to extend its domain to triples where z = 0. The process can be reversed by setting z = 1, or

g(x,y)=f(x,y,1).\,

The equation f(x, y, z) = 0 can then be thought of as the homogeneous form of g(x, y) = 0 and it defines the same curve when restricted to the Euclidean plane. For example, the homogeneous form of the equation of the line ax + by + c = 0 is ax + by + cz = 0.^[6]

Other dimensions

The discussions in the preceding sections apply analogously to projective spaces other than the plane. So the points on the projective line may be represented by pairs of coordinates (x, y), not both zero. In this case point at infinity in this case is (1, 0). Similarly the points in projective n-space are represented by (n + 1)-tuples.^[7]

Alternate definitions

Another definition of project space is as equivalence classes. For non-zero element of R³, Define (x₁, y₁, z₁)~(x₂, y₂, z₂) to mean there is a non-zero λ so that (x₁, y₁, z₁)=(λx₂, λy₂, λz₂). Then ~ is an equivalence relations and the projective plane can be defined as the equivalence classes of R³ − {0}. If (x, y, z) is one of elements of the equivalence class p then these are taken to be homogeneous coordinates of p.

Lines in this space are defined to be sets of solutions of equations of the form ax + by + cz = 0 where not all of a, b and c are zero. The condition ax + by + cz = 0 depends only on the equivalence class of (x, y, z) so the equation defines a set of points in the projective line. The mapping (x, y)→(x, y, 1) defines and inclusion from the Euclidean plane to the projective plane and the complement of the image is the set of points with z=0. This is the equation of a line according to the definition and the complement is called the line at infinity.

The equivalence classes p are the lines through the origin with the origin removed. The origin does not really play an essential part in the previous discussion so it can be added back in without changing the properties of the projective plane. This produces a variation on the definition, namely the projective plane is defined as the set of lines in R³ that pass through the origin and the coordinates of a non-zero element (x, y, z) of a line are taken to be homogeneous coordinates of the line. These lines are now interpreted as points in the projective plane.

Again, this discussion applies analogously to other dimensions. So the projective space of dimension n can be defined as the set of lines through the origin in Rⁿ.^[8]

Elements other than points

The equation of a line in the projective plane may be given as sx + ty + uz = 0 where s, t and u are constants. Each triple (s, t, u) determines a line, the line determined is unchanged if it is multiplied by a nonzero scalar, and at least one of s, t and u must be different than 0. So the triple (s, t, u) may be taken to be homogeneous coordinates of a line in the projective plane, that is line coordinates as opposed to point coordinates. If in sx + ty + uz = 0 the letters s, t and u are taken as variables and x, y and z are taken as constants then equation becomes an equation of a set of lines in the space of all lines in the plane. Geometrically it represents the set of lines that pass though the point (x, y, z) and may be interpreted as the equation of the point in line-coordinates. In the same way, planes in 3-space may be given sets of four homogeneous coordinates, and so on for higher dimensions.^[9]

Duality

The same relation, sx + ty + uz = 0, may be regarded either the equation of a line or the equation of a point. In general, there is no difference either algebraically or logically between the homogeneous coordinates of points and lines. So the plane geometry with points as the fundamental elements and the plane geometry with lines as the fundamental element are equivalent except for interpretation. This leads to the concept of duality in projective geometry, the principle that the roles of points and lines can be interchanged in a theorem in projective geometry and the result will also be a theorem. Analogously, the theory of points in projective 3-space is dual to the theory of planes in projective 3-space, and so on for higher dimensions.^[10]

Plücker coordinates

Assigning coordinates to lines in projective 3-space is more complicated since it would seem that at total of 8 coordinates, either the coordinates of two points which lie on the line or two planes whose intersection is the line. A useful method, due to Julius Plücker, creates a set of six coordinate as the determinants x_iy_j − x_jy_i (1 ≤ i < j ≤ 4) from the homogeneous coordinates of two points (x₁, x₂, x₃, x₄) and (y₁, y₂, y₃, y₄) on the line. The Plücker embedding is the generalization of this to create homogeneous coordinates of elements of any dimension m in a projective space of dimension n.^[11]^[12]

Application to Bézout's theorem

Bézout's theorem predicts that the number of points of intersection of two curves is equal to the product of their degrees (assuming an algebraically complete field and with certain conventions followed for counting intersection multiplicities). Bézout's theorem predicts there is one point of intersection of two lines and in general this is true, but when the lines are parallel the point of intersection is infinite. Homogeneous coordinates can be used to locate the point of intersection in this case. Similarly, Bézout's theorem predicts that a line will intersect a conic at two points, but in some cases one or both of the points is infinite and homogeneous coordinates must be used to locate them. For example, y = x² and x = 0 have only one point of intersection in the finite plane. To find the other point of intersection, convert the equations into homogeneous form, yz = x² and x = 0. This produces x = yz = 0 and, assuming not all of x, y and z are 0, the solutions are x = y = 0, z ≠ 0 and x = z = 0, y ≠ 0. This first solution is the point (0, 0) in Cartesian coordinates, the finite point of intersection. The second solutions gives the homogeneous coordinates (0, 1, 0) which corresponds to the direction of the y-axis. For the equations xy = 1 and x = 0 there are no finite points of intersection. Converting the equations into homogeneous form gives xy = z² and x = 0. Solving produces the equation z² = 0 which has a double root at z = 0. From the original equation, x = 0, so y ≠ 0 since at least one coordinate must be non-zero. Therefore (0, 1, 0) is the point of intersection counted with multiplicity 2 in agreement with the theorem.^[13]

Circular points

The homogeneous form for the equation of a circle is x² + y² + 2axz + 2byz + cz². The intersection of this curve with the line at infinity can be found by setting z = 0. This produces the equation x² + y² = 0 which has two solutions in the complex projective plane, (1, i, 0) and (1, −i, 0). These points are called the circular points at infinity and can be regarded as the common points of intersection of all circles. This can be generalized to curves of higher order as circular algebraic curves.^[14]

Change of coordinate systems

Just as the selection of axes in the Cartesian coordinate is somewhat arbitrary, the selection of a single system of homogeneous coordinates out of all possible systems is somewhat arbitrary. Therefore it is useful to know how the different systems are related to each other.

Let (x, y, z) be the homogeneous coordinates of a point in the projective plane and for a fixed matrix

A={\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}},

with det(A)≠0, define a new set of coordinates (X, Y, Z) by the equation

{\begin{pmatrix}X\\Y\\Z\end{pmatrix}}=A{\begin{pmatrix}x\\y\\z\end{pmatrix}}.

Multiplication of (x, y, z) by a scalar results in the multiplication of (X, Y, Z) by the same scalar, and X, Y and Z cannot be all 0 unless x, y and z are all zero since A is nonsingular. So (X, Y, Z) are a new system of homogeneous coordinates for points in the projective plane. If z is fixed at 1 then

X=ax+by+c,\,Y=dx+ey+f,\,Z=gx+hy+i

are proportional to the signed distances from the point to the lines

ax+by+c=0,\,dx+ey+f=0,\,gx+hy+i=0.

(The signed distance is the distance multiplied a sign 1 or −1 depending on which side of the line the point lies.) Note that for a = b = 0 the value of X is simply a constant, and similarly for Y and Z.

The three lines,

ax+by+cz=0,\,dx+ey+fz=0,\,gx+hy+iz=0

in homogeneous coordinates, or

X=0,\,Y=0,\,Z=0

in the (X, Y, Z) system, form a triangle called the triangle of reference for the system.^[15]

Barycentric coordinates

Möbius' original formulation of homogeneous coordinates specified the position of a point as the center of mass (or barycenter) of a system of three point masses placed at the vertices of a fixed triangle. Points within the triangle are represented by positive masses and points outside the triangle are represented by allowing negative masses. Multiplying the masses in the system by a scalar does not affect the center of mass, so this is a special case of a system of homogeneous coordinates.

Trilinear coordinates

Let l, m, n be three lines in the plane and define a set of coordinates X, Y and Z of a point p as the signed distances from p to these three lines. These are called the trilinear coordinates of p with respect to the triangle. Strictly speaking these are not homogeneous, since the values of X, Y and Z are determined exactly, not just up to proportionality. There is a linear relationship between them however, so these coordinates can be made homogeneous by allowing multiples of (X, Y, Z) to represent the same point. More generally, X, Y and Z can be defined as constants p, r and q times the distances to l, m and n, resulting in a different system of homogeneous coordinates with the same triangle of reference. This is, in fact, the most general type of system of homogeneous coordinates for points in the plane if none of the lines is the line at infinity.^[16]

Use in computer graphics

Homogeneous coordinates are ubiquitous in computer graphics because allow common operations such as translation, rotation, scaling and perspective projection to be implemented as matrix operations. 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.^{[citation needed]}

For example, in perspective projection a position in space is associated with the line from it to a fixed point called the center of projection. The point is then mapped to a plane by finding the point of intersection of that plane and the line. This produces an accurate representation of how a three dimensional object appears to the eye. In the simplest situation, the center of projection is the origin and points are mapped to the plane z = 1, working for the moment in Cartesian coordinates. The for a given point in space, (x, y, z), the point where the line and the plane intersect is (x/z, y/z, 1). Dropping the now superfluous z coordinate, this becomes (x/z, y/z). In homogeneous coordinates, the point (x, y, z) is represented by (xw, yw, zw, w) and the point it maps to on the plane is represented by (xw, yw, zw), so projection can be represented in matrix form as

{\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix}}.

Matrices representing other geometric transformations can be combined with this and each other by matrix multiplication. As a result, any perspective projection of space can be represented as a single matrix.^[17]^[18]

References

^ O'Connor, John J.; Robertson, Edmund F., "August Ferdinand Möbius", MacTutor History of Mathematics Archive, University of St Andrews
^ Smith, David Eugene (1906). History of Modern Mathematics. J. Wiley & Sons. p. 53.
^ For the section: Jones pp. 120–122
^ Used in Woods.
^ Used in Miranda.
^ For the section: Miranda p. 14, Jones p. 120
^ Bôcher pp. 13–14
^ For the section: Cox, Little, O'Shea p. 360–362
^ Bôcher, pp 107–108 (adapted to the plane according to the footnote on p. 108)
^ Woods pp. 2, 40
^ Wilczynski p. 50
^ Bôcher p. 110
^ Follows Jones pp. 117–118, 122 with simplified examples.
^ Jones p. 204
^ Briot & Bouquet
^ Jones pp. 452 ff.
^ Mortenson, Michael E. (1999). Mathematics for Computer Graphics Applications. Industrial Press Inc. p. 318. ISBN 083113111X.
^ McConnell, Jeffrey J. (2006). Computer Graphics: Theory into Practice. Jones & Bartlett Learning. p. 120. ISBN 0763722502.

Weisstein, Eric W. "Homogeneous Coordinates". MathWorld.
Briot, Charles; Bouquet, Jean Claude (1896). Elements of Analytical Geometry of Two Dimensions. trans. J.H. Boyd. Werner school book company. p. 380.
Wilczynski, Ernest Julius (1906). Projective Differential Geometry of Curves and Ruled Surfaces. B.G. Teubner.
Bôcher, Maxime (1907). Introduction to Higher Algebra. Macmillan. pp. 11ff.
Jones, Alfred Clement (1912). An Introduction to Algebraical Geometry. Clarendon.
Woods, Frederick S. (1922). Higher Geometry. Ginn and Co. pp. 27ff.
Miranda, Rick (1995). Algebraic Curves and Riemann Surfaces. AMS Bookstore. p. 13. ISBN 0821802682.
Stillwell, John (2002). Mathematics and its History. Springer. pp. 134ff. ISBN 0387953361.
Cox, David A.; Little, John B.; O'Shea, Donal (2007). Ideals, Varieties, and Algorithms. Springer. p. 357. ISBN 0387356509.

External links

[1]

[1] O'Connor, John J.; Robertson, Edmund F., "August Ferdinand Möbius", MacTutor History of Mathematics Archive, University of St Andrews

[2] Smith, David Eugene (1906). History of Modern Mathematics. J. Wiley & Sons. p. 53.

[3] For the section: Jones pp. 120–122

[4] Used in Woods.

[5] Used in Miranda.

[6] For the section: Miranda p. 14, Jones p. 120

[7] Bôcher pp. 13–14

[8] For the section: Cox, Little, O'Shea p. 360–362

[9] Bôcher, pp 107–108 (adapted to the plane according to the footnote on p. 108)

[10] Woods pp. 2, 40

[11] Wilczynski p. 50

[12] Bôcher p. 110

[13] Follows Jones pp. 117–118, 122 with simplified examples.

[14] Jones p. 204

[15] Briot & Bouquet

[16] Jones pp. 452 ff.

[17] Mortenson, Michael E. (1999). Mathematics for Computer Graphics Applications. Industrial Press Inc. p. 318. ISBN 083113111X.

[18] McConnell, Jeffrey J. (2006). Computer Graphics: Theory into Practice. Jones & Bartlett Learning. p. 120. ISBN 0763722502.

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