The Point at Infinity

Homogeneous coordinates provide a convenient and effective way to transfer points from one coordinate system to the corresponding point of an alternative coordinate system. Infinite region in one coordinate system is often converted to a finite region in the alternative system. If transfer method is incorrect, lines parallelism cannot be saved. However, after the transformation the intersection points are again at the points of intersection. This property is used to determine the homogeneous coordinate representation of points at infinity.

Consider a pair of intersecting lines defined by the equations:

(1.60)

Lines intersect at a point with coordinates x = 3/5, y = 2/5. We write the equation of the form x + y - 1 = 0, 2x - 3y = 0 and represent them in matrix form:

(1.61)

or:

(1.62)

If the matrix [ M *] is square matrix, then the intersection can be obtained by matrix inversion. Change the system of initial equations as follows:

(1.63)

or in matrix form:

(1.64)

i.e.:

(1.65)

Square matrix, the inverse of this, is:

(1.66)

Multiplying both sides by [ M ]^-1 and considering that [ M ] [ M ]^-1 = [ I ] is the identity matrix we obtain:

(1.67)

Thus, the intersection point again has the coordinates x = 3/5, y = 2/5.

Consider now two parallel lines, defined as follows:

(1.68)

By the definition of Euclidean geometry, the intersection point of two parallel lines is at infinity. Continuing the previous arguments, we compute the intersection point of these lines, given in matrix form:

(1.65)

However, despite the fact that the matrix is square, it has not inverse matrix, as two of its lines are identical. This matrix is singular. We can get a different formulation with a reversible matrix. It is possible by rewriting the system of equations as follows:

(1.66)

or in matrix form:

(1.67)

In this case, the matrix is ​​not singular and there exists a matrix inverse:

(1.68)

Multiplying both sides of the expression by the inverse matrix, we obtain:

The resulting homogeneous coordinates [1 -1 0] define the intersection point of two parallel lines, i.e. point at infinity. In particular, they represent a given point in the direction [1 -1] for two-dimensional space. Here are some examples:

[1 0 0] – a point on the positive x -axis,

[-1 0 0] – a point on the negative x -axis,

[0 1 0] – a point on the positive y -axis,

[0 -1 0] – a point on the negative y -axis,

[1 1 0] – along the line y = x in the direction [1 1].

Vector with homogeneous component h = 0 is indeed the point at infinity, and can also be interpreted as a moving to the limit (Table 1.1).

Table 1.1

Homogeneous coordinates for a point [4 3]

Consider the line y* = (3/4)x* and the point [ X Y h ] = [4 3 1]. Recall that in the homogeneous coordinates there is no single view of the coordinate of the vector (Table 1.1). Point [4 3 1] is presented in the homogeneous coordinates in all directions. Note that in this table for h → 0, the ratio x*/y* is equal to 3/4, as required to maintain the equation. In addition, we note that the next pair (x*, y*), all of whose points lie on the line y* = (3/4)x*, is rapidly approaching to infinity. Thus, the limit for h → 0 is a point of infinity, given in homogeneous coordinates as [ X Y H ] = [4 3 0].

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