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Representing Lines

The representation usually used for a line in two dimensions is of the form
where m is the gradient of the line and c is the intercept of the line with the y axis (Fig 20).


Fig. 20 Line representation

An alternative representation of a line is
where r is the perpendicular distance from the line to the origin and tex2html_wrap_inline3368 is the angle the line makes with the x axis, as shown in Fig 20.

The latter form has the advantage that the gradient m, with a range tex2html_wrap_inline3374 has been replaced by the range of angles tex2html_wrap_inline3376.

This is easier to deal with computationally.

(This will be important later -- see Hough Transforms).

Another alternative representation of an edge or line (again, see Fig 20) is by the vector pair tex2html_wrap_inline3378, where tex2html_wrap_inline3380 is a direction vector (usually normalised) along the edge and tex2html_wrap_inline3382 is a vector from the origin to the closest point on the line.

Thus, the length of tex2html_wrap_inline3382 is the perpendicular distance of the line from the origin.

This form of line representation is useful for both two- and three-dimensional lines, and indeed for three-dimensional lines this form is preferable.

Another advantage of this form of line representation is that the line can be parametrised.

Thus, we can specify the position of any point on the line, such as the end of an edge, by its distance t along the line. Therefore the coordinates of a point tex2html_wrap_inline3388 tex2html_wrap_inline3390 or tex2html_wrap_inline3381 are

tex2html_wrap_inline2984 David Marshall 1994-1997