Parametric plot

A parametric plot is a plot among functions of a parameter.

Examples of parametric plots

An example of a parametric plot is the plot of ordered pairs

:$(rcos(t),rsin(t)),\; r>0\; ,$.

As the parameter "t" increases, the parametric plot traces out a circle of radius "r", in the counter clockwise direction (i.e. in the direction of positive angle).

Related concepts

A comparametric plot is a special kind of parametric plot in which a function is plotted against a version of itself with a coordinate transformed parameter. For example, the comparametric plot $(f(t),f(at+b))\; ,$ defines a plot of f against a coordinate transformed (translated and dilated) version of f.

Of particular interest are plots of functions against dilated or contracted versions of themselves, e.g.

:$(f(q),f(2q))\; ,$

which have many practical applications in signal processing for the comparison of differently measured signals of the same quantity.

A common example of this application is in mathematical photography where various frames of video or image sequences have overlapping subject matter. In regions of overlap, once registered (using algebraic projective geometry) the result is differently exposed pictures of the same subject matter. When pictures differ only in exposure, there is a comparametric relationship between the pictures, modulo some noise factor. Recent advances in comparametric theory include Frank Candocia's seminal work on piecewise linear comparametric plots.

The theory of comparametric equations is used extensively in image processing, radar, and audio signal processing.