slopescommon

Slopescommon is a term used informally in mathematics and data analysis to describe the set or collection of slope values that are common to multiple linear relationships within a dataset or model. It is not a standard or universally defined notion, but it can be useful for identifying parallel structures or dominant directions among a group of lines.

Mathematically, consider a collection of lines expressed as y = m_i x + b_i. The slopescommon is the subset of slopes {m_i} that occur with a specified frequency threshold: for a chosen threshold k, slopescommon = {m : at least k lines share slope m}. When k = 2, slopescommon consists of slopes that appear in at least two lines. Geometrically, each element of slopescommon corresponds to a parallel class that appears more than once in the collection.

Computation and properties can be straightforward: compute the slope of each line (with a tolerance for measurement error if slopes are approximate), tally occurrences, and select those with frequency above the threshold. Slopescommon is invariant to translations of the lines (shifting intercepts does not change slopes), and its behavior under coordinate transformations depends on the transformation; uniform scaling preserves slopes, while nonuniform affine changes can alter them.

Applications include pattern recognition, clustering of linear trends, and diagnostic analysis in regression contexts to detect redundant or parallel relationships. It can also aid in the interpretation of architectural plans, road networks, or any domain where parallelism among linear components is of interest.

See also: slope, parallelism, linear regression, line fitting.