Logaxc

Logaxc is a mathematical transform that defines a class of generalized logarithmic-axial mappings used to map multi-dimensional data with high dynamic range into a more tractable form for analysis and visualization. It combines logarithmic scaling along coordinates with an optional rotation that aligns the data along chosen axes.

Formally, for x in R^n, logaxc(x) = R · log(|x| + 1), where |x| is the component-wise absolute value, log is the natural logarithm, and R is an orthogonal matrix representing axial orientation. A parametric variant introduces a base or exponent p, yielding logaxc_p(x) = R · log_p(|x|^p + 1). The transform is invertible on the nonzero positive orthant, via x_i = s_i (e^{y_i} - 1) with s_i = sign(x_i).

Applications include data visualization of skewed datasets, feature engineering for machine learning, and signal processing tasks that benefit from simultaneous magnitude and axis-aligned structure.

The term logaxc appears in theoretical discussions and early software prototypes from the 2010s and 2020s. It remains a niche concept, with multiple variants defined differently across sources. See also: logarithmic transform, power transform, and orthogonal rotation.