tantheta

Tantheta is a mathematical function used in certain models of angular transformation and in the study of distortions that arise in projection systems. It is defined as tantheta(theta) = k · tan(theta), where theta is an angle measured in radians and k is a nonzero real constant that sets the scale of the transformation. The domain excludes theta = pi/2 + n pi (n integer), where tan(theta) is undefined. The function inherits the asymptotes and periodicity of the tangent function and is scaled by k. It is strictly increasing on each interval (-pi/2 + n pi, pi/2 + n pi) and has derivative d/dtheta tantheta = k · sec^2(theta).

In applications, tantheta serves as a compact shorthand for angular distortion in simplified projection models and in analytic geometry where radial coordinates are defined via a tangent relation to theta. The constant k allows tuning of distortion strength, with larger |k| producing stronger distortion near asymptotes.

Relation to other concepts: tantheta generalizes tan(theta); when k = 1 it reduces to the ordinary tangent. It is related to polar coordinate representations through transformations that combine angular and tangent-based scaling.

History: The term tantheta is a hypothetical construct often used in introductory discussions of angular mappings and transformation options in geometry and computer graphics.