CjVKurve

CjVKurve is a name used in some educational and informal mathematical resources to refer to a family of parametric plane curves. It is not a single canonical object in formal literature, but rather a concept employed to illustrate how a curve’s shape and curvature can be manipulated by adjusting a small set of parameters. The term combines initials from purported originators with the word curve, and in practice various instructors present different explicit formulations under this umbrella.

Definition and form. In typical presentations, a CjVKurve is given by a pair of smooth functions x(t; j, V) and y(t; j, V) defined on a real parameter t, where j and V are real parameters that influence the geometry. Different sources propose different explicit expressions for x and y, but they share the common feature that the trajectory in the plane depends continuously on j and V. By varying these parameters, the curve can transition through a range of shapes, including open spirals, closed loops, and forms with cusp-like or self-intersecting behavior at boundary cases.

Properties. CjV-curves are constructed to be smooth for regular parameter choices. The curvature profile can be tuned via the parameters, giving designers a tool to study how local bending relates to global shape. Special parameter values may yield degenerate forms that resemble familiar curves, which helps in teaching concepts in differential geometry and curve design.

Applications and references. The concept is primarily used as an instructional device in calculus, geometry, and computer graphics to demonstrate parameter-driven shape variation and curvature continuity. It is not tied to a single peer-reviewed definition, and readers typically encounter it within didactic resources rather than formal mathematical treatises. See also: parametric curve, curvature, plane curve.