The Cross Curve

Introduction:

The cross curve is the curve formed by replacing the exponents 2 in the equation for an ellipse by -2. The asymptotes on the graph above are not actually part of the graph; an implicit plot gives a more accurate representation of the curve:

Definition:

Implicitly, it is defined by

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature:

kappa[t] = [click for formula]