covertices

Covertices are geometric points associated with conic sections, specifically the endpoints of the minor axis in an ellipse or the endpoints of the conjugate axis in a hyperbola. They reflect the dual relationship between the major (or transverse) axis and the perpendicular axis.

In an ellipse, the covertices lie on the ellipse at the ends of the minor axis. For an ellipse centered at (h, k) with semi-major axis a and semi-minor axis b (and a ≥ b), the standard equation is (x−h)^2/a^2 + (y−k)^2/b^2 = 1 when the major axis is horizontal. The covertices are at (h, k±b). If the major axis is vertical, the covertices are at (h±b, k). Thus covertices are the endpoints of the minor axis through the center.

In a hyperbola, the covertices are the endpoints of the conjugate axis. For a hyperbola centered at (h, k) with transverse axis along the x-direction (x−h)^2/a^2 − (y−k)^2/b^2 = 1, the vertices are (h±a, k) and the covertices are (h, k±b). If the transverse axis is vertical, the vertices are (h, k±a) and the covertices are (h±b, k). The conjugate axis and its endpoints do not lie on the hyperbola itself.

Rotations add complexity: for rotated ellipses or hyperbolas, the covertices still lie along the axis perpendicular to the main axis through the center, but their coordinates require applying the rotation to the axis directions.

In summary, covertices denote the endpoints of the minor axis for an ellipse or the conjugate axis for a hyperbola, serving as a complementary reference to the vertices.