with their Inversions

The hyperbolas are computed with this

parametrization:

c(t) = [a * cosh(t), b * sinh(t)]

Note: (x/a)^2 - (y/b)^2 = 1 ("implicit equation")

The point & tangent construction is shown:

Intersect the ray from the left focal point with the

circle. Connect this point S with the right focal

point F. The symmetry line between S and F is the

hyperbola tangent where the ray meets the hyperbola.

Inversion(vector) = (radius/|vector|)^2 * vector

The inverted curves are lemniscates

parametrization:

c(t) = [a * cosh(t), b * sinh(t)]

Note: (x/a)^2 - (y/b)^2 = 1 ("implicit equation")

The point & tangent construction is shown:

Intersect the ray from the left focal point with the

circle. Connect this point S with the right focal

point F. The symmetry line between S and F is the

hyperbola tangent where the ray meets the hyperbola.

Inversion(vector) = (radius/|vector|)^2 * vector

The inverted curves are lemniscates

Construction: Intersect ray from left focal point LF with circle
of radius 2a around LF. Connect intersection point P

with right focal point RF. The symmetry line between P and RF is tangent to the hyperbola where it intersects the ray from LF.

with right focal point RF. The symmetry line between P and RF is tangent to the hyperbola where it intersects the ray from LF.