and , let be the midpoint of . Then the lemniscate of diameter can also be defined as the set of points , , , together with the locus of the points such that is a right angle (cf. Thales' theorem and its converse). The following theorem about angles occurring in the lemniscate is due to German mathematician Gerhard Christoph Hermann Vechtmann, who described it 1843 in his dissertation on lemniscates. and are the foci of the lemniscate, is the midpoint of the line segment and is any point on the lemniscate outside the line connecting and . The normal of the lemniscate in intersects the line connecting and in . Now the interior angle of the triangle at is one third of the triangle's exterior angle at (see also angle trisection). In addition the interior angle at is twice the interior angle at . Further properties The lemniscate is symmetric to the line connecting its foci and and as well to the perpendicular bisector of the line segment . The lemniscate is symmetric to the midpoint of the line segment . The area enclosed by the lemniscate is . The lemniscate is the circle inversion of a hyperbola and vice versa. The two tangents at the midpoint are perpendicular, and each of them forms an angle of with the line connecting and . The planar cross-section of a standard torus tangent to its inner equator is a lemniscate. The curvature at is . The maximum curvature, which occurs at , is therefore . Applications Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models. See also Lemniscate of Booth Lemniscate of Gerono Gauss's constant Lemniscatic elliptic function Cassini oval Notes References External links "Lemniscate of Bernoulli" at The MacTutor History of Mathematics archive "Lemniscate of Bernoulli" at MathCurve. Coup d'œil sur la lemniscate de Bernoulli (in French) Plane curves Algebraic curves Spiric sections