One possible source is the book
- Lawden, D. F. Elliptic functions and applications, New York, NY etc.: Springer-Verlag, 1989, doi:10.1007/978-1-4757-3980-0,
(esp. Ch. 4 Geometrical Applications). In partiqular it contains the proof of addition theorem for Jacobi elliptic functions based on the formulae of spherical trigonometry. Probably many such gems are collected in old books. In partiqular this (Legendre's) proof is presented in the book
- Cayley, A. An elementary treatise on elliptic functions 2nd ed. Dover Publications, Inc., New York, 1961 (p. 27) (Internet Archive version).
One more collection is the dissertation
- Snape, J. Applications of elliptic functions in classical and algebraic geometry University of Durham, 2004.
Also there is a special book
- Dragović, V. & Radnović, M. Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. Basel: Birkhäuser, 2011, doi:10.1007/978-3-0348-0015-0
about geometry, billiards and (hyper)elliptic curves.
