Derivation of Geomagnetic field lines (after interaction with solar wind)
Short answer: Tsyganenko was not the first to attempt an observation based mathematical model, and older purely theoretical models also exist.
The first mathematical magnetic field model was created from measurements made on Earth's surface by Carl Friedrich Gauss in the 1830s, when he derived the mathematical techniques we still use. For the solar wind and interactions between the Earth's field and the solar wind, we're talking the advent of the space age when we were first able to make measurements out in space and begin to mathematically describe them, though the idea of the solar wind had been suggested (not by that name) in the early 1900s, and theoretical models exist from the 1960s at least.
Wikipedia has a brief summary of the history of major magnetospheric and solar wind discoveries, that suggests things kicked off with satellite measurements made in 1958/9. Tsyganenko cites a NASA study from 1975 as the first empirical model of Earth's magnetosphere and its interaction with the solar wind, though direct measurements apparently weren't made in the magneto-tail until 1983 (from the first Wiki page).
An aside for clarity: the dipolar nature of Earth's magnetic field isn't due to the interaction with the solar wind - the field generated by the geodynamo in Earth's outer core contributes the bulk of Earth's magnetosphere and has a more complicated shape when considered at the Earth's surface, for example.
Magnetic fields have a strong dependence on distance from their source, and smaller spatial features attenuate more rapidly (think like with sound waves - you hear low frequencies far away, but the high frequencies attenuate more quickly), so the approximation as a dipole gets more accurate as you move further away from the source of the field.
So when considering the interaction between the solar wind and Earth's field, which occurs at several Earth radii out in space (the exact distance changes as it depends on the strength of Earth's field pushing out, and the strength of the solar wind pushing back) the Earth's field is roughly dipolar, and then gets distorted with a squashed bow-shock and elongated magneto-tail by the action of the solar wind.