It eschews angles entirely, sticking to ratios. It avoids square roots by sticking to "quadrances" (squared distance; i.e. pythagoras/euclidean-distance without taking square roots).
I highly recommend Wildberger's extensive Youtube channels too https://www.youtube.com/@njwildberger and https://www.youtube.com/@WildEggmathematicscourses
He's quite contrarian, so I'd take his informal statements with a pinch of salt (e.g. that there's no such thing as Real numbers; the underlying argument is reasonable, but the grand statements lose all that nuance); but he ends up approaching many subjects from an interesting perspective, and presents lots of nice connections e.g. between projective geometry, linear algebra, etc.
There are certain drawbacks. If the solution involves non-algebraic numbers there is no getting away from the transcendental numbers (that ultimately get approximated by algebraic numbers).
srean•52m ago
I wholeheartedly agree with the point being made in the post. I had commented about this in the recent asin() post but deleted thinking it might not be of general interest.
If you care about angles and rotations in the plane, it is often profitable to represent an angle not by a scalar such as a degree or a radian but as a tuple
or as a complex number.This way one can often avoid calls to expensive trigonometric functions. One may need calls to square roots and general polynomial root finding.
In Python you can represent an angle as a unit complex numbers and the runtime will do the computations for you.
For example, if you needed the angular bisector of an angle subtended at the origin (you can translate the vertex there and later undo the translation), the bisector is just the geometric mean of the arms of the angle
Along with stereographic transform and its inverse you can do a lot.This is directly related to the field of algebraic numbers.
With complex numbers you get translations, scaled rotations and reflections. Sufficient for Euclidean geometry.