Rational Trigonometry - Summer of Math Exposition

My entry in The Summer of Math Exposition.

If you are interested and want to learn more about Rational Trigonometry check out Norman J Wildberger's series on the subject:

If you want to contribute the wild egg wiki needs your help:

[acegikmo]

I'd never really considered the computational advantages in rational trig, that's fascinating! Thanks for the video, I should read into this more and see if there's some neat use cases in my work

[hhbxxeo]

If I'm not mistaken, the spread is the weight of the normalised bivector deduced from the two lines.
It's thus the square of the sinus of the angle and is also a normalised determinant (in the sense of calculated with normalised vectors).
The bivector still retains the attitude and orientation while the spread (as a number) doesn't.

[annwan]

Awesome video, gotta read more on the subject

[ilonachan]

Is there no visual intuition for what the spread _means?_ Because while sin and cos are hard to explicitly compute, they are quite easy to understand in terms of their geometric meaning... but where's the spread in the diagram?

Also, as far as I can tell, the spread is not invariant under rotation! That makes it pretty much useless for some applications where angles provide easy insights; if I rotate a triangle about in 2d space the angles stay fixed, but the spread somehow does not????

Last note: The definition of the spread seems familiar. Initially I overlooked the minus sign and thought "oh that's the scalar product in disguise". Turns out it's more complex than that (pun not intended), something with the conjugate of (a_2, b_2) maybe, but I'm sure the circle is still lurking there. Would have been neat to see where exactly.

UPDATE: The spread between line 1 with slope theta_1 and line 2 with slope theta_1+theta_2 comes out to be cos²(2theta_1+theta_2). This still hasn't given me any insight into where the circle is, but it confirms that the spread does in fact depend on theta_1, which it really should not do.

[blacky]

the triple spread is far more complicated than the simple sum of interiors, there definitly is a reason for the current system to existiert over "rational" trig.

Also, i find your law of cos weird. I know it as a^2 + b^2 - 2abcos() = c^2.

[fredg]

0:43 A very difficult problem only solvable using a formula from calculus ? (1)Put a point on one of your lines. (2)Draw a right-angled triangle. (3)Use the arctangent function. Alternatively you can use vectors and either use the dot product followed by an arccosine or the cross product followed by an arcsine. Your explanation about the "spread" doesn't lead to a much simpler formula, and it doesn't gives you an angle at the end. As mentionned in another comment, it gives you a value that is never shown on the figure. Watching your video we don't get the sense of what does it mean to have a spread of 0.5 or 1. What are the minimum or maximum values that a spread can take ? What is the value of this spread when the two lines are parallel ? 0, infinity, another value ?

[jasiu]

While it seems nice to have all trig be rational, I recommend extreme caution wit NJ Wildberger's views. He seems not to accept existence of irrational numbers, or any kind of infinity. By modern math standards, that's a fringe point of view.