What are the limitations of using quadrance and spread in trigonometry?

[bayan]

does anyone know what they include?

here is a prefrence.

thanks

 

[dduardo]

I just saw that. Here is a pdf explaining the technique:

http://web.maths.unsw.edu.au.nyud.net:8090/~norman/papers/Chapter1.pdf

 

[amcavoy]

Is this actually going to be taken seriously? I mean, will it be taught instead of "classical" trigonometry?

 

[Hurkyl]

While it's interesting, I find it hard to take it seriously.

First off, classical trigonometry (and length) has the huge advantages of being coordinate-free and additive. You can solve problems directly from the diagrams, whereas it could take a good amount of time to simply set up a problem in a coordinate-based approach.

Secondly, it touts as a "feature" that it treats corresponding angles identically. However, now the student is faced with multiple putative solutions (possibly a great many) from which 'e must distill the desired solution.

Finally, we already have well-understood tools for doing coordinate geometry: the dot and cross products. They do at least the same thing, but have the huge, giant, overwhelming advantage of being linear -- we can use much of our algebraic intuition when working with them.

 

[rachmaninoff]

Meh. All he's done is mearly replaced $\theta$ with $\cos \theta$. He's using (normalized) scalar (inner) products, without referring to them by name. I searched the pdf for "inner" and "scalar" - is the author unaware that he's rediscovered well-known math and is claiming it as his own? Is he that stupid?

Just skimming the pdf, his "quadrance" is in no way different from "length"; and his "spread" is the 'ratio of quadrances' - i.e., the cosine, or the normalized scalar product. And there are many basic algebraic errors throughout! :grumpy:

"quadrance" = length;
"spread" = cosine

 

[arildno]

Besides, saying that the "quadrance" concept is more basic or natural than the distance concept because it doesn't involve square roots, is just plain silly.
Has he never heard of an unmarked ruler??

Just because analytic geometry will use square roots in order to express distances in terms of Cartesian coordinates, does not in any way change the fact that "length" or "distance" arguably remains (one of) the most fundamental concepts in geometry.

 

[amcavoy]

"Forty-five degrees becomes a spread of 1/2, while thirty and sixty degrees become respectively spreads of 1/4 and 3/4. What could be simpler than that?"

$\sin^2{\left(45^o\right)}=\frac{1}{4}$

$\sin^2{\left(60^o\right)}=\frac{3}{4}$

I don't see why this is new.

 

[LeonhardEuler]

The "quadrance" is just the length squared and the "spread" is defined as the ratio of the quadrance of the opposite leg of a triangle to the quadrance of its hypatenous. In other words it is $sin^2{\theta}$. I see a number of disadvantages to this system. First, you can not specify a unique point in the plane by giving its quadrance from the origin and spread from an axis. This will give four possible points. Second, it seems kind of unweildly for physical situations. If you have a partilcle moving at constant velocity, its "speed" defined as $\frac{dQ}{dt}$ depends on where you place the origin and changes as the particle moves. If you have an object rotating without the influence of external torques, then $\frac{dS}{dt}$ is constantly changing, while $\frac{d\theta}{dt}$ is constant. Third, the "spread" of two lines can not be greater than 1. In other words, no angles of greater than 90° are allowed. Triangles containing these angles must be split into two triangles to be dealt with. The "simple formula" he gives for the spread of two lines is nothing more than the magnitude of the cross-product of two vectors along these lines divided by their magnitudes and squared.