Vtyjkyuk

(Background: this is inspired by Chebyshev polynomials and expanding a function as a Chebyshev series.)

Solving for $ z $ gives
$$
z=xy pm sqrt(1-x^2)(1-y^2),
$$
where $-1leq x,y leq 1$. Now choose the positive square root and call it $ xoplus y$.
(Alternatively, we can define $xoplus y=cos(arccos(x)+arccos(y))$.)

We can check commutativity, identity ($1$), inverses ($-x$). Associativity is might be a bit harder, but I reckon it works.

The defining equation is a cubic, hence, it reminds me of the group structure of an elliptic curve, but an elliptic curve has two variables and the group law is defined by intersection with a line, while this equation has three variable, one is the "sum" of the other two.

Question
Has anyone seen a structure like this before?

At first glance it seems to be the same as Cayley's nodal cubic surface. Although there it is described by the (projective) equation $$wxy+xyz+wxz+wyz=0.$$ How, then, do obtain my version of it (if, in fact, they are the same.)

Here is a page which seems to derive my equation from Cayley's.

This is better understood by considering the roots of unity. Suppose
$, u_n = r_n + s_nsqrt-1 ,$ for $, n=1,2,3 ,$ be three roots of unity such that $, u_3 = u_1 u_2. ,$ Then
$, r_1^2 + r_2^2 + r_3^2 - 2 r_1 r_2 r_3 = 1. ,$ The same equation is true if $, u_3 = u_1/u_2 ,$ or $, u_3 = u_2/u_1. ,$ Or more symmetrically if $, u_1u_2u_3 = 1.,$ The reason is that given only the real part of a root of unity, there are two possible values for the imaginary part. The roots of unity form a multiplicative group, but if we only have the real part, the group operation is two-valued.

The case of addition of points on elliptic curves is very appropriate. The group operation there is related to three points on a line, but not in the obvious way. That is, the third point is not the sum of the other two, but rather the negative of the sum. Another way to state it is that the sum of all three points is the identity. Also, given only the $,x,$ coordinate of a point, there are two values for the $,y,$ coordinate. There is some similarity between "adding" points on the unit circle and points on an elliptic curve.

Added: for real $x,y,z geq 1$ we get all
$$x = cosh t, ; ; y = cosh u, ; ; z = cosh ; (t+u).$$
Actually, as $cosh$ is even, we could demand $t+u+v=0$ with solution
$$x = cosh t, ; ; y = cosh u, ; ; z = cosh v.$$
Easy to check that this satisfies $x^2 + y^2 + z^2 = 1 + 2 xyz.$
Multiplying the variables by $i$ preserves the identity, while $cosh it = cos t,$ so we get
$$x = cos t, ; ; y = cos u, ; ; z = cos ; (t+u)$$
for $|x|,|y|,|z| leq 1.$

Not sure yet whether having, say, $z geq 1$ forces both $x,y geq 1.$ Notice that there is no benefit to having two negative values, we can just negate both.. Of course, you might like negative numbers. Matter of taste.
We might have the entire surface with these and then $(-x,-y,-z)$
for the $cosh$ part. Note the gradient being $0$ at the point $(1,1,1)$
of the surface.

There are three involutions; by alternating these one may travel around the surface.

$$ (x,y,z) mapsto ( 2yz-x,y,z) $$
$$ (x,y,z) mapsto (x,2zx-y,z) $$
$$ (x,y,z) mapsto (x,y,2xy-z) $$

Not sure why you are interested in $|x| leq 1.$ For any $t$ we get a solution
$$ (t,t,1) , $$ then the third involution takes us to
$$ (t,t,2t^2 - 1) $$ A different involution ( and re-ordering) takes us to
$$ ( t, 2t^2 - 1, 4 t^3 - 3 t ) $$
which reminds me of $(cos theta, ; cos 2 theta, ; cos 3 theta)$
but also the more useful
$$(cosh w, ; cosh 2 w, ; cosh 3 w).$$
This aspect is very successful: given integers $1 leq m < n,$ and positive real $w,$ we get an ordered solution
$$ ( cosh mw, ; cosh nw, ; ; cosh , (m+n)w ) $$
which explains the repetitions of numbers such as $5042,$ which is $cosh 7w$ when $cosh w = 2.$

Here are some solutions with positive integer entries that are distinct:

26 7 2
97 26 2
99 17 3
244 31 4
362 97 2
362 26 7
485 49 5
577 99 3
846 71 6
1351 362 2
1351 97 7
1921 244 4
2024 127 8
2889 161 9
3219 1933 1727
3363 577 3
3363 99 17
3510 2145 1998
3551 3287 2025
3614 3218 1663
3970 199 10
4015 3409 1727
4095 3569 3087
4097 3203 2947
4127 3417 3047
4237 4115 2095
4247 4183 2177
4299 4149 4095
4446 3130 927
4754 4665 582
4801 485 5
5042 1351 2
5042 362 7
5042 97 26

You also can have

• $(2,3,6pmsqrt24)$ so all three are greater than 1.

• $(-2,-3,6pmsqrt24$

• $(1,y,y)$

• $(-1,y,-y)$

Remember $cos(a+b)=cos acos b -sin asin b$ so the choice of square-root might get tricky.