Creating new symbolic identities

I have a matrix expression involving three real symbolic variables called x, y, and z, which are three elements of a vector v in R3. It is known that v is constrained to be a unit vector, so that x^2 + y^2 + z^2 is always equal to 1. Is there a way to tell MATLAB about this identity (x^2 + y^2 + z^2 = 1) so that whenever it encounters the expression x^2 + y^2 + z^2, it substitutes a 1 for that expression?

I tried using subs(S, x^2 + y^2 + z^2, 1), but that doesn't work ... it appears the second argument to a three-argument subs() call has to be a single variable.

I tried the following trick, inspired by the conversion from Cartesian to Spherical coordinates:

syms a b real;
s1 = subs(S, x, cos(a) * sin(b));
s2 = subs(s1, y, sin(a) * sin(b));
s3 = subs(s2, z, cos(b));

and that actually performs the x^2 + y^2 + z^2 = 1 substitution, but has the unfortunate side effect that all remaining expressions are now in terms of a and b instead of x, y, and z.

There has to be a way to do this, right? Corallary: I can't be the first person who has wanted something like this, can I?

Answer 1

[Christopher Creutzig]

Works for me:

>> syms x y z
>> S = x^2+y^2+z^2+x+y+z+1

S =

x^2 + x + y^2 + y + z^2 + z + 1

>> subs(S, x^2+y^2+z^2, 1)

ans =

x + y + z + 2

Here's something else you might want to try, since it also handles cases that are a bit more “hidden”:

>> assume(x^2+y^2+z^2==1)
>> S

S =

x^2 + x + y^2 + y + z^2 + z + 1

>> simplify(S)

ans =

x + y + z + 2

>> simplify(x^3+x*y^2-x)

ans =

-x*z^2