trigonometric parametrization

Question

I am trying to figure out a pattern. I will start with examples. $$\text{Let } PD(\text{Set } A):= \text{Parametric Description of }A$$ $$ A:=\{(x,y)\in \mathbb R ^2|x^2+y^2 =1 \} $$ $$PD(A): x=\cos(t), y=\sin(t)$$ $$B:=\{(x,y,z)\in \mathbb R ^3|x^2+y^2+z^2 =1 \}$$$$PD(B): x=\sin(t)\sin(u), y=\sin(t)\cos(u),z=\cos(t)$$$$C:=\{(w,x,y,z)\in \mathbb R ^4|w^2+x^2+y^2+z^2 =1 \}$$ $$PD(C): w=\sin(t)\sin(u)\sin(v),x=\sin(t)\sin(u)\cos(v), y=\sin(t)\cos(u),z=\cos(t)$$Now I'm not sure exactly how to parameterize:$$D:=\{(v,w,x,y,z)\in \mathbb R ^5|v^2+w^2+x^2+y^2+z^2 =1 \}$$ I am seeing some pattern but I am not sure if it is enough to find a parameterization for D. I know why the parameterizations for A,B and C work from using trig identities, but I am still not able to produce my own. The parameterizations shown are examples from my Calculus 3 class.

Answer 1

I cannot produce a formula, but here is the parametrization: $ \\ PD(D): \\v=sin(r)sin(s)sin(t)sin(u)\\w=sin(r)sin(s)sin(t)cos(u) \\ x=sin(r)sin(s)cos(t) \\ y=sin(r)cos(s) \\z=cos(r)$

Here is the pattern that I followed:There is 1 less parameter than the number of variables. The first variable(in this case $v$) is set to $sin(p_1)sin(p_2)...sin(p_n)$ The second variable is set to $sin(p_1)sin(p_2)...sin(p_{n-1})cos(p_n)$. In other words it is set to the same thing as the first variable, but the last term is changed from $sin(p_n)$ to $cos(p_n)$. The rest of the variables parameterizations can be found by seing how the equation collapses as such: $$v^2+w^2 = sin^2(s)sin^2(t)sin^2(u)$$therefore, the only logical parameterization for x would be $sin(s)sin(t)cos(u)$, because $v^2+w^2+x^2$ will collapse to $sin^2(s)sin^2(t)$.