- #1
member 428835
- Homework Statement
- Given ##r(s) = \sin(c s)/c## we define the surface ##S = \langle r(s) \cos(\phi), r(s) \sin(\phi), (1 - \cos(cs))/c \rangle##. Compute a normal vector ##n## to ##S##.
- Relevant Equations
- a surface normal is found for a parametric surface ##S(s,\phi)## via ##n = d_sS \times d_\phi S##
Not HW, but seems to fit here.
I compute $$n.S = \frac{(-1+\cos(c s))}{c^2} \sin(c s) \neq 0$$
I use the following in Mathematica:
If you copy-paste this you should get the same output, where ##c = \cos(\alpha)##. What am I doing wrong? What is ##n##? Any help is greatly appreciated!
I compute $$n.S = \frac{(-1+\cos(c s))}{c^2} \sin(c s) \neq 0$$
I use the following in Mathematica:
Code:
r[s_, \[Alpha]_] := Sin[Cos[\[Alpha]] s]/Cos[\[Alpha]]
z[s_, \[Alpha]_] := (1 - Cos[Cos[\[Alpha]] s])/Cos[\[Alpha]]
x[s_, \[CurlyPhi]_, \[Alpha]_] := r[s, \[Alpha]] Cos[\[CurlyPhi]]
y[s_, \[CurlyPhi]_, \[Alpha]_] := r[s, \[Alpha]] Sin[\[CurlyPhi]]
z[s_, \[CurlyPhi]_, \[Alpha]_] := z[s, \[Alpha]]
S[s_, \[CurlyPhi]_, \[Alpha]_] := {x[s, \[CurlyPhi], \[Alpha]],
y[s, \[CurlyPhi], \[Alpha]], z[s, \[CurlyPhi], \[Alpha]]}
Cross[D[S[s, \[CurlyPhi], \[Alpha]], s],
D[S[s, \[CurlyPhi], \[Alpha]], \[CurlyPhi]]].S[
s, \[CurlyPhi], \[Alpha]] // FullSimplify
If you copy-paste this you should get the same output, where ##c = \cos(\alpha)##. What am I doing wrong? What is ##n##? Any help is greatly appreciated!