Are Maps on a Sphere Homotopic if They Avoid Antipodal Points?

[R136a1]

If I take two arbitrary continuous maps $f,g:S^n\rightarrow S^n$ such that $f(x) \neq -g(x)$ for any $x\in S^n$, then $f$ and $g$ are homotopic.

How do I show this result? I really don't see how to use the condition that $f$ and $g$ never occupy two antipodal points. Any hint would be appreciated.

 

[WannabeNewton]

Consider the straight line homotopy $H(x,t) = f(x) + t(g(x) - f(x))$. Try to force the straight line homotopy onto the sphere; you'll see how the constraint on $f(x)$ and $g(x)$ not occupying antipodal points comes into play.

 

[R136a1]

Thanks a lot, miss!

So, my idea is to take

H(x,t) = \frac{f(x) + t(g(x) - f(x))}{\|f(x) + t(g(x) - f(x))\|}

I guess the constraint on $f$ and $g$ comes into play because we don't want the denominator to vanish? But I have troubles proving this rigorously. Assume that the denominator is $0$, then

(t-1)f(x) = tg(x)

I'm pretty stuck now!

 

[WannabeNewton]

$f(x)$ and $g(x)$ are both elements of $S^{n}$; take the norm of both sides.

 

[R136a1]

$f(x)$ and $g(x)$ are both elements of $S^{n}$; take the norm of both sides.

Wow, I didn't think of that! Why are you so smart?