The unit sphere in R^3 is arcwise connected

**Pinkk** · September 15th 2010, 06:31 PM

So I attempted to find the intersection of the unit sphere with a plane that contains any two arbitrary points $a, b$ on the sphere and the origin, but I only came up with a really convoluted expression that I am not even sure can be used to find a function $f: [0,1] \to \mathbb{R}^{3}$ such that $f(0) = a, f(1) = b$ and for all $t\in [0,1]$ , $f(t)$ is on the unit sphere. Any help would be appreciated.

**Failure** · September 15th 2010, 08:38 PM

Originally Posted by Pinkk

So I attempted to find the intersection of the unit sphere with a plane that contains any two arbitrary points $a, b$ on the sphere and the origin, but I only came up with a really convoluted expression that I am not even sure can be used to find a function $f: [0,1] \to \mathbb{R}^{3}$ such that $f(0) = a, f(1) = b$ and for all $t\in [0,1]$ , $f(t)$ is on the unit sphere. Any help would be appreciated.

You can parametrize the sphere with two angles $\vartheta, \varphi$ so that you get $x=r\sin\vartheta \cos\varphi, y = r\sin \vartheta \sin\varphi, z=r\cos\vartheta$ .
This is a map $g:\, [0;2\pi)\times [-\pi/2;\pi/2]\to \in\mathbb{R}^3, (\varphi,\vartheta)\mapsto (x,y,z)$ .
Assuming that you have coordinates $(\varphi_a,\vartheta_a), (\varphi_b,\vartheta_b)$ for two given points $a,b$ on the unit sphere, you next parametrize the line segment connecting the points $(\varphi_a,\vartheta_a), (\varphi_b,\vartheta_b)$ that lie in the rectangular area of the domain of these two parameters. This gives you the map $h:\, [0;1]\to [0;2\pi)\times [-\pi/2;\pi/2]$ .
Now a parametrization of the arc connecting a and b on the sphere is $f\,: [0;1]\to \mathbb{R}^3, t\mapsto g(h(t))$ , I think.

**Pinkk** · September 15th 2010, 09:05 PM

What would a parameterization of the line segment look like, simply $h(t) = (\varphi_a,\vartheta_a) + t(\varphi_b - \varphi_a,\vartheta_b - \vartheta_a)$ ?

**Failure** · September 15th 2010, 10:55 PM

Originally Posted by Pinkk

What would a parameterization of the line segment look like, simply $h(t) = (\varphi_a,\vartheta_a) + t(\varphi_b - \varphi_a,\vartheta_b - \vartheta_a)$ ?

Sure, why not? Since the domain of the parametrization of the sphere by $g:\, (\varphi,\vartheta)\to\mathbb{R}^3$ is a rectangle and thus convex, that parametrization h(t) of the line segment is sure to give you a line segment that remains completely within the domain of g.

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