I have a unit sphere in Cartesian coordinates:
$x^2 + y^2 + z^2 = 1$
or in spherical coordinates:
$x = \rho \sin(\phi) \cos(\theta)\\
y = \rho \sin(\phi) \sin(\theta)\\
z = \rho \cos(\phi)$
I select a point $P$ on the surface of the sphere, in the coordinate system that is more convenient. I want to identify the points, on the surface of the sphere, on the circle of radius $r$ centred in $P$, where $r$ is the distance of the shortest arc on the surface of the sphere from each point to $P$ (i.e. $r$ is the great arc between two points on a unit sphere).
I expect this problem to be pretty simple but I haven't found any clear resource on the web. Any idea? Essentially what I need is the equation in 3D of a "circular section" of a sphere, possibly as a function of one angle.
Best Answer
I'm not entirely sure if you're asking for a closed expression that gives the points, or just a description of these points and am going to assume the latter.
The first step in solving this problem is to change it into an easier one. We may rotate the sphere in such a way the point $P$ becomes the north pole $N=(0,0,1)$. If we call this rotation (which may be expressed as a $3\times3$ matrix) $R$, a point $Q$ is on the circle with radius $r$ around $P$ iff $R(Q)$ is on the circle with radius $r$ around $N$. This is due to the fact that a rotation of the sphere does not change distances on the sphere.
The next step is to consider these particular circles of radius $r$ around $N$. The best way to see this is to draw a sketch of the intersection of the sphere with the $yz$-plane, i.e. a circle of radius 1 around the origin. We can also see to points $P_1, P_2$ on the circle at distance $r$ from $N$. (I'll assume $r$ is small enough for this to make sense) These points should be such that $P_1$ is obtained from $P_2$ by reflecting around the $z$-axis. The spherical coordinate $\theta$ corresponding to these points can now easily be obtained: this angle is given as an angle wrt the $z$-axis, while $r$ is precisely the arc length corresponding to this angle on a circle with radius 1. This leads to the conclusion that $r=\theta$ in this special case.
The next thing to notice, is that all points at distance $r$ from $N$ have the same $z$-coordinate. By projecting either $P_1$ or $P_2$ at this $z$-coordinate (which again can be done insightful in the same picture), one finds this height to be $z_0=\cos(\theta)=\cos(r)$. Moreover, yet again from considering the same picture, the required circle is a circle parallel to the $xy$-plane with center $(0,0,z_0)$ and radius $\sin(\theta)=\sin(r)$.
We may thus conclude that the points at distance $r$ from $N$ are of the form $(\sin(r)\cos(\phi),\sin(r)\sin(\phi),cos(r))$ with $\phi\in[0,2\pi)$. Using the rotation matrix $R$ from before, we thus get the set of points satisfying your original question: $$\{R^{-1}(\sin(r)\cos(\phi),\sin(r)\sin(\phi),cos(r))\mid \phi\in[0,2\pi)\}.$$