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Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from where you stand.

Q1. For a given $d \in (0,\pi]$, what are the shortest such paths $\gamma(d)$?

Two other ways to define $\gamma(d)$:

1. The shortest path $\gamma$ such that every point on $S$ is no more than a distance $d$ from some point of $\gamma$.
2. The shortest path $\gamma$ such that the union of disks of radius $d$ centered on every point of $\gamma$ cover $S$.

For $d \in [\pi/2,\pi]$, $\gamma(d)$ is an arc of a great circle of length $2\pi - 2d$. So, for $d=\pi$, $\gamma$ is a single point; for $d=3\pi/4$, $\gamma$ is an arc of length $\pi/2$; for $d=\pi/2$, $\gamma$ is a semicircle. As $d$ approaches zero, it might be that $\gamma$ is a type of spiral, e.g.:
          
           _{(Image from grabcad.com.)}
What is quite unclear to me is when $d$ is less than but close to $\pi/2$. For example, suppose $d = \frac{5}{12}\pi = 75^\circ$. The union of disks of this radius centered on the equatorial semicircle that works for $\pi/2$ is shown in (a) below: a strip connecting the north and south poles is uncovered (blue circles are boundaries of disks of radius $d$ centered on $\gamma$):
   
One can construct a candidate $\gamma$ as in (b) above: extend the semicircle at each end so that all points on the equator are reached, and add short vertical sections, red to reach the north pole, and green to reach the south pole. It does not seem likely that such sharp turns yield the shortest $\gamma$.

Perhaps this question can be answered even without precise understanding of $\gamma(d)$ for all $d$:

Q2. Does $\gamma(d)$ vary continuously with respect to $d$?

I originally wanted to explore this for any (smooth, closed, compact) surface in $\mathbb{R}^3$, but it already seems not so straightforward for the sphere.

Q3. Has this question been studied before? It feels classical.

The question could also be posed for $(d{-}1)$-dimensional spheres in $\mathbb{R}^d$, again seeking an optimal inspection path for each $d$. (My interest in $\mathbb{R}^3$ stems from two related concepts: Voronoi diagrams on a surface, and the cut locus with respect to a path, although neither seems to help here.) Thanks for any pointers or ideas!

---

(Related earlier MO questions: [Optimal paintbrush geodesics][1]; [Shortest closed curve to inspect a sphere][2].)

Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from where you stand.

Q1. For a given $d \in (0,\pi]$, what are the shortest such paths $\gamma(d)$?

Two other ways to define $\gamma(d)$:

1. The shortest path $\gamma$ such that every point on $S$ is no more than a distance $d$ from some point of $\gamma$.
2. The shortest path $\gamma$ such that the union of disks of radius $d$ centered on every point of $\gamma$ cover $S$.

For $d \in [\pi/2,\pi]$, $\gamma(d)$ is an arc of a great circle of length $2\pi - 2d$. So, for $d=\pi$, $\gamma$ is a single point; for $d=3\pi/4$, $\gamma$ is an arc of length $\pi/2$; for $d=\pi/2$, $\gamma$ is a semicircle. As $d$ approaches zero, it might be that $\gamma$ is a type of spiral, e.g.:
          
           _{(Image from grabcad.com.)}
What is quite unclear to me is when $d$ is less than but close to $\pi/2$. For example, suppose $d = \frac{5}{12}\pi = 75^\circ$. The union of disks of this radius centered on the equatorial semicircle that works for $\pi/2$ is shown in (a) below: a strip connecting the north and south poles is uncovered (blue circles are boundaries of disks of radius $d$ centered on $\gamma$):
   
One can construct a candidate $\gamma$ as in (b) above: extend the semicircle at each end so that all points on the equator are reached, and add short vertical sections, red to reach the north pole, and green to reach the south pole. It does not seem likely that such sharp turns yield the shortest $\gamma$.

Perhaps this question can be answered even without precise understanding of $\gamma(d)$ for all $d$:

Q2. Does $\gamma(d)$ vary continuously with respect to $d$?

I originally wanted to explore this for any (smooth, closed, compact) surface in $\mathbb{R}^3$, but it already seems not so straightforward for the sphere.

Q3. Has this question been studied before? It feels classical.

The question could also be posed for $(d{-}1)$-dimensional spheres in $\mathbb{R}^d$, again seeking an optimal inspection path for each $d$. (My interest in $\mathbb{R}^3$ stems from two related concepts: Voronoi diagrams on a surface, and the cut locus with respect to a path, although neither seems to help here.) Thanks for any pointers or ideas!

---

9Aug2022. This issue was touched again by Fejes Tóth's Exploring a planet problem by Zhao, Yufei. "Exploring a planet, revisited." American Mathematical Monthly (2022): 1-3. But the Tóth version uses $n$ great-circle satellites. Related to the Shortest closed curve to inspect a sphere, but quite distinct. As far as I know, my original questions remain unsolved.

 

---

(Related earlier MO questions: Optimal paintbrush geodesics; Shortest closed curve to inspect a sphere.)

Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from where you stand.

Q1. For a given $d \in (0,\pi]$, what are the shortest such paths $\gamma(d)$?

Two other ways to define $\gamma(d)$:

1. The shortest path $\gamma$ such that every point on $S$ is no more than a distance $d$ from some point of $\gamma$.
2. The shortest path $\gamma$ such that the union of disks of radius $d$ centered on every point of $\gamma$ cover $S$.

For $d \in [\pi/2,\pi]$, $\gamma(d)$ is an arc of a great circle of length $2\pi - 2d$. So, for $d=\pi$, $\gamma$ is a single point; for $d=3\pi/4$, $\gamma$ is an arc of length $\pi/2$; for $d=\pi/2$, $\gamma$ is a semicircle. As $d$ approaches zero, it seems that $\gamma$ should be some type of spiral, e.g.:
           SpiralOnSphere http://cs.smith.edu/~orourke/MathOverflow/SpiralOnSphere.jpg
           _{(Image from grabcad.com.)}
What is quite unclear to me is when $d$ is less than but close to $\pi/2$. For example, suppose $d = \frac{5}{12}\pi = 75^\circ$. The union of disks of this radius centered on the equatorial semicircle that works for $\pi/2$ is shown in (a) below: a strip connecting the north and south poles is uncovered (blue circles are boundaries of disks of radius $d$ centered on $\gamma$):
    SpherePaint12 http://cs.smith.edu/~orourke/MathOverflow/SpherePaint12.jpg
One can construct a candidate $\gamma$ as in (b) above: extend the semicircle at each end so that all points on the equator are reached, and add short vertical sections, red to reach the north pole, and green to reach the south pole. It does not seem likely that such sharp turns yield the shortest $\gamma$.

Perhaps this question can be answered even without precise understanding of $\gamma(d)$ for all $d$:

Q2. Does $\gamma(d)$ vary continuously with respect to $d$?

I originally wanted to explore this for any (smooth, closed, compact) surface in $\mathbb{R}^3$, but it already seems not so straightforward for the sphere.

Q3. Has this question been studied before? It feels classical.

The question could also be posed for $(d{-}1)$-dimensional spheres in $\mathbb{R}^d$, again seeking an optimal inspection path for each $d$. (My interest in $\mathbb{R}^3$ stems from two related concepts: Voronoi diagrams on a surface, and the cut locus with respect to a path, although neither seems to help here.) Thanks for any pointers or ideas!

---

(Related earlier MO questions: [Optimal paintbrush geodesics][1]; [Shortest closed curve to inspect a sphere][2].)

Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from where you stand.

Q1. For a given $d \in (0,\pi]$, what are the shortest such paths $\gamma(d)$?

Two other ways to define $\gamma(d)$:

1. The shortest path $\gamma$ such that every point on $S$ is no more than a distance $d$ from some point of $\gamma$.
2. The shortest path $\gamma$ such that the union of disks of radius $d$ centered on every point of $\gamma$ cover $S$.

For $d \in [\pi/2,\pi]$, $\gamma(d)$ is an arc of a great circle of length $2\pi - 2d$. So, for $d=\pi$, $\gamma$ is a single point; for $d=3\pi/4$, $\gamma$ is an arc of length $\pi/2$; for $d=\pi/2$, $\gamma$ is a semicircle. As $d$ approaches zero, it might be that $\gamma$ is a type of spiral, e.g.:
          
           _{(Image from grabcad.com.)}
What is quite unclear to me is when $d$ is less than but close to $\pi/2$. For example, suppose $d = \frac{5}{12}\pi = 75^\circ$. The union of disks of this radius centered on the equatorial semicircle that works for $\pi/2$ is shown in (a) below: a strip connecting the north and south poles is uncovered (blue circles are boundaries of disks of radius $d$ centered on $\gamma$):
   
One can construct a candidate $\gamma$ as in (b) above: extend the semicircle at each end so that all points on the equator are reached, and add short vertical sections, red to reach the north pole, and green to reach the south pole. It does not seem likely that such sharp turns yield the shortest $\gamma$.

Perhaps this question can be answered even without precise understanding of $\gamma(d)$ for all $d$:

Q2. Does $\gamma(d)$ vary continuously with respect to $d$?

I originally wanted to explore this for any (smooth, closed, compact) surface in $\mathbb{R}^3$, but it already seems not so straightforward for the sphere.

Q3. Has this question been studied before? It feels classical.

The question could also be posed for $(d{-}1)$-dimensional spheres in $\mathbb{R}^d$, again seeking an optimal inspection path for each $d$. (My interest in $\mathbb{R}^3$ stems from two related concepts: Voronoi diagrams on a surface, and the cut locus with respect to a path, although neither seems to help here.) Thanks for any pointers or ideas!

---

(Related earlier MO questions: [Optimal paintbrush geodesics][1]; [Shortest closed curve to inspect a sphere][2].)

Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from where you stand.

Q1. For a given $d \in (0,\pi]$, what are the shortest such paths $\gamma(d)$?

Two other ways to define $\gamma(d)$:

1. The shortest path $\gamma$ such that every point on $S$ is no more than a distance $d$ from some point of $\gamma$.
2. The shortest path $\gamma$ such that the union of disks of radius $d$ centered on every point of $\gamma$ cover $S$.

For $d \in [\pi/2,\pi]$, $\gamma(d)$ is an arc of a great circle of length $2\pi - 2d$. So, for $d=\pi$, $\gamma$ is a single point; for $d=3\pi/4$, $\gamma$ is an arc of length $\pi/2$; for $d=\pi/2$, $\gamma$ is a semicircle. As $d$ approaches zero, it seems that $\gamma$ should be some type of spiral, e.g.:
           SpiralOnSphere http://cs.smith.edu/~orourke/MathOverflow/SpiralOnSphere.jpg
           _{(Image from grabcad.com.)}
What is quite unclear to me is when $d$ is less than but close to $\pi/2$. For example, suppose $d = \frac{5}{12}\pi = 75^\circ$. The union of disks of this radius centered on the equatorial semicircle that works for $\pi/2$ is shown in (a) below: a strip connecting the north and south poles is uncovered (blue circles are boundaries of disks of radius $d$ centered on $\gamma$):
    SpherePaint12 http://cs.smith.edu/~orourke/MathOverflow/SpherePaint12.jpg
One can construct a candidate $\gamma$ as in (b) above: extend the semicircle at each end so that all points on the equator are reached, and add short vertical sections, red to reach the north pole, and green to reach the south pole. It does not seem likely that such sharp turns yield the shortest $\gamma$.

Perhaps this question can be answered even without precise understanding of $\gamma(d)$ for all $d$:

Q2. Does $\gamma(d)$ vary continuously with respect to $d$?

I originally wanted to explore this for any (smooth, closed, compact) surface in $\mathbb{R}^3$, but it already seems not so straightforward for the sphere.

Q3. Has this question been studied before? It feels classical.

The question could also be posed for $(d{-}1)$-dimensional spheres in $\mathbb{R}^d$, again seeking an optimal inspection path for each $d$. (My interest in $\mathbb{R}^3$ stems from two related concepts: Voronoi diagrams on a surface, and the cut locus with respect to a path, although neither seems to help here.) Thanks for any pointers or ideas!

---

(Related earlier MO questions: [Optimal paintbrush geodesics][1]; [Shortest closed curve to inspect a sphere][2].)

Suppose you would like to "inspect" every point of a unit-radius sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$ on $S$, but you can only see a distance $d$ from where you stand.

Q1. For a given $d \in (0,\pi]$, what are the shortest such paths $\gamma(d)$?

Two other ways to define $\gamma(d)$:

1. The shortest path $\gamma$ such that every point on $S$ is no more than a distance $d$ from some point of $\gamma$.
2. The shortest path $\gamma$ such that the union of disks of radius $d$ centered on every point of $\gamma$ cover $S$.

For $d \in [\pi/2,\pi]$, $\gamma(d)$ is an arc of a great circle of length $2\pi - 2d$. So, for $d=\pi$, $\gamma$ is a single point; for $d=3\pi/4$, $\gamma$ is an arc of length $\pi/2$; for $d=\pi/2$, $\gamma$ is a semicircle. As $d$ approaches zero, it seems that $\gamma$ should be some type of spiral, e.g.:
           SpiralOnSphere http://cs.smith.edu/~orourke/MathOverflow/SpiralOnSphere.jpg
           _{(Image from grabcad.com.)}
What is quite unclear to me is when $d$ is less than but close to $\pi/2$. For example, suppose $d = \frac{5}{12}\pi = 75^\circ$. The union of disks of this radius centered on the equatorial semicircle that works for $\pi/2$ is shown in (a) below: a strip connecting the north and south poles is uncovered (blue circles are boundaries of disks of radius $d$ centered on $\gamma$):
    SpherePaint12 http://cs.smith.edu/~orourke/MathOverflow/SpherePaint12.jpg
One can construct a candidate $\gamma$ as in (b) above: extend the semicircle at each end so that all points on the equator are reached, and add short vertical sections, red to reach the north pole, and green to reach the south pole. It does not seem likely that such sharp turns yield the shortest $\gamma$.

Perhaps this question can be answered even without precise understanding of $\gamma(d)$ for all $d$:

Q2. Does $\gamma(d)$ vary continuously with respect to $d$?

I originally wanted to explore this for any (smooth, closed, compact) surface in $\mathbb{R}^3$, but it already seems not so straightforward for the sphere.

Q3. Has this question been studied before? It feels classical.

The question could also be posed for $(d{-}1)$-dimensional spheres in $\mathbb{R}^d$, again seeking an optimal inspection path for each $d$. (My interest in $\mathbb{R}^3$ stems from two related concepts: Voronoi diagrams on a surface, and the cut locus with respect to a path, although neither seems to help here.) Thanks for any pointers or ideas!

---

(Related earlier MO questions: [Optimal paintbrush geodesics][1]; [Shortest closed curve to inspect a sphere][2].)