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edited Oct 31 2011 at 6:35 |
Imagine I take a sheet of paper , and use a pin to generate an $N$x$M$ rectangular array of small holescovering the sheet. Next, I then fold the sheet of paper from left-to-right into to form a cylinder of radius $r_c$ and height $h_c$, with where there are $N$ pinholes around its circumference and $M$ pinholes from the top of the cylinder to the bottom. No edge-effects from the folding process are discernible.
Say
Now, say I pick a coordinate, $C$ C$, in the three-dimensional space inside the cylinder, . $C$ is some distance from the bottom of the cylinder, $A$, and some distance from the central-axis of the cylinder, $B$. Next, I then proceed to shine a laser(, or thread a very thin string ) between two pinholesin the wall of the cylinder, $(p_1, p_2)$, which comes such that the beam or the string is as close as possible to $C$. Here, the laser (or the string), which string can be treated as a one-dimensional line between chord in the two pinholesinterior of the cylinder.
How do I choose $(p_1, p_2)$ to generate a line containing a coordinate $C*$ as close as possible to my chosen coordinate $C$? In general, how well can I do as a function of the density of the pinhole array , and the position of the coordinate in terms of $A$ and $B$? Pressing my luck, in terms of minimizing the (straight-line) difference between $C$ and $C^*$, are there between better geometries for the pinholes than a rectangular array?
For convenience purposes, this (http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html) Wolfram Mathworld site derives the minimum distance, in three-dimensional space, between an arbitrary coordinate and a line/vector.
Update: First of all, thanks to Joseph O'Rourke for the awesome graphic! Secondly, I would be very interested in an analysis of worst-case delta with excluded regions, say, at the top and bottom of the cylinder (as Gerhard Paseman suggested). Also, if this helps simplify things, I would be very interested in a version of this question where one slices the cylinder along a line of symmetry from top-to-bottom, generating two equivalent sheets of paper with pinholes held a distance $L$ apart.
Update 2: Joseph O'Rourke states a two-dimensional variant of this problem in Part 2 (P2) of his question "Chord arrangement that avoids confining small or large disks", (http://mathoverflow.net/questions/76980/chord-arrangement-that-avoids-confining-small-or-large-disks).
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edited Oct 10 2011 at 12:24 |
A possible point of motivation: Let's say I want to design an artificial vascular system for growing some special type of tissue, but my capillaries / blood vessels can only be straight-line segments. This is perhaps because these are straight-forward to manufacture at the relevant size scale, or because bending/slack can lead to tearing. Let's also say that, if a single cell dies from starvation, it's neighbors are likely to perish as well, leading to significant tissue damage. If cells are small relative to the size of the cylindrical or cubic chamber, and can deform around the straight-line segments representing capillaries... (1) - For a fixed-size chamber and a fixed number of straight-line segment capillaries, how do I minimize the average distance between, say, the center-of-mass of any cell and a blood vessel? (2) - If I can only thread capillaries through a chamber with a square rectangular grid of pinholes / peepholes, how well can I do for the cylinder mentioned in the earlier problem description, or cube with two equivalent/parallel faces containing pinholes seperated by a fixed distance?
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edited Oct 3 2011 at 17:24 |
Imagine I take a sheet of paper, and use a pin to generate an $N$x$M$ rectangular array of small holes covering the sheet. Next, I fold the sheet of paper from left-to-right into a cylinder of radius $r_c$ and height $h_c$, with $N$ pinholes around its circumference and $M$ pinholes from the top of the cylinder to the bottom. No edge-effects from the folding process are discernible.
Say I pick a coordinate, $C$ in three-dimensional space inside the cylinder, some distance from the bottom of the cylinder, $A$, and some distance from the central-axis of the cylinder, $B$. Next, I shine a laser (or thread a very thin string) between two pinholes in the wall of the cylinder, $(p_1, p_2)$, which comes as close as possible to $C$. Here, the laser (or the string), which can be treated as a one-dimensional line between the two pinholes.
How do I choose $(p_1, p_2)$ to generate a line containing a coordinate $C*$ as close as possible to my chosen coordinate $C$? In general, how well can I do as a function of the density of the pinhole array, and the position of the coordinate in terms of $A$ and $B$? Pressing my luck, in terms of minimizing the (straight-line) difference between $C$ and $C^*$, are there between geometries for the pinholes than a rectangular array?
For convenience purposes, this (http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html) Wolfram Mathworld site derives the minimum distance, in three-dimensional space, between an arbitrary coordinate and a line/vector.
Update: First of all, thanks to Joseph O'Rourke for the awesome graphic! Secondly, I would be very interested in an analysis of worst-case delta with excluded regions, say, at the top and bottom of the cylinder (as Gerhard Paseman suggested). Also, if this helps simplify things, I would be very interested in a version of this question where one slices the cylinder along a line of symmetry from top-to-bottom, generating two equivalent sheets of paper with pinholes held a distance $L$ apart.
Update 2: Joseph O'Rourke states a two-dimensional variant of this problem in Part 2 (P2) of his question "Chord arrangement that avoids confining small or large disks", (http://mathoverflow.net/questions/76980/chord-arrangement-that-avoids-confining-small-or-large-disks).
A possible point of motivation: Let's say I want to design an artificial vascular system for growing some special type of tissue, but my capillaries / blood vessels can only be straight-line segments. This is perhaps because these are straight-forward to manufacture at the relevant size scale, or because bending/slack can lead to tearing. Let's also say that, if a single cell dies from starvation, it's neighbors are likely to perish as well, leading to significant tissue damage.
If cells are small relative to the size of the cylindrical or cubic chamber, and can deform around the straight-line segments representing capillaries...
(1) - For a fixed-size chamber and a fixed number of straight-line segment capillaries, how do I minimize the average distance between, say, the center-of-mass of any cell and a blood vessel?
(2) - If I can only thread capillaries through a chamber with a square rectangular grid of pinholes / peepholes, how well can I do for the cylinder mentioned in the earlier problem description, or cube with two equivalent/parallel faces containing pinholes seperated by a fixed distance?
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edited Oct 3 2011 at 17:04 |
A possible point of motivation: Let's say I want to design an artificial vascular system for growing some special type of tissue, but my capillaries / blood vessels can only be straight-line segments. This is perhaps because these are straight-forward to manufacture at the relevant size scale, or because bending/slack can lead to tearing. If cells are small relative to the size of the cylindrical or cubic chamber, and can deform around the straight-line segments representing capillaries... (1) - For a fixed-size chamber and a fixed number of straight-line segment capillaries, how do I minimize the average distance between, say, the center-of-mass of any cell and a blood vessel? (2) - If I can only thread capillaries through a chamber with a square rectangular grid of pinholes / peepholes, how well can I do for the cylinder mentioned in the earlier problem description, or cube with two equivalent/parallel faces containing pinholes seperated by a fixed distance?
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edited Oct 2 2011 at 23:02 |
Imagine I take a sheet of paper, and use a pin to generate an $N$x$M$ rectangular array of small holes covering the sheet. Next, I fold the sheet of paper from left-to-right into a cylinder of radius $r_c$ and height $h_c$, with $N$ pinholes around its circumference and $M$ pinholes from the top of the cylinder to the bottom. No edge-effects from the folding process are discernible.
Say I pick a coordinate, $C$ in three-dimensional space inside the cylinder, some distance from the bottom of the cylinder, $A$, and some distance from the central-axis of the cylinder, $B$. Next, I shine a laser (or thread a very thin string) between two pinholes in the wall of the cylinder, $(p_1, p_2)$, which comes as close as possible to $C$. Here, the laser (or the string), which can be treated as a one-dimensional line between the two pinholes.
How do I choose $(p_1, p_2)$ to generate a line containing a coordinate $C*$ as close as possible to my chosen coordinate $C$? In general, how well can I do as a function of the density of the pinhole array, and the position of the coordinate in terms of $A$ and $B$? Pressing my luck, in terms of minimizing the (straight-line) difference between $C$ and $C^*$, are there between geometries for the pinholes than a rectangular array?
For convenience purposes, this (http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html) Wolfram Mathworld site derives the minimum distance, in three-dimensional space, between an arbitrary coordinate and a line/vector.
Update: First of all, thanks to Joseph O'Rourke for the awesome graphic! Secondly, I would be very interested in an analysis of worst-case delta with excluded regions, say, at the top and bottom of the cylinder (as Gerhard Paseman suggested). Also, if this helps simplify things, I would be very interested in a version of this question where one slices the cylinder along a line of symmetry from top-to-bottom, generating two equivalent sheets of paper with pinholes held a distance $L$ apart.
For convenience purposes,
Update 2: Joseph O'Rourke states a two-dimensional variant of this (http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html) Wolfram Mathworld site derives the minimum distance, problem in three-dimensional spacePart 2 (P2) of his question "Chord arrangement that avoids confining small or large disks", between an arbitrary coordinate and a line/vector. (http://mathoverflow.net/questions/76980/chord-arrangement-that-avoids-confining-small-or-large-disks).
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edited Oct 2 2011 at 17:11 |
Imagine I take a sheet of paper, and use a pin to generate an $N$x$M$ rectangular array of small holes covering the sheet. Next, I fold the sheet of paper from left-to-right into a cylinder of radius $r_c$ and height $h_c$, with $N$ pinholes around its circumference and $M$ pinholes from the top of the cylinder to the bottom. No edge-effects from the folding process are discernible.
Say I pick a coordinate, $C$ in three-dimensional space inside the cylinder, some distance from the bottom of the cylinder, $A$, and some distance from the central-axis of the cylinder, $B$. Next, I shine a laser (or thread a very thin string) between two pinholes in the wall of the cylinder, $(p_1, p_2)$, which comes as close as possible to $C$. Here, the laser (or the string), which can be treated as a one-dimensional line between the two pinholes.
How do I choose $(p_1, p_2)$ to generate a line containing a coordinate $C*$ as close as possible to my chosen coordinate $C$? In general, how well can I do as a function of the density of the pinhole array, and the position of the coordinate in terms of $A$ and $B$? Pressing my luck, in terms of minimizing the (straight-line) difference between $C$ and $C^*$, are there between geometries for the pinholes than a rectangular array?
Update: First of all, thanks to Joseph O'Rourke for the awesome graphic! Secondly, I would be very interested in an analysis of worst-case delta with excluded regions, say, at the top and bottom of the cylinder (as Gerhard Paseman suggested). Also, if this helps simplify things, I would be very interested in a version of this question where one slices the cylinder along a line of symmetry from top-to-bottom, generating two equivalent sheets of paper with pinholes held a distance $L$ apart.
For convenience purposes, this (http://mathworld.wolfram.com/Point-LineDistance3-Dimensional.html) Wolfram Mathworld site derives the minimum distance, in three-dimensional space, between an arbitrary coordinate and a line/vector.
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edited Oct 2 2011 at 9:42 |
Imagine I take a sheet of paper, and use a pin to generate an $N$x$M$ rectangular array of small holes covering the sheet. Next, I fold the sheet of paper from left-to-right into a cylinder of radius $r_c$ and height $h_c$, with $N$ pinholes around its circumference and $M$ pinholes from the top of the cylinder to the bottom. No edge-effects from the folding process are discernible.
Say I pick a coordinate, $C$ in three-dimensional space inside the cylinder, some distance from the bottom of the cylinder, $A$, and some distance from the central-axis of the cylinder, $B$. Next, I shine a laser (or thread a very thin string) between two pinholes in the wall of the cylinder, $(p_1, p_2)$, which comes as close as possible to $C$. Here, the laser (or the string), which can be treated as a one-dimensional line between the two pinholes.
How do I choose $(p_1, p_2)$ to generate a line containing a coordinate $C*$ as close as possible to my chosen coordinate $C$? In general, how well can I do as a function of the density of the pinhole array, and the position of the coordinate in terms of $A$ and $B$? Pressing my luck, in terms of minimizing the (straight-line) difference between $C$ and $C^*$, are there between geometries for the pinholes than a rectangular array?
Update: First of all, thanks to Joseph O'Rourke for the awesome graphic! Secondly, I would be very interested in an analysis of worst-case delta with excluded regions, say, at the top and bottom of the cylinder (as Gerhard Paseman suggestedsuggested). Also, if this helps simplify the problemthings, I would be very interested in a version of this question where instead of a one slices the cylinder along a line of symmetry from top-to-bottom, one has generating two equivalent sheets of paper with pinholes held a distance $L$ apart.
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edited Oct 2 2011 at 7:17 |
Imagine I take a sheet of paper, and use a pin to generate an $N$x$M$ rectangular array of small holes covering the sheet. Next, I fold the sheet of paper from left-to-right into a cylinder of radius $r_c$ and height $h_c$, with $N$ pinholes around its circumference and $M$ pinholes from the top of the cylinder to the bottom. No edge-effects from the folding process are discernible.
Say I pick a coordinate, $C$ in three-dimensional space inside the cylinder, some distance from the bottom of the cylinder, $A$, and some distance from the central-axis of the cylinder, $B$. Next, I shine a laser (or thread a very thin string) between two pinholes in the wall of the cylinder, $(p_1, p_2)$, which comes as close as possible to $C$. Here, the laser (or the string), which can be treated as a one-dimensional line between the two pinholes.
How do I choose $(p_1, p_2)$ to generate a line containing a coordinate $C*$ as close as possible to my chosen coordinate $C$? In general, how well can I do as a function of the density of the pinhole array, and the position of the coordinate in terms of $A$ and $B$? Pressing my luck, in terms of minimizing the (straight-line) difference between $C$ and $C^*$, are there between geometries for the pinholes than a rectangular array?
Update: First of all, thanks to Joseph O'Rourke for the awesome graphic! Secondly, I would be very interested in an analysis of worst-case delta with excluded regions, say, at the top and bottom of the cylinder as Gerhard Paseman suggested. Also, if this helps simplify the problem, I would be interested in a version of this question where instead of a cylinder, one has two equivalent sheets of paper with pinholes held a distance $L$ apart.
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asked Oct 1 2011 at 21:09 |
Threading pinholes in the wall of cylinder to pass through an internal coordinate
Imagine I take a sheet of paper, and use a pin to generate an $N$x$M$ rectangular array of small holes covering the sheet. Next, I fold the sheet of paper from left-to-right into a cylinder of radius $r_c$ and height $h_c$, with $N$ pinholes around its circumference and $M$ pinholes from the top of the cylinder to the bottom. No edge-effects from the folding process are discernible.
Say I pick a coordinate, $C$ in three-dimensional space inside the cylinder, some distance from the bottom of the cylinder, $A$, and some distance from the central-axis of the cylinder, $B$. Next, I shine a laser (or thread a very thin string) between two pinholes in the wall of the cylinder, $(p_1, p_2)$, which comes as close as possible to $C$. Here, the laser (or the string), which can be treated as a one-dimensional line between the two pinholes.
How do I choose $(p_1, p_2)$ to generate a line containing a coordinate $C*$ as close as possible to my chosen coordinate $C$? In general, how well can I do as a function of the density of the pinhole array, and the position of the coordinate in terms of $A$ and $B$? Pressing my luck, in terms of minimizing the (straight-line) difference between $C$ and $C^*$, are there between geometries for the pinholes than a rectangular array?
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