General equation for projection of regular grid onto a line? The Next CEO of Stack...
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General equation for projection of regular grid onto a line?
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$begingroup$
I have a regular grid of points in $xy$, say a square grid, and I want to make an orthonogal projection onto a line through the origin, with slope $tan alpha$:
I would like to derive a mathematical expression for the positions of the projected points on the line.
Can I write down a general expression for the orthogonal projection matrix of a bunch of points onto a straight line of known slope and intercept?
geometry rotations projection
asked Mar 17 at 17:29
SuperCiociaSuperCiocia
295213
$endgroup$
add a comment |
$begingroup$
I have a regular grid of points in $xy$, say a square grid, and I want to make an orthonogal projection onto a line through the origin, with slope $tan alpha$:
I would like to derive a mathematical expression for the positions of the projected points on the line.
Can I write down a general expression for the orthogonal projection matrix of a bunch of points onto a straight line of known slope and intercept?
geometry rotations projection
asked Mar 17 at 17:29
SuperCiociaSuperCiocia
295213
$endgroup$
add a comment |
$begingroup$
I have a regular grid of points in $xy$, say a square grid, and I want to make an orthonogal projection onto a line through the origin, with slope $tan alpha$:
I would like to derive a mathematical expression for the positions of the projected points on the line.
Can I write down a general expression for the orthogonal projection matrix of a bunch of points onto a straight line of known slope and intercept?
geometry rotations projection
asked Mar 17 at 17:29
SuperCiociaSuperCiocia
295213
$endgroup$
I have a regular grid of points in $xy$, say a square grid, and I want to make an orthonogal projection onto a line through the origin, with slope $tan alpha$:
I would like to derive a mathematical expression for the positions of the projected points on the line.
Can I write down a general expression for the orthogonal projection matrix of a bunch of points onto a straight line of known slope and intercept?
geometry rotations projection
geometry rotations projection
asked Mar 17 at 17:29
SuperCiociaSuperCiocia
295213
asked Mar 17 at 17:29
SuperCiociaSuperCiocia
295213
asked Mar 17 at 17:29
SuperCiociaSuperCiocia
295213
asked Mar 17 at 17:29
SuperCiociaSuperCiocia
295213
asked Mar 17 at 17:29
SuperCiociaSuperCiocia
295213
295213
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes.
First you should determine the unit vector describing the line of projection.
In your case it is $v=[cos alpha sin alpha]^T$.
And now every point $P$ described by the vector $p$ can be projected to the line with the projection matrix $vv^T$ to the vector $p_p=vv^Tp$.
If you have bunch of points use simply matrix constructed from column vectors describing these points
$[p_{p1} p_{p2} dots p_{pn}]=vv^T[p_1 p_2 dots p_n]$.
answered Mar 19 at 11:46
WidawensenWidawensen
4,72831446
$endgroup$
$begingroup$
Yes it’s good thanks. I was trying to figure out whether or not I should ask a new question. What I want to know if the spacing between the projected points on the line. I can easily do that with your formula. The thing is if I change the slope I will also have NEW points in the original grid that will be projected. That will give new spacings and I don’t know how to incorporate this.
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
Have you got any ideas about this or should I ask a new question?
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
@SuperCiocia Interestingly, but I was thinking also about development of your question, only in my case I was interested when we would have integer coordinates of projected points (evidently at least $ cosalpha$ and $sinalpha$ should have been rational like $[0.6 0.8]$. It is interesting how many points of grid can be projected into projected points which are still on grid. But this is more question from number theory.
$endgroup$
– Widawensen
Mar 21 at 7:58
$begingroup$
@SuperCiocia Returning to the problem of spacing I'm not sure what you want to investigate: equal spacing between projected points or maybe the maximum points which are projected into the same point on line? When the line is rotating spacing is changing, that's for sure - different questions can be asked based on the phenomenon of changing spacing, we could visualize the effects with some kind of histogram which can be more or less discrete. Of course in this case the grid should be somehow limited, say $100 times 100$
$endgroup$
– Widawensen
Mar 21 at 8:06
$begingroup$
I have a new question about this now.
$endgroup$
– SuperCiocia
Mar 21 at 15:56
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
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active
oldest
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$begingroup$
Yes.
First you should determine the unit vector describing the line of projection.
In your case it is $v=[cos alpha sin alpha]^T$.
And now every point $P$ described by the vector $p$ can be projected to the line with the projection matrix $vv^T$ to the vector $p_p=vv^Tp$.
If you have bunch of points use simply matrix constructed from column vectors describing these points
$[p_{p1} p_{p2} dots p_{pn}]=vv^T[p_1 p_2 dots p_n]$.
answered Mar 19 at 11:46
WidawensenWidawensen
4,72831446
$endgroup$
$begingroup$
Yes it’s good thanks. I was trying to figure out whether or not I should ask a new question. What I want to know if the spacing between the projected points on the line. I can easily do that with your formula. The thing is if I change the slope I will also have NEW points in the original grid that will be projected. That will give new spacings and I don’t know how to incorporate this.
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
Have you got any ideas about this or should I ask a new question?
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
@SuperCiocia Interestingly, but I was thinking also about development of your question, only in my case I was interested when we would have integer coordinates of projected points (evidently at least $ cosalpha$ and $sinalpha$ should have been rational like $[0.6 0.8]$. It is interesting how many points of grid can be projected into projected points which are still on grid. But this is more question from number theory.
$endgroup$
– Widawensen
Mar 21 at 7:58
$begingroup$
@SuperCiocia Returning to the problem of spacing I'm not sure what you want to investigate: equal spacing between projected points or maybe the maximum points which are projected into the same point on line? When the line is rotating spacing is changing, that's for sure - different questions can be asked based on the phenomenon of changing spacing, we could visualize the effects with some kind of histogram which can be more or less discrete. Of course in this case the grid should be somehow limited, say $100 times 100$
$endgroup$
– Widawensen
Mar 21 at 8:06
$begingroup$
I have a new question about this now.
$endgroup$
– SuperCiocia
Mar 21 at 15:56
add a comment |
$begingroup$
Yes.
First you should determine the unit vector describing the line of projection.
In your case it is $v=[cos alpha sin alpha]^T$.
And now every point $P$ described by the vector $p$ can be projected to the line with the projection matrix $vv^T$ to the vector $p_p=vv^Tp$.
If you have bunch of points use simply matrix constructed from column vectors describing these points
$[p_{p1} p_{p2} dots p_{pn}]=vv^T[p_1 p_2 dots p_n]$.
answered Mar 19 at 11:46
WidawensenWidawensen
4,72831446
$endgroup$
$begingroup$
Yes it’s good thanks. I was trying to figure out whether or not I should ask a new question. What I want to know if the spacing between the projected points on the line. I can easily do that with your formula. The thing is if I change the slope I will also have NEW points in the original grid that will be projected. That will give new spacings and I don’t know how to incorporate this.
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
Have you got any ideas about this or should I ask a new question?
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
@SuperCiocia Interestingly, but I was thinking also about development of your question, only in my case I was interested when we would have integer coordinates of projected points (evidently at least $ cosalpha$ and $sinalpha$ should have been rational like $[0.6 0.8]$. It is interesting how many points of grid can be projected into projected points which are still on grid. But this is more question from number theory.
$endgroup$
– Widawensen
Mar 21 at 7:58
$begingroup$
@SuperCiocia Returning to the problem of spacing I'm not sure what you want to investigate: equal spacing between projected points or maybe the maximum points which are projected into the same point on line? When the line is rotating spacing is changing, that's for sure - different questions can be asked based on the phenomenon of changing spacing, we could visualize the effects with some kind of histogram which can be more or less discrete. Of course in this case the grid should be somehow limited, say $100 times 100$
$endgroup$
– Widawensen
Mar 21 at 8:06
$begingroup$
I have a new question about this now.
$endgroup$
– SuperCiocia
Mar 21 at 15:56
add a comment |
$begingroup$
Yes.
First you should determine the unit vector describing the line of projection.
In your case it is $v=[cos alpha sin alpha]^T$.
And now every point $P$ described by the vector $p$ can be projected to the line with the projection matrix $vv^T$ to the vector $p_p=vv^Tp$.
If you have bunch of points use simply matrix constructed from column vectors describing these points
$[p_{p1} p_{p2} dots p_{pn}]=vv^T[p_1 p_2 dots p_n]$.
answered Mar 19 at 11:46
WidawensenWidawensen
4,72831446
$endgroup$
Yes.
First you should determine the unit vector describing the line of projection.
In your case it is $v=[cos alpha sin alpha]^T$.
And now every point $P$ described by the vector $p$ can be projected to the line with the projection matrix $vv^T$ to the vector $p_p=vv^Tp$.
If you have bunch of points use simply matrix constructed from column vectors describing these points
$[p_{p1} p_{p2} dots p_{pn}]=vv^T[p_1 p_2 dots p_n]$.
answered Mar 19 at 11:46
WidawensenWidawensen
4,72831446
answered Mar 19 at 11:46
WidawensenWidawensen
4,72831446
answered Mar 19 at 11:46
WidawensenWidawensen
4,72831446
answered Mar 19 at 11:46
WidawensenWidawensen
4,72831446
4,72831446
$begingroup$
Yes it’s good thanks. I was trying to figure out whether or not I should ask a new question. What I want to know if the spacing between the projected points on the line. I can easily do that with your formula. The thing is if I change the slope I will also have NEW points in the original grid that will be projected. That will give new spacings and I don’t know how to incorporate this.
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
Have you got any ideas about this or should I ask a new question?
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
@SuperCiocia Interestingly, but I was thinking also about development of your question, only in my case I was interested when we would have integer coordinates of projected points (evidently at least $ cosalpha$ and $sinalpha$ should have been rational like $[0.6 0.8]$. It is interesting how many points of grid can be projected into projected points which are still on grid. But this is more question from number theory.
$endgroup$
– Widawensen
Mar 21 at 7:58
$begingroup$
@SuperCiocia Returning to the problem of spacing I'm not sure what you want to investigate: equal spacing between projected points or maybe the maximum points which are projected into the same point on line? When the line is rotating spacing is changing, that's for sure - different questions can be asked based on the phenomenon of changing spacing, we could visualize the effects with some kind of histogram which can be more or less discrete. Of course in this case the grid should be somehow limited, say $100 times 100$
$endgroup$
– Widawensen
Mar 21 at 8:06
$begingroup$
I have a new question about this now.
$endgroup$
– SuperCiocia
Mar 21 at 15:56
add a comment |
$begingroup$
Yes it’s good thanks. I was trying to figure out whether or not I should ask a new question. What I want to know if the spacing between the projected points on the line. I can easily do that with your formula. The thing is if I change the slope I will also have NEW points in the original grid that will be projected. That will give new spacings and I don’t know how to incorporate this.
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
Have you got any ideas about this or should I ask a new question?
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
@SuperCiocia Interestingly, but I was thinking also about development of your question, only in my case I was interested when we would have integer coordinates of projected points (evidently at least $ cosalpha$ and $sinalpha$ should have been rational like $[0.6 0.8]$. It is interesting how many points of grid can be projected into projected points which are still on grid. But this is more question from number theory.
$endgroup$
– Widawensen
Mar 21 at 7:58
$begingroup$
@SuperCiocia Returning to the problem of spacing I'm not sure what you want to investigate: equal spacing between projected points or maybe the maximum points which are projected into the same point on line? When the line is rotating spacing is changing, that's for sure - different questions can be asked based on the phenomenon of changing spacing, we could visualize the effects with some kind of histogram which can be more or less discrete. Of course in this case the grid should be somehow limited, say $100 times 100$
$endgroup$
– Widawensen
Mar 21 at 8:06
$begingroup$
I have a new question about this now.
$endgroup$
– SuperCiocia
Mar 21 at 15:56
$begingroup$
Yes it’s good thanks. I was trying to figure out whether or not I should ask a new question. What I want to know if the spacing between the projected points on the line. I can easily do that with your formula. The thing is if I change the slope I will also have NEW points in the original grid that will be projected. That will give new spacings and I don’t know how to incorporate this.
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
Yes it’s good thanks. I was trying to figure out whether or not I should ask a new question. What I want to know if the spacing between the projected points on the line. I can easily do that with your formula. The thing is if I change the slope I will also have NEW points in the original grid that will be projected. That will give new spacings and I don’t know how to incorporate this.
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
Have you got any ideas about this or should I ask a new question?
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
Have you got any ideas about this or should I ask a new question?
$endgroup$
– SuperCiocia
Mar 20 at 14:49
$begingroup$
@SuperCiocia Interestingly, but I was thinking also about development of your question, only in my case I was interested when we would have integer coordinates of projected points (evidently at least $ cosalpha$ and $sinalpha$ should have been rational like $[0.6 0.8]$. It is interesting how many points of grid can be projected into projected points which are still on grid. But this is more question from number theory.
$endgroup$
– Widawensen
Mar 21 at 7:58
$begingroup$
@SuperCiocia Interestingly, but I was thinking also about development of your question, only in my case I was interested when we would have integer coordinates of projected points (evidently at least $ cosalpha$ and $sinalpha$ should have been rational like $[0.6 0.8]$. It is interesting how many points of grid can be projected into projected points which are still on grid. But this is more question from number theory.
$endgroup$
– Widawensen
Mar 21 at 7:58
$begingroup$
@SuperCiocia Returning to the problem of spacing I'm not sure what you want to investigate: equal spacing between projected points or maybe the maximum points which are projected into the same point on line? When the line is rotating spacing is changing, that's for sure - different questions can be asked based on the phenomenon of changing spacing, we could visualize the effects with some kind of histogram which can be more or less discrete. Of course in this case the grid should be somehow limited, say $100 times 100$
$endgroup$
– Widawensen
Mar 21 at 8:06
$begingroup$
@SuperCiocia Returning to the problem of spacing I'm not sure what you want to investigate: equal spacing between projected points or maybe the maximum points which are projected into the same point on line? When the line is rotating spacing is changing, that's for sure - different questions can be asked based on the phenomenon of changing spacing, we could visualize the effects with some kind of histogram which can be more or less discrete. Of course in this case the grid should be somehow limited, say $100 times 100$
$endgroup$
– Widawensen
Mar 21 at 8:06
$begingroup$
I have a new question about this now.
$endgroup$
– SuperCiocia
Mar 21 at 15:56
$begingroup$
I have a new question about this now.
$endgroup$
– SuperCiocia
Mar 21 at 15:56
add a comment |
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