Mixing
Find coordinates of a 2D plane within a 3D plane
$begingroup$
I'm not sure this is the right place to ask this question, if not I do apologies and I will move on. I am asking this question as a programmer, however it seemed entirely maths based.
Image one is sitting still and holds in front of their face a sheet of graph paper with grid lines. This piece of paper represents a 2D plane, however although it exists inside of another 3D environment, it can be move about such that it's axis may not line up in any meaningful way.
Given 3 points where both the real world X,Y,Z values are known and the relative X,Y coordinates of their position within the paper drawn grid and given that two of these points share the same virtual 2D Y value but differ on X and the third point shares neither the same X nor Y value but does indeed exist on the same virtual 2D plane.
How can I then transform any given coordinate of one set to the other. Say I want to know where square (12,20) on the graph paper exists in the real world and vice versa.
Also, say I have a set of real world coordinates that is not on my 2D plane but directly above it. Is there a way and if so, how would i: traverse "downwards" through 3D space perpendicular to the find the 2D point directly below.
geometry
asked Aug 4 '15 at 7:33
user1611172user1611172
63
$endgroup$
add a comment |
$begingroup$
I'm not sure this is the right place to ask this question, if not I do apologies and I will move on. I am asking this question as a programmer, however it seemed entirely maths based.
Image one is sitting still and holds in front of their face a sheet of graph paper with grid lines. This piece of paper represents a 2D plane, however although it exists inside of another 3D environment, it can be move about such that it's axis may not line up in any meaningful way.
Given 3 points where both the real world X,Y,Z values are known and the relative X,Y coordinates of their position within the paper drawn grid and given that two of these points share the same virtual 2D Y value but differ on X and the third point shares neither the same X nor Y value but does indeed exist on the same virtual 2D plane.
How can I then transform any given coordinate of one set to the other. Say I want to know where square (12,20) on the graph paper exists in the real world and vice versa.
Also, say I have a set of real world coordinates that is not on my 2D plane but directly above it. Is there a way and if so, how would i: traverse "downwards" through 3D space perpendicular to the find the 2D point directly below.
geometry
asked Aug 4 '15 at 7:33
user1611172user1611172
63
$endgroup$
$begingroup$
How is your plane defined? You can't know what you want without some sort of description of this plane.
$endgroup$
– A.P.
Aug 4 '15 at 7:44
$begingroup$
Let's say I have an arena or perhaps an enclosed box. The box is 1,000cm on all sides. There is also a 100x100cm piece of paper with 1cm grid lines drawn on it. Therefor this sheet of paper represents a 2D plane that we define by the 1cm grid. This sheet of paper may be arbitraliy oriented within the box. I know of only 3 points on the box according to the criteria in the question. For example lets say the grid square (50,50)(45,60),(55,60) have known coordinates within the 3D box/arena. Is there a generlised way I can then derive the 3D coords of any given 2D coord on the sheet?
$endgroup$
– user1611172
Aug 4 '15 at 8:00
$begingroup$
Not unless you can tell us how the plane is given. How do you define its orientation and position in 3D space, for example?
$endgroup$
– A.P.
Aug 4 '15 at 9:23
$begingroup$
For example, if the position of your sheet of "paper" is given as two opposite points of a rectangle, you can use those to compute the parametric equations of a plane, which give you what you want; or you can compute its Cartesian equation, which immediately gives you what you want because it is of the form $a X + b Y + c Z = d$. Still, this plane must be defined somehow, because otherwise you couldn't draw it...
$endgroup$
– A.P.
Aug 4 '15 at 10:09
$begingroup$
Edit: Thank you for your responses and time :) If I know the global X,Y,Z coordinates of 5 points on the 2D plan where the points form two lines parallel the x,y axis of the 2D plane. For example 2D grid coordinates (10,10)(-10,0),(0,0)(10,0)(0,-10) (imagine they form a plus sign shape on the center of the grid) have known 3D coordinates can I then use that to define the plane and derive 3D coordinates from arbitrarily given 2D coordinates.
$endgroup$
– user1611172
Aug 5 '15 at 4:20
add a comment |
$begingroup$
I'm not sure this is the right place to ask this question, if not I do apologies and I will move on. I am asking this question as a programmer, however it seemed entirely maths based.
Image one is sitting still and holds in front of their face a sheet of graph paper with grid lines. This piece of paper represents a 2D plane, however although it exists inside of another 3D environment, it can be move about such that it's axis may not line up in any meaningful way.
Given 3 points where both the real world X,Y,Z values are known and the relative X,Y coordinates of their position within the paper drawn grid and given that two of these points share the same virtual 2D Y value but differ on X and the third point shares neither the same X nor Y value but does indeed exist on the same virtual 2D plane.
How can I then transform any given coordinate of one set to the other. Say I want to know where square (12,20) on the graph paper exists in the real world and vice versa.
Also, say I have a set of real world coordinates that is not on my 2D plane but directly above it. Is there a way and if so, how would i: traverse "downwards" through 3D space perpendicular to the find the 2D point directly below.
geometry
asked Aug 4 '15 at 7:33
user1611172user1611172
63
$endgroup$
I'm not sure this is the right place to ask this question, if not I do apologies and I will move on. I am asking this question as a programmer, however it seemed entirely maths based.
Image one is sitting still and holds in front of their face a sheet of graph paper with grid lines. This piece of paper represents a 2D plane, however although it exists inside of another 3D environment, it can be move about such that it's axis may not line up in any meaningful way.
Given 3 points where both the real world X,Y,Z values are known and the relative X,Y coordinates of their position within the paper drawn grid and given that two of these points share the same virtual 2D Y value but differ on X and the third point shares neither the same X nor Y value but does indeed exist on the same virtual 2D plane.
How can I then transform any given coordinate of one set to the other. Say I want to know where square (12,20) on the graph paper exists in the real world and vice versa.
Also, say I have a set of real world coordinates that is not on my 2D plane but directly above it. Is there a way and if so, how would i: traverse "downwards" through 3D space perpendicular to the find the 2D point directly below.
geometry
geometry
asked Aug 4 '15 at 7:33
user1611172user1611172
63
asked Aug 4 '15 at 7:33
user1611172user1611172
63
asked Aug 4 '15 at 7:33
user1611172user1611172
63
asked Aug 4 '15 at 7:33
user1611172user1611172
63
asked Aug 4 '15 at 7:33
user1611172user1611172
63
63
$begingroup$
How is your plane defined? You can't know what you want without some sort of description of this plane.
$endgroup$
– A.P.
Aug 4 '15 at 7:44
$begingroup$
Let's say I have an arena or perhaps an enclosed box. The box is 1,000cm on all sides. There is also a 100x100cm piece of paper with 1cm grid lines drawn on it. Therefor this sheet of paper represents a 2D plane that we define by the 1cm grid. This sheet of paper may be arbitraliy oriented within the box. I know of only 3 points on the box according to the criteria in the question. For example lets say the grid square (50,50)(45,60),(55,60) have known coordinates within the 3D box/arena. Is there a generlised way I can then derive the 3D coords of any given 2D coord on the sheet?
$endgroup$
– user1611172
Aug 4 '15 at 8:00
$begingroup$
Not unless you can tell us how the plane is given. How do you define its orientation and position in 3D space, for example?
$endgroup$
– A.P.
Aug 4 '15 at 9:23
$begingroup$
For example, if the position of your sheet of "paper" is given as two opposite points of a rectangle, you can use those to compute the parametric equations of a plane, which give you what you want; or you can compute its Cartesian equation, which immediately gives you what you want because it is of the form $a X + b Y + c Z = d$. Still, this plane must be defined somehow, because otherwise you couldn't draw it...
$endgroup$
– A.P.
Aug 4 '15 at 10:09
$begingroup$
Edit: Thank you for your responses and time :) If I know the global X,Y,Z coordinates of 5 points on the 2D plan where the points form two lines parallel the x,y axis of the 2D plane. For example 2D grid coordinates (10,10)(-10,0),(0,0)(10,0)(0,-10) (imagine they form a plus sign shape on the center of the grid) have known 3D coordinates can I then use that to define the plane and derive 3D coordinates from arbitrarily given 2D coordinates.
$endgroup$
– user1611172
Aug 5 '15 at 4:20
add a comment |
$begingroup$
How is your plane defined? You can't know what you want without some sort of description of this plane.
$endgroup$
– A.P.
Aug 4 '15 at 7:44
$begingroup$
Let's say I have an arena or perhaps an enclosed box. The box is 1,000cm on all sides. There is also a 100x100cm piece of paper with 1cm grid lines drawn on it. Therefor this sheet of paper represents a 2D plane that we define by the 1cm grid. This sheet of paper may be arbitraliy oriented within the box. I know of only 3 points on the box according to the criteria in the question. For example lets say the grid square (50,50)(45,60),(55,60) have known coordinates within the 3D box/arena. Is there a generlised way I can then derive the 3D coords of any given 2D coord on the sheet?
$endgroup$
– user1611172
Aug 4 '15 at 8:00
$begingroup$
Not unless you can tell us how the plane is given. How do you define its orientation and position in 3D space, for example?
$endgroup$
– A.P.
Aug 4 '15 at 9:23
$begingroup$
For example, if the position of your sheet of "paper" is given as two opposite points of a rectangle, you can use those to compute the parametric equations of a plane, which give you what you want; or you can compute its Cartesian equation, which immediately gives you what you want because it is of the form $a X + b Y + c Z = d$. Still, this plane must be defined somehow, because otherwise you couldn't draw it...
$endgroup$
– A.P.
Aug 4 '15 at 10:09
$begingroup$
Edit: Thank you for your responses and time :) If I know the global X,Y,Z coordinates of 5 points on the 2D plan where the points form two lines parallel the x,y axis of the 2D plane. For example 2D grid coordinates (10,10)(-10,0),(0,0)(10,0)(0,-10) (imagine they form a plus sign shape on the center of the grid) have known 3D coordinates can I then use that to define the plane and derive 3D coordinates from arbitrarily given 2D coordinates.
$endgroup$
– user1611172
Aug 5 '15 at 4:20
$begingroup$
How is your plane defined? You can't know what you want without some sort of description of this plane.
$endgroup$
– A.P.
Aug 4 '15 at 7:44
$begingroup$
How is your plane defined? You can't know what you want without some sort of description of this plane.
$endgroup$
– A.P.
Aug 4 '15 at 7:44
$begingroup$
Let's say I have an arena or perhaps an enclosed box. The box is 1,000cm on all sides. There is also a 100x100cm piece of paper with 1cm grid lines drawn on it. Therefor this sheet of paper represents a 2D plane that we define by the 1cm grid. This sheet of paper may be arbitraliy oriented within the box. I know of only 3 points on the box according to the criteria in the question. For example lets say the grid square (50,50)(45,60),(55,60) have known coordinates within the 3D box/arena. Is there a generlised way I can then derive the 3D coords of any given 2D coord on the sheet?
$endgroup$
– user1611172
Aug 4 '15 at 8:00
$begingroup$
Let's say I have an arena or perhaps an enclosed box. The box is 1,000cm on all sides. There is also a 100x100cm piece of paper with 1cm grid lines drawn on it. Therefor this sheet of paper represents a 2D plane that we define by the 1cm grid. This sheet of paper may be arbitraliy oriented within the box. I know of only 3 points on the box according to the criteria in the question. For example lets say the grid square (50,50)(45,60),(55,60) have known coordinates within the 3D box/arena. Is there a generlised way I can then derive the 3D coords of any given 2D coord on the sheet?
$endgroup$
– user1611172
Aug 4 '15 at 8:00
$begingroup$
Not unless you can tell us how the plane is given. How do you define its orientation and position in 3D space, for example?
$endgroup$
– A.P.
Aug 4 '15 at 9:23
$begingroup$
Not unless you can tell us how the plane is given. How do you define its orientation and position in 3D space, for example?
$endgroup$
– A.P.
Aug 4 '15 at 9:23
$begingroup$
For example, if the position of your sheet of "paper" is given as two opposite points of a rectangle, you can use those to compute the parametric equations of a plane, which give you what you want; or you can compute its Cartesian equation, which immediately gives you what you want because it is of the form $a X + b Y + c Z = d$. Still, this plane must be defined somehow, because otherwise you couldn't draw it...
$endgroup$
– A.P.
Aug 4 '15 at 10:09
$begingroup$
For example, if the position of your sheet of "paper" is given as two opposite points of a rectangle, you can use those to compute the parametric equations of a plane, which give you what you want; or you can compute its Cartesian equation, which immediately gives you what you want because it is of the form $a X + b Y + c Z = d$. Still, this plane must be defined somehow, because otherwise you couldn't draw it...
$endgroup$
– A.P.
Aug 4 '15 at 10:09
$begingroup$
Edit: Thank you for your responses and time :) If I know the global X,Y,Z coordinates of 5 points on the 2D plan where the points form two lines parallel the x,y axis of the 2D plane. For example 2D grid coordinates (10,10)(-10,0),(0,0)(10,0)(0,-10) (imagine they form a plus sign shape on the center of the grid) have known 3D coordinates can I then use that to define the plane and derive 3D coordinates from arbitrarily given 2D coordinates.
$endgroup$
– user1611172
Aug 5 '15 at 4:20
$begingroup$
Edit: Thank you for your responses and time :) If I know the global X,Y,Z coordinates of 5 points on the 2D plan where the points form two lines parallel the x,y axis of the 2D plane. For example 2D grid coordinates (10,10)(-10,0),(0,0)(10,0)(0,-10) (imagine they form a plus sign shape on the center of the grid) have known 3D coordinates can I then use that to define the plane and derive 3D coordinates from arbitrarily given 2D coordinates.
$endgroup$
– user1611172
Aug 5 '15 at 4:20
add a comment |
1 Answer
1
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$begingroup$
For simplicity, suppose you know the coordinates in 3D space $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ of (the image of) $(1,0)$ and $(0,1)$, respectively. Then every point $(a,b)$ in 2D space can be written as
$$
(a,b) = a(1,0) + b(0,1)
$$
and the corresponding point in 3D space has coordinates
$$
a(x_1,y_1,z_1) + b(x_2,y_2,z_2) = (ax_1 + bx_2, ay_1 + b y_2, az_2 + bz_2)
$$
If instead you know the 3D coordinates of, say, $(10,0)$ and $(0,10)$, then you can use the same trick after writing
$$
(a,b) = frac{a}{10}(10,0) + frac{b}{10}(0,10)
$$
answered Aug 5 '15 at 8:33
A.P.A.P.
8,18021840
$endgroup$
add a comment |
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1 Answer
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$begingroup$
For simplicity, suppose you know the coordinates in 3D space $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ of (the image of) $(1,0)$ and $(0,1)$, respectively. Then every point $(a,b)$ in 2D space can be written as
$$
(a,b) = a(1,0) + b(0,1)
$$
and the corresponding point in 3D space has coordinates
$$
a(x_1,y_1,z_1) + b(x_2,y_2,z_2) = (ax_1 + bx_2, ay_1 + b y_2, az_2 + bz_2)
$$
If instead you know the 3D coordinates of, say, $(10,0)$ and $(0,10)$, then you can use the same trick after writing
$$
(a,b) = frac{a}{10}(10,0) + frac{b}{10}(0,10)
$$
answered Aug 5 '15 at 8:33
A.P.A.P.
8,18021840
$endgroup$
add a comment |
$begingroup$
For simplicity, suppose you know the coordinates in 3D space $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ of (the image of) $(1,0)$ and $(0,1)$, respectively. Then every point $(a,b)$ in 2D space can be written as
$$
(a,b) = a(1,0) + b(0,1)
$$
and the corresponding point in 3D space has coordinates
$$
a(x_1,y_1,z_1) + b(x_2,y_2,z_2) = (ax_1 + bx_2, ay_1 + b y_2, az_2 + bz_2)
$$
If instead you know the 3D coordinates of, say, $(10,0)$ and $(0,10)$, then you can use the same trick after writing
$$
(a,b) = frac{a}{10}(10,0) + frac{b}{10}(0,10)
$$
answered Aug 5 '15 at 8:33
A.P.A.P.
8,18021840
$endgroup$
add a comment |
$begingroup$
For simplicity, suppose you know the coordinates in 3D space $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ of (the image of) $(1,0)$ and $(0,1)$, respectively. Then every point $(a,b)$ in 2D space can be written as
$$
(a,b) = a(1,0) + b(0,1)
$$
and the corresponding point in 3D space has coordinates
$$
a(x_1,y_1,z_1) + b(x_2,y_2,z_2) = (ax_1 + bx_2, ay_1 + b y_2, az_2 + bz_2)
$$
If instead you know the 3D coordinates of, say, $(10,0)$ and $(0,10)$, then you can use the same trick after writing
$$
(a,b) = frac{a}{10}(10,0) + frac{b}{10}(0,10)
$$
answered Aug 5 '15 at 8:33
A.P.A.P.
8,18021840
$endgroup$
For simplicity, suppose you know the coordinates in 3D space $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ of (the image of) $(1,0)$ and $(0,1)$, respectively. Then every point $(a,b)$ in 2D space can be written as
$$
(a,b) = a(1,0) + b(0,1)
$$
and the corresponding point in 3D space has coordinates
$$
a(x_1,y_1,z_1) + b(x_2,y_2,z_2) = (ax_1 + bx_2, ay_1 + b y_2, az_2 + bz_2)
$$
If instead you know the 3D coordinates of, say, $(10,0)$ and $(0,10)$, then you can use the same trick after writing
$$
(a,b) = frac{a}{10}(10,0) + frac{b}{10}(0,10)
$$
answered Aug 5 '15 at 8:33
A.P.A.P.
8,18021840
answered Aug 5 '15 at 8:33
A.P.A.P.
8,18021840
answered Aug 5 '15 at 8:33
A.P.A.P.
8,18021840
answered Aug 5 '15 at 8:33
A.P.A.P.
8,18021840
8,18021840
add a comment |
add a comment |
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$begingroup$
How is your plane defined? You can't know what you want without some sort of description of this plane.
$endgroup$
– A.P.
Aug 4 '15 at 7:44
$begingroup$
Let's say I have an arena or perhaps an enclosed box. The box is 1,000cm on all sides. There is also a 100x100cm piece of paper with 1cm grid lines drawn on it. Therefor this sheet of paper represents a 2D plane that we define by the 1cm grid. This sheet of paper may be arbitraliy oriented within the box. I know of only 3 points on the box according to the criteria in the question. For example lets say the grid square (50,50)(45,60),(55,60) have known coordinates within the 3D box/arena. Is there a generlised way I can then derive the 3D coords of any given 2D coord on the sheet?
$endgroup$
– user1611172
Aug 4 '15 at 8:00
$begingroup$
Not unless you can tell us how the plane is given. How do you define its orientation and position in 3D space, for example?
$endgroup$
– A.P.
Aug 4 '15 at 9:23
$begingroup$
For example, if the position of your sheet of "paper" is given as two opposite points of a rectangle, you can use those to compute the parametric equations of a plane, which give you what you want; or you can compute its Cartesian equation, which immediately gives you what you want because it is of the form $a X + b Y + c Z = d$. Still, this plane must be defined somehow, because otherwise you couldn't draw it...
$endgroup$
– A.P.
Aug 4 '15 at 10:09
$begingroup$
Edit: Thank you for your responses and time :) If I know the global X,Y,Z coordinates of 5 points on the 2D plan where the points form two lines parallel the x,y axis of the 2D plane. For example 2D grid coordinates (10,10)(-10,0),(0,0)(10,0)(0,-10) (imagine they form a plus sign shape on the center of the grid) have known 3D coordinates can I then use that to define the plane and derive 3D coordinates from arbitrarily given 2D coordinates.
$endgroup$
– user1611172
Aug 5 '15 at 4:20