[Math question] Why do 3 non collinear points lie in a plane?
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Hi guys,
so as the title says and in terms of geometry of course, why do 3 non collinear points lie in a distinct plane? I mean why can't these 3 points lie in multiple planes?
If somebody could try to show me visually maybe with aide of a paint tool such as ms paint,pinta, gimp or even photoshop that would be awesome. I just can't seem to figure this out and if we add a fourth non collinear point then these 4 points may not lie in one distinct plane.
So yeah in other words, if 3 non collinear points define a plane why doesn't 4 non collinear points not definitively define a plane?
thanks :)
so as the title says and in terms of geometry of course, why do 3 non collinear points lie in a distinct plane? I mean why can't these 3 points lie in multiple planes?
If somebody could try to show me visually maybe with aide of a paint tool such as ms paint,pinta, gimp or even photoshop that would be awesome. I just can't seem to figure this out and if we add a fourth non collinear point then these 4 points may not lie in one distinct plane.
So yeah in other words, if 3 non collinear points define a plane why doesn't 4 non collinear points not definitively define a plane?
thanks :)
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| if 3 non collinear points define a plane why doesn't 4 non collinear points |
Take a sheet of paper. Flex it along one diagonal. Convince yourself that you have 4 points (the four corners) and the sides join the points with straight lines, but the paper isn't flat (planar).
Cut your paper along that diagonal so that you have a triangle (3 non-collinear points). If the sides joining those points have to be straight then there is only one plane passing through them.
Interestingly, 4 points may not define a unique plane; however the vector area spanning them is unique and well-defined. It works out as half the vector cross product of the diagonals.
| so as the title says and in terms of geometry of course, why do 3 non collinear points lie in a distinct plane? I mean why can't these 3 points lie in multiple planes? |
it is an extrapolation of 2 distinct points creating a line. 2d, 3d, does not matter, if you have 2 points, and you connect them with the shortest possible connection, it will be a line through them.
a plane is the same thing. if you have 3 points, you can draw a line through each pair. If all 3 are not on one line, then the lines you draw form.. what?
If you do the same with a 4th point, though, it becomes possible to do ... what?
answers.. 1, you get a triangle, a 2d object. Its not possible to make it 3-d. The slice in which it is 2-d, that is your plane.
2, it adds a dimension. now you can draw a pyramid, or some other 3d thing, and its not always a plane. The extra point can be anywhere, but just as the third point off a line extends 1-d to 2-d, the 4th point takes 2-d to 3d.
if you don't see this, start back a 1 point. what is 1 point? all it defines is that point, nowhere else exists in the space created by it. Its a 0-d space, really.
now take 2 points, what is that? It defines both points and all points exactly between them: that is a line segment, or you can extend out past them as far as you need, that is a line.
Now add a point, and its back to what I said above.. the points between them become an area, a triangle. it takes a 4th point to get a volume....
playing with paper can be instructive, but there are pitfalls too. If you put 3 points on a sheet of paper, its plane, but if you wad/fold up the paper a bit into a 3-dish thing, its hard to see that they STILL make a plane, just not the same one, and not one that you can look at (its no longer the sheet of paper itself as the plane).
a plane is the same thing. if you have 3 points, you can draw a line through each pair. If all 3 are not on one line, then the lines you draw form.. what?
If you do the same with a 4th point, though, it becomes possible to do ... what?
answers.. 1, you get a triangle, a 2d object. Its not possible to make it 3-d. The slice in which it is 2-d, that is your plane.
2, it adds a dimension. now you can draw a pyramid, or some other 3d thing, and its not always a plane. The extra point can be anywhere, but just as the third point off a line extends 1-d to 2-d, the 4th point takes 2-d to 3d.
if you don't see this, start back a 1 point. what is 1 point? all it defines is that point, nowhere else exists in the space created by it. Its a 0-d space, really.
now take 2 points, what is that? It defines both points and all points exactly between them: that is a line segment, or you can extend out past them as far as you need, that is a line.
Now add a point, and its back to what I said above.. the points between them become an area, a triangle. it takes a 4th point to get a volume....
playing with paper can be instructive, but there are pitfalls too. If you put 3 points on a sheet of paper, its plane, but if you wad/fold up the paper a bit into a 3-dish thing, its hard to see that they STILL make a plane, just not the same one, and not one that you can look at (its no longer the sheet of paper itself as the plane).
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Meh, yer all makin' it too hard.
1. Get yerself a magic marker.
2. Go to your kitchen table.
3. (Check to see that your wife is not nearby.)
4. Put three distinct dots on the tabletop.
(If you cannot connect them to make a triangle, you did it wrong.)
That's your plane.
5. Put a dot on the ceiling.
Since your table does not intersect with your ceiling, the dot on the ceiling is not in the tabletop's plane.
You could have put another dot on the table, but the important point is this:
→ There is no guarantee that the fourth dot is in the same plane as the other three dots. ←
Hence, four dots do not definitively define a plane. They could, but they probably don't.
Hope this helps.
1. Get yerself a magic marker.
2. Go to your kitchen table.
3. (Check to see that your wife is not nearby.)
4. Put three distinct dots on the tabletop.
(If you cannot connect them to make a triangle, you did it wrong.)
That's your plane.
5. Put a dot on the ceiling.
Since your table does not intersect with your ceiling, the dot on the ceiling is not in the tabletop's plane.
You could have put another dot on the table, but the important point is this:
→ There is no guarantee that the fourth dot is in the same plane as the other three dots. ←
Hence, four dots do not definitively define a plane. They could, but they probably don't.
Hope this helps.
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Starting to make sense now,
I'm going to go with the table example as it seems to be the easiest to grasp for a beginner( I wish I would have payed more attention in geometry class when I was a kid lol)
Okay so I have an understanding of what a plane is, it's just an invisible area in space(3 dimensions) but the actual plane is 2d, so you can place it flat on a surface but this doesn't mean the plane is rotated flat on a surface, just that you could.
- https://math.stackexchange.com/questions/3743058/why-do-three-non-collinears-points-define-a-plane - user David Stork posts a good image of an example of different planes.
Going back to the table example let's say we draw three dots on the table and decide to put a rubix cube above the 3 dots, we then draw a dot on the top surface/plane of the rubix cube. We have 4 dots now ( 3 on the table and 1 on the cube), we could have a plane that has the dot of the cube and the two dots from the table but we couldn't have a plane that contains the dot from the cube and the 3 dots on the table all these dots would not be coplanar.
I'm going to use the image from the link I provided, it's pretty obvious from the picture to tell what plane the 3 points are on and actually that's quite a good example to prove that 3 points define a distinct plane but unfortunately not all images are like this one when dealing with geometry, you are more than likely going to encounter like this - https://i.imgur.com/MkwkClD.png
how do you go about figuring out which plane points are coplanar on a diagram like the one above( latter link).
are A,G,M and I coplanar? and how can you tell if they are?
Also @Helios, yeah re-reading that. I think I could have chosen my words a little better haha
I'm going to go with the table example as it seems to be the easiest to grasp for a beginner( I wish I would have payed more attention in geometry class when I was a kid lol)
Okay so I have an understanding of what a plane is, it's just an invisible area in space(3 dimensions) but the actual plane is 2d, so you can place it flat on a surface but this doesn't mean the plane is rotated flat on a surface, just that you could.
- https://math.stackexchange.com/questions/3743058/why-do-three-non-collinears-points-define-a-plane - user David Stork posts a good image of an example of different planes.
Going back to the table example let's say we draw three dots on the table and decide to put a rubix cube above the 3 dots, we then draw a dot on the top surface/plane of the rubix cube. We have 4 dots now ( 3 on the table and 1 on the cube), we could have a plane that has the dot of the cube and the two dots from the table but we couldn't have a plane that contains the dot from the cube and the 3 dots on the table all these dots would not be coplanar.
I'm going to use the image from the link I provided, it's pretty obvious from the picture to tell what plane the 3 points are on and actually that's quite a good example to prove that 3 points define a distinct plane but unfortunately not all images are like this one when dealing with geometry, you are more than likely going to encounter like this - https://i.imgur.com/MkwkClD.png
how do you go about figuring out which plane points are coplanar on a diagram like the one above( latter link).
are A,G,M and I coplanar? and how can you tell if they are?
Also @Helios, yeah re-reading that. I think I could have chosen my words a little better haha
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| are A,G,M and I coplanar? |
The question in that image is also ambiguous, for the same reason. It's unclear whether it's posing a trick question or wording a real question poorly.
I meant are the points A,G,M and I (together) coplanar, but yeah the image does leave a lot open to interpretation.
Sorry, I missed the "I".
Yes, AGMI are coplanar. AGM are colinear, so there's a single line (l) that contains all three. For any line and a point lying outside of it there's a single plane that contains both. Since such a plane contains both I and l, and l contains AGM, that plane contains all four points.
Yes, AGMI are coplanar. AGM are colinear, so there's a single line (l) that contains all three. For any line and a point lying outside of it there's a single plane that contains both. Since such a plane contains both I and l, and l contains AGM, that plane contains all four points.
the table example is a good start, but now put a point on the wall, the floor, and the ceiling.
its difficult to see/imagine such a plane, and why it is a plane. I mean, a table itself is mostly a plane, so putting any number of points on it is going to be coplanar... that won't quite get you all the way to the finish line.
I don't know a really good way to fully visualize it without a computer... put 3 points in space in some software that can fill in between them and see that no matter how you rotate you get a 2d slice...
you can do it with math, various ways... take the normals at each point and observe they are all parallel (it may not pass a PHD proof of plane but its easy to see for understanding purposes and easy to do). Or I think you can back into it with a 'lets assume its not a plane, then what ... ... oh, that can't happen, therefore...)
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Okay so I have an understanding of what a plane is, it's just an invisible area in space(3 dimensions) but the actual plane is 2d, so you can place it flat on a surface but this doesn't mean the plane is rotated flat on a surface, just that you could.
A plane is a 2-d slice of a 3-d space in the same way that a line is a 1-d thing that can live in a 2-d space. If you go with that idea, a plane is a line that has been extruded in any direction perpendicular to the original line. (it does not have to be 90 degrees, but start there).
its difficult to see/imagine such a plane, and why it is a plane. I mean, a table itself is mostly a plane, so putting any number of points on it is going to be coplanar... that won't quite get you all the way to the finish line.
I don't know a really good way to fully visualize it without a computer... put 3 points in space in some software that can fill in between them and see that no matter how you rotate you get a 2d slice...
you can do it with math, various ways... take the normals at each point and observe they are all parallel (it may not pass a PHD proof of plane but its easy to see for understanding purposes and easy to do). Or I think you can back into it with a 'lets assume its not a plane, then what ... ... oh, that can't happen, therefore...)
-----------------------
Okay so I have an understanding of what a plane is, it's just an invisible area in space(3 dimensions) but the actual plane is 2d, so you can place it flat on a surface but this doesn't mean the plane is rotated flat on a surface, just that you could.
A plane is a 2-d slice of a 3-d space in the same way that a line is a 1-d thing that can live in a 2-d space. If you go with that idea, a plane is a line that has been extruded in any direction perpendicular to the original line. (it does not have to be 90 degrees, but start there).
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Starting to make sense, I think (I hope at least) that I'm getting the basics of it under wraps.
The main issue I was facing was understanding how and why 3 points can identify a unique plane( all 3 points will be on that unique plane) and why four points can't. So here is my brief understanding and my terminology or choice of words may not be precise in a mathematical sense but I'll express what I think is the answer to that question.
So if you have 2 points that are on the same plane (collinear also obviously) you can rotate many planes around those 2 points so it's pretty obvious these two points could sit in a multitude of planes. But if we have a third point that isn't collinear only one plane will go through all these 3 points, now with 4 points gets a little murky, let's say those 3 said points are A,B,C and they all lay on the same plane well then point D may or may not be on the same plane as A,B, or C or in other words we will not be able to connect the four dots in a way that doesn't bend or distort the plane, that's basically correct, right?
A question comes to mind. If we have 3 points, no matter where those 3 points are placed on the x,y,z axis, will they always form a plane(that is not bent or distorted obviously)?
This is really interesting,I'd like to try it. I'm obviously learning geometry for a reason, to understand and possibly make a couple of simple video games and also maybe get into physics sims. As far as scope, what would I need to search up on to understand normals of a point, or in other words is there a certain sub topic in math that focuses on or covers this?
Thanks :)
The main issue I was facing was understanding how and why 3 points can identify a unique plane( all 3 points will be on that unique plane) and why four points can't. So here is my brief understanding and my terminology or choice of words may not be precise in a mathematical sense but I'll express what I think is the answer to that question.
So if you have 2 points that are on the same plane (collinear also obviously) you can rotate many planes around those 2 points so it's pretty obvious these two points could sit in a multitude of planes. But if we have a third point that isn't collinear only one plane will go through all these 3 points, now with 4 points gets a little murky, let's say those 3 said points are A,B,C and they all lay on the same plane well then point D may or may not be on the same plane as A,B, or C or in other words we will not be able to connect the four dots in a way that doesn't bend or distort the plane, that's basically correct, right?
A question comes to mind. If we have 3 points, no matter where those 3 points are placed on the x,y,z axis, will they always form a plane(that is not bent or distorted obviously)?
| you can do it with math, various ways... take the normals at each point and observe they are all parallel (it may not pass a PHD proof of plane but its easy to see for understanding purposes and easy to do). Or I think you can back into it with a 'lets assume its not a plane, then what ... ... oh, that can't happen, therefore...) |
This is really interesting,I'd like to try it. I'm obviously learning geometry for a reason, to understand and possibly make a couple of simple video games and also maybe get into physics sims. As far as scope, what would I need to search up on to understand normals of a point, or in other words is there a certain sub topic in math that focuses on or covers this?
Thanks :)
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| Adam2016 wrote: |
|---|
| A question comes to mind. If we have 3 points, no matter where those 3 points are placed on the x,y,z axis, will they always form a plane |
If they are not collinear then they will lie in one unique plane. If they are collinear then they will lie in an infinity of planes.
| that is not bent |
There is no such thing as a bent plane (by definition).
| why 3 points can identify a unique plane( all 3 points will be on that unique plane) and why four points can't. |
The equation of a plane has three independent parameters. If you impose 4 constraints (i.e. that it passes through four points) then you are (usually) overspecifying the problem.
A plane has normal that is perpendicular to all lines that are on the plane: https://web.ma.utexas.edu/users/m408m/Display12-5-4.shtml
Distance of point from plane: https://web.ma.utexas.edu/users/m408m/Display12-5-5.shtml
The distance is 0 if the fourth point is on the same plane as the first three.
Distance of point from plane: https://web.ma.utexas.edu/users/m408m/Display12-5-5.shtml
The distance is 0 if the fourth point is on the same plane as the first three.
| [ The equation of a plane has three independent parameters. If you impose 4 constraints (i.e. that it passes through four points) then you are (usually) overspecifying the problem. |
let's say we have 2 (both implicitly and obviously collinear) points that lie in a plane S, and we have another set of points that lie in another plane(again both implicitly and obviously collinear) will those 4 points be coplanar? (I'm guessing yes because a real life example of this would be the top of a cube and the bottom of a cube ( https://www.researchgate.net/figure/The-schematic-of-assessing-the-coplanar-lines-by-the-volume-of-the-tetrahedron-The-red_fig3_329550309 ) ).
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| let's say we have 2 (both implicitly and obviously collinear) points that lie in a plane S, and we have another set of points that lie in another plane(again both implicitly and obviously collinear) will those 4 points be coplanar? |
By the way, you don't need to clarify that two points are colinear. There's no way for two points to not be colinear.
Maybe the simplest way to understand it is to look at it inductively. "A set of n linearly independent points specifies a unique (n - 1)-dimensional space."
1 point specifies only itself, a dimensionless region.
2 points specify a line, a 1-D region.
3 points specify a plane, a 2-D region.
4 points specify a volume, a 3-D region.
5 points specify a 4-D hypervolume.
etc.
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This is really interesting,I'd like to try it...
https://mathworld.wolfram.com/NormalVector.html#:~:text=The%20normal%20vector%2C%20often%20simply,pointing%20normal%20are%20usually%20distinguished.
specifically the section
The equation of a plane with normal vector n=(a,b,c) passing through the point (x_0,y_0,z_0) is given by a(x-x_0)+b(y-y_0)+c(z-z_0)=0.
or
http://www.leadinglesson.com/problem-on-finding-a-normal-vector-to-a-surface
points don't have normals. all your points in this topic are in a plane, planes do have a normal vector, you will see when you brows the links.
things that are not clear right off..
-any point on the plane can be picked to find a normal.
-all the points and their normals are parallel, just as they are all 90 degrees off the plane.
- some of the topics go off about getting normals on surfaces (not always a plane). This is advanced, ignore for now. They also go a bit into calculus, gradients is 3-d / multivariable calc, its sort of the slope of a surface.
https://mathworld.wolfram.com/NormalVector.html#:~:text=The%20normal%20vector%2C%20often%20simply,pointing%20normal%20are%20usually%20distinguished.
specifically the section
The equation of a plane with normal vector n=(a,b,c) passing through the point (x_0,y_0,z_0) is given by a(x-x_0)+b(y-y_0)+c(z-z_0)=0.
or
http://www.leadinglesson.com/problem-on-finding-a-normal-vector-to-a-surface
points don't have normals. all your points in this topic are in a plane, planes do have a normal vector, you will see when you brows the links.
things that are not clear right off..
-any point on the plane can be picked to find a normal.
-all the points and their normals are parallel, just as they are all 90 degrees off the plane.
- some of the topics go off about getting normals on surfaces (not always a plane). This is advanced, ignore for now. They also go a bit into calculus, gradients is 3-d / multivariable calc, its sort of the slope of a surface.
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| adam2016 wrote: |
|---|
| As far as scope, what would I need to search up on to understand normals of a point, or in other words is there a certain sub topic in math that focuses on or covers this? |
Have you done vector algebra yet? It will all make sense when you do.
Search for cross product.
If one has 2 line segments that lie in a plane, the cross product will produce a normal to that plane. The order of the cross product is important; cross product is not associative: a cross b is different to b cross a. The difference is that the sense of the direction is opposite. vectors can have the same direction but opposite sense, as in (0,0,1) is the opposite sense to (0,0,-1), but these directions are the same. There is a right hand rule for determining the sense of the cross product: if vector A is represented by the index finger of the right hand; vector B is the second finger held at an angle to the index finger; the thumb pointing up is the direction and sense of the cross product. If one reverses the order, then the thumb points down: the opposite sense.
Example: A (1,0,0) - the X axis ; B (0,1,0) the Y axis: A cross B is (0,0,1) Positive Z axis. B cross A is (0,0,-1) negative Z axis.
To compare cross products from using different line segments, calc the unit vector (magnitude 1.0) for each cross product result. If the the points are all in the same plane, the cross product unit vectors should be equal as long as the sense is the same.
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| jonnin wrote: |
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| The equation of a plane with normal vector n=(a,b,c) passing through the point (x_0,y_0,z_0) is given by a(x-x_0)+b(y-y_0)+c(z-z_0)=0. |
| TheIdeasMan wrote: |
|---|
| Example: A (1,0,0) - the X axis ; B (0,1,0) the Y axis: A cross B is (0,0,1) Positive Z axis. |
With this example
n = (a,b,c) = (0,0,1) and point P (0,0,0) is on the "XY-plane". All points (x,y,0) are on the "XY-plane". |
|
Result: Point must have
z == 0 to be on the "XY plane". True for all (x,y,0).Lets take another plane. It has same normal n, but point Q (2,5,3.14) is on it.
Lets fill those values int equation:
|
|
Result: all points (x,y,3.14) are on this plane.
There is no point that would be on both of these planes.
| TheIdeasMan wrote: |
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| the cross product unit vectors should be equal as long as the sense is the same. |
Totally agree that the cross product is an invaluable tool in constructing a normal.
Just to add that there is no need to normalise to unit vectors or worry about the sense/orientation. Fundamentally, you just need to show that they are parallel and that can be achieved quickly by showing that the cross product of the two vectors is 0. (In this example that amounts to a cross product of cross products!)
Only slight downside is the usual problem in computing of comparing two floating-point numbers for equality.
| No, not necessarily. Imagine four arbitrary points A, B, C, and D on a plane. They're obviously coplanar, right? Now lift D off the plane. They're obviously not coplanar anymore, right? A and B lie on a plane, while C and D lie on a different plane (actually both pairs of points are intersected by infinitely many planes, but let's ignore that). Then it's possible for two pairs of points to lie on two different planes and not be coplanar. |
That makes sense, as you said if A,B,C and D were coplanar and you moved point D so it is no longer on the same plane as A,B and C, in relation to two lines are parallel , A and B are coplanar and collinear(all two points will be as you said), C and D are also both coplanar and collinear but a plane (flat 2d) cannot go through all points A,B,C and D with distorting the plane, right?
This can also be said for other points in the same example, this will hold true also for the lines AC and BD. But if we did not move D and left it on the same plane as A,B and C then obviously AB will be coplanar with line CD, same as AC to BD and so on, right?
With all that being said, again let's take the point D and move off the plane that A,B and C lie on. If we take three points A,B(both coplanar) and D(off the plane) then A,B and C are still coplanar as a plane can be places through any 3 points, right?
| Have you done vector algebra yet? It will all make sense when you do. |
I did some manipulation of matrices a few years back in college although an emphasis wasn't placed on them, but overall I would say I haven't really delved too much into vector algebra yet apart from playing around with simple 2d vectors with SDL to move objects(such as move an object in the direction of another object or fire a bullet towards the other object), so things such as vector normalization, magnitude, addition and subtraction etc.
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