Point vs vector confusion.
What's the difference between point and vector? I've been reading things on the internet but still confuse. Some guy on gamedev.net forum archive said that it depends on the coordinate system. In that case let's assume I only know cartesian coordinate system in 2D space.
A point is represented in (x,y) what about a vector? I always thought that a polar coordinate is a vector but now I think I'm wrong.
PS; If you browse at google or wikipedia, you get a bunch of stuff.
http://en.wikipedia.org/wiki/Vector
http://en.wikipedia.org/wiki/Vector_%28mathematics_and_physics%29
but my instinct tells me that Euclidean vector is what I'm look for. Maybe I just need to eat then sleep.
A point is represented in (x,y) what about a vector? I always thought that a polar coordinate is a vector but now I think I'm wrong.
PS; If you browse at google or wikipedia, you get a bunch of stuff.
http://en.wikipedia.org/wiki/Vector
http://en.wikipedia.org/wiki/Vector_%28mathematics_and_physics%29
but my instinct tells me that Euclidean vector is what I'm look for. Maybe I just need to eat then sleep.
A point is a geometric entity, a vector is an algebraic entity ( a single row or column matrix )
You can represent points as vectors.
Notice that there are several types of vectors, in physics the starting point of a vector may not be on the origin.
You can represent points as vectors.
Notice that there are several types of vectors, in physics the starting point of a vector may not be on the origin.
Vector is a difference of two points. A->B = B - A. Every point is a vector. O->A = A - O = A, where O is the origin with coordinates (0;0). It is most comfortable to represent a vector as it's x, y, etc. components, but you can also represent it as an angle and length.
First of all, the difference depends on the context. In software engineering, the difference is usually ignored. For example, in my computational geometry course we ignored the difference.
In math though the stuff is perceived differently. You have sets of objects that have additional structure (relations, operations, etc) that you augment, until you express enough detail to solve your problems with quantification. I am a bit rusty, but here is what I seem to recollect.
First, you've got points, for which you can test various relations - identity, collinearity, coplanarity - basically incidences. No point is more central than another, no point is more distant or less distant to any other.
Then you've got vector spaces. A vector space captures the notion of those incidences, but also provides ratio between vectors, and sum of vectors. Affine space is the set of points from the previous paragraph, and a corresponding set of vectors. There is a function that maps each pair of points to a vector, like hamsterman said. So, the vectors capture the relationship between two points, and provide additional structure in the form of ratio between two such relationships. This way you can say that a certain point is closer or further to another point on some line.
Also, vectors can be added together to form other vectors. This means that you can have coordinate systems. You can choose one point to be the center of the universe (people believed it was Rome), and some vectors to be in a sense unitary, and express the remaining points with numerical coordinates.
Then you've got Euclidean spaces. Those are vector spaces with dot product. Essentially, this means that you also have the cosine of the angle between vectors, and norms, distances. But most importantly - you know which vectors are perpendicular and which are unitary in length. This allows you to use orthonormal coordinate systems. Euclidean spaces for affine spaces are called Euclidean point spaces, at least in the part of the world that was under Russian influence for a period of time. The dot product of the coordinates is then the dot product of the vectors. This means that you can explore (to some extent) the angles between the line segments in your objects using the coordinates of their end points. This is something that does not hold for coordinates in affine spaces.
And at last, you've got oriented Euclidean space. This simply means that some angles are positive and some negative. This allows you to take the sine of the angles for example. The cosine is agnostic to direction, but the sine is not. Now you can extract quantities such as the perp dot product from the coordinates in order to fully explore the angles (with direction).
All this may seem like a lot of nonsense. After all, don't we all work with numbers. And the numbers always have dot product, and they are all vectors, and vectors in the end of the day end up being points. So, who needs this gradation.
The answer depends. For example, to put the objects in ordered associative container you need to have strict weak ordering. In other words - to have comparison operator that orders them in a certain way. But there may be no natural way to perform such comparison for this class of objects. Instead, you can still perform deep member-wise or bit-pattern comparison, which makes no sense in the real world, but still orders the objects and allows you to use the container.
The same is true for points. In the computer they are always expressed with numbers, and they always have dot products. This allows some of the algorithms to run in all cases. But in the real world, the angles that those coordinates provide may make no sense. So, for example, rotation may be something idiotic.
Take for example the model of dynamical system, that involves quantities such as say temperature and concentration of chemicals over time. The model of the dynamical system takes one point, which is sample of those quantities, and produces another point, which is sample of those quantities in the following moment. Still, can you take the inner product between the vectors of those points and extract some "angle"? Would a rotation of the point correspond to something meaningful? Finally no. For intermediate computations it may be necessary, but it will make little sense in the final result, except in plots and such.
Now, if the coordinates describe conventional geometry or quantities of the same kind (not mixed temperature and concentration, but only temperature, or only concentration), then angles, orientation and all the stuff can be derived from the numbers. And this is what the above categories describe - which property can be encoded in the numerical representation and which is just numerical nonsense.
Regards
In math though the stuff is perceived differently. You have sets of objects that have additional structure (relations, operations, etc) that you augment, until you express enough detail to solve your problems with quantification. I am a bit rusty, but here is what I seem to recollect.
First, you've got points, for which you can test various relations - identity, collinearity, coplanarity - basically incidences. No point is more central than another, no point is more distant or less distant to any other.
Then you've got vector spaces. A vector space captures the notion of those incidences, but also provides ratio between vectors, and sum of vectors. Affine space is the set of points from the previous paragraph, and a corresponding set of vectors. There is a function that maps each pair of points to a vector, like hamsterman said. So, the vectors capture the relationship between two points, and provide additional structure in the form of ratio between two such relationships. This way you can say that a certain point is closer or further to another point on some line.
Also, vectors can be added together to form other vectors. This means that you can have coordinate systems. You can choose one point to be the center of the universe (people believed it was Rome), and some vectors to be in a sense unitary, and express the remaining points with numerical coordinates.
Then you've got Euclidean spaces. Those are vector spaces with dot product. Essentially, this means that you also have the cosine of the angle between vectors, and norms, distances. But most importantly - you know which vectors are perpendicular and which are unitary in length. This allows you to use orthonormal coordinate systems. Euclidean spaces for affine spaces are called Euclidean point spaces, at least in the part of the world that was under Russian influence for a period of time. The dot product of the coordinates is then the dot product of the vectors. This means that you can explore (to some extent) the angles between the line segments in your objects using the coordinates of their end points. This is something that does not hold for coordinates in affine spaces.
And at last, you've got oriented Euclidean space. This simply means that some angles are positive and some negative. This allows you to take the sine of the angles for example. The cosine is agnostic to direction, but the sine is not. Now you can extract quantities such as the perp dot product from the coordinates in order to fully explore the angles (with direction).
All this may seem like a lot of nonsense. After all, don't we all work with numbers. And the numbers always have dot product, and they are all vectors, and vectors in the end of the day end up being points. So, who needs this gradation.
The answer depends. For example, to put the objects in ordered associative container you need to have strict weak ordering. In other words - to have comparison operator that orders them in a certain way. But there may be no natural way to perform such comparison for this class of objects. Instead, you can still perform deep member-wise or bit-pattern comparison, which makes no sense in the real world, but still orders the objects and allows you to use the container.
The same is true for points. In the computer they are always expressed with numbers, and they always have dot products. This allows some of the algorithms to run in all cases. But in the real world, the angles that those coordinates provide may make no sense. So, for example, rotation may be something idiotic.
Take for example the model of dynamical system, that involves quantities such as say temperature and concentration of chemicals over time. The model of the dynamical system takes one point, which is sample of those quantities, and produces another point, which is sample of those quantities in the following moment. Still, can you take the inner product between the vectors of those points and extract some "angle"? Would a rotation of the point correspond to something meaningful? Finally no. For intermediate computations it may be necessary, but it will make little sense in the final result, except in plots and such.
Now, if the coordinates describe conventional geometry or quantities of the same kind (not mixed temperature and concentration, but only temperature, or only concentration), then angles, orientation and all the stuff can be derived from the numbers. And this is what the above categories describe - which property can be encoded in the numerical representation and which is just numerical nonsense.
Regards
In simple terms, a vector has direction and magnitude. A point (a certain distance from an origin) is often used to describe a vector.
Last edited on
Thanks to all of you, I think I understand it now.
Conclusion:
Vectors are most often represented as a point relative to the origin but vectors may not start the origin. Knowing that vectors are often assume to be relative to the origin, I can also represent a vector in polar coordinates (angle and magnitude).
Is that right?
| Bazzy wrote: |
|---|
| Notice that there are several types of vectors, in physics the starting point of a vector may not be on the origin. |
| simeonz wrote: |
|---|
| All this may seem like a lot of nonsense. After all, don't we all work with numbers. And the numbers always have dot product, and they are all vectors, and vectors in the end of the day end up being points. So, who needs this gradation. |
| hamsterman wrote: |
|---|
| It is most comfortable to represent a vector as it's x, y, etc. components, but you can also represent it as an angle and length. |
| moorecm wrote: |
|---|
| A point (a certain distance from an origin) is often used to describe a vector. |
Conclusion:
Vectors are most often represented as a point relative to the origin but vectors may not start the origin. Knowing that vectors are often assume to be relative to the origin, I can also represent a vector in polar coordinates (angle and magnitude).
Is that right?
According to what I have been taught, points just do not have coordinates. Neither polar coordinates, nor Cartesian coordinates, nothing. A point is a freestanding carefree entity that exists in the endless universe/multiverse. It can form lines and planes with other points and those lines and planes can form intersections in the form of other lines, planes and points, but that's all. No coordinates, no computing, nothing.
Vectors are this virtual reality thing. They are not n-tuples of numbers as you may think. They are math abstraction, like numbers themselves. Vectors in finite dimensional vector spaces can be expressed as weighted sum of other vectors, and this allows you to use the weights as coordinates. But you have to fix some vectors as being the "basis" and express the others from those.
Affine spaces deal with the problem of expressing the difference between two points as a vector. Then, the points can be mapped into vectors after choosing one point as the origin. After that, if you also choose few vectors as the basis, you can map the vector space into n-tuples of numbers, i.e. coordinates. The combination of origin point and vector basis is called coordinate system. The vector space acts as a proxy between the point space and the coordinates, so to speak.
What I was rambling about in my previous post is that not all coordinate systems are orthonormal and with standard orientation. Consequently, the coordinates may have limited interpretation. But this is math stuff, so I won't go into it again.
In summary, a point has no coordinates, but can be tested for various geometrical incidences with other points. However, to make the stuff computable, you need coordinates. Vectors provide the means to get the coordinates (although they are not coordinates), and affine spaces provide the means to map points into vectors. The former requires the choice of basis, and the latter requires the choice of origin.
Now, how does this stuff relate to say, geodetic coordinates (latitude, longitude, height), I don't know. I mean, the formulas you can use are easily accessible on the web. But how vectors interact with non-Euclidean geometries as they are called, I am not aware yet.
Regards
Vectors are this virtual reality thing. They are not n-tuples of numbers as you may think. They are math abstraction, like numbers themselves. Vectors in finite dimensional vector spaces can be expressed as weighted sum of other vectors, and this allows you to use the weights as coordinates. But you have to fix some vectors as being the "basis" and express the others from those.
Affine spaces deal with the problem of expressing the difference between two points as a vector. Then, the points can be mapped into vectors after choosing one point as the origin. After that, if you also choose few vectors as the basis, you can map the vector space into n-tuples of numbers, i.e. coordinates. The combination of origin point and vector basis is called coordinate system. The vector space acts as a proxy between the point space and the coordinates, so to speak.
What I was rambling about in my previous post is that not all coordinate systems are orthonormal and with standard orientation. Consequently, the coordinates may have limited interpretation. But this is math stuff, so I won't go into it again.
In summary, a point has no coordinates, but can be tested for various geometrical incidences with other points. However, to make the stuff computable, you need coordinates. Vectors provide the means to get the coordinates (although they are not coordinates), and affine spaces provide the means to map points into vectors. The former requires the choice of basis, and the latter requires the choice of origin.
Now, how does this stuff relate to say, geodetic coordinates (latitude, longitude, height), I don't know. I mean, the formulas you can use are easily accessible on the web. But how vectors interact with non-Euclidean geometries as they are called, I am not aware yet.
Regards
| points just do not have coordinates |
As far as I am aware, it is just as permissible to introduce a basis and write points in coordinates as vectors (although strictly speaking the bases for the vectors and the points are different though related).
| how vectors interact with non-Euclidean geometries as they are called, I am not aware yet |
If by this you mean genuinely curved spaces (as in general relativity) then the definition of vectors is profoundly different to any previous definition of vectors in algebra and so forth.
If by this you merely mean the polar coordinate representation of Euclidean space, there may still be a problem. If we're talking about the distance between two points, this is easy in Cartesian coordinates. However, in Polar coordinates, the basis changes round the sphere so a so called "straight line" (geodesic) between the points is not so easy to calculate the length of. You can find the formula on Wikipedia here http://en.wikipedia.org/wiki/Polar_coordinates#Integral_calculus_.28Arc_Length.29.
There is however, an integration involved. Generally integration is needed to calculate length. Euclidean (and Minkowskian) geometry is the exception to this where one can find the length of straight lines alone without integration.
Sorry if this is poorly explained. Hope it helps.
Thanks Xander314. Interesting post.
I would guess it is just a matter of nomenclature.
My analytic geometry course treated the stuff in decreasing levels of abstraction. And I admit, I kind of liked that. Vectors had no intrinsic coordinates. Some fundamental basis (and coordinate system) is usually assumed to exist before the problem is approached and I believe that's why engineers treat points and vectors and n-tuples as equivalent. Coordinate systems are given treatment after the fact.
I was taught on the other hand, that no basis is anymore fundamental than another. That depends on the mathematical model. Vectors require basis to be turned into coordinates, and points require origin to be turned into vectors. And you can not turn two points into a vector just like that. You need affine space to supply mapping between points and vectors, which is also in a sense arbitrary. But this is an extremely purist school of thought, which admittedly is sometimes overdoing things.
Hmm. I am not sure what I was thinking, but I am definitely lacking in this department. If I fix the radial coordinate, I imagined I will get some curved space. A sphere. I believe they call such continuous surfaces on which you can try to recreate some analogy of Euclidean geometry manifolds. Or am wrong? I admit I have only skimmed over the Wikipedia article. I am just blabbering here.
Anyway, after reading a bit about them, I frequently think to myself that people assume they are walking on a plane, but it is actually a curved surface which they locally approximate with a plane. Those problems in physics from school are actually very different from this perspective. "How much time the train will travel from point A to point B?" The answer may end up being a multitude of answers related by a period.
My thought was, I don't know how you define bases in such space. What vectors would you use? For a sphere or ellipsoid you may derive lengths and areas and stuff by converting the geometry to Cartesian coordinates and integration as you say. But do you have vectors anymore? Do you have the same definition of bases? Do you adjust the notion of collinearity? I guess you need to adjust something, but it's way over my head.
Regards
| As far as I am aware, it is just as permissible to introduce a basis and write points in coordinates as vectors (although strictly speaking the bases for the vectors and the points are different though related). |
I would guess it is just a matter of nomenclature.
My analytic geometry course treated the stuff in decreasing levels of abstraction. And I admit, I kind of liked that. Vectors had no intrinsic coordinates. Some fundamental basis (and coordinate system) is usually assumed to exist before the problem is approached and I believe that's why engineers treat points and vectors and n-tuples as equivalent. Coordinate systems are given treatment after the fact.
I was taught on the other hand, that no basis is anymore fundamental than another. That depends on the mathematical model. Vectors require basis to be turned into coordinates, and points require origin to be turned into vectors. And you can not turn two points into a vector just like that. You need affine space to supply mapping between points and vectors, which is also in a sense arbitrary. But this is an extremely purist school of thought, which admittedly is sometimes overdoing things.
| If by this you mean genuinely curved spaces (as in general relativity) then the definition of vectors is profoundly different to any previous definition of vectors in algebra and so forth. If by this you merely mean the polar coordinate representation of Euclidean space, ... |
Hmm. I am not sure what I was thinking, but I am definitely lacking in this department. If I fix the radial coordinate, I imagined I will get some curved space. A sphere. I believe they call such continuous surfaces on which you can try to recreate some analogy of Euclidean geometry manifolds. Or am wrong? I admit I have only skimmed over the Wikipedia article. I am just blabbering here.
Anyway, after reading a bit about them, I frequently think to myself that people assume they are walking on a plane, but it is actually a curved surface which they locally approximate with a plane. Those problems in physics from school are actually very different from this perspective. "How much time the train will travel from point A to point B?" The answer may end up being a multitude of answers related by a period.
My thought was, I don't know how you define bases in such space. What vectors would you use? For a sphere or ellipsoid you may derive lengths and areas and stuff by converting the geometry to Cartesian coordinates and integration as you say. But do you have vectors anymore? Do you have the same definition of bases? Do you adjust the notion of collinearity? I guess you need to adjust something, but it's way over my head.
Regards
| Conclusion: Vectors are most often represented as a point relative to the origin but vectors may not start the origin. Knowing that vectors are often assume to be relative to the origin, I can also represent a vector in polar coordinates (angle and magnitude). Is that right? |
That is my understanding, yes. They're not as difficult as they're being made out to be in here. :P
@moorecm
It's a question of terminology.
You address the definition of vector as directed line segment. Like here:
http://en.wikipedia.org/wiki/Euclidean_vector
But even then, you have to discriminate between free vectors and bound vectors. Bound vectors are directed line segments and free vectors are equivalence classes for the free vectors that have the same length and direction. Consequently, the bound vectors have starting point, while the free vectors do not have one per se.
Also, vectors are not necessarily used to describe entities in geometry. For example, they can represent states of a system. This is the more general interpretation I was addressing. Like here:
http://en.wikipedia.org/wiki/Vector_space
It's a question of terminology.
You address the definition of vector as directed line segment. Like here:
http://en.wikipedia.org/wiki/Euclidean_vector
But even then, you have to discriminate between free vectors and bound vectors. Bound vectors are directed line segments and free vectors are equivalence classes for the free vectors that have the same length and direction. Consequently, the bound vectors have starting point, while the free vectors do not have one per se.
Also, vectors are not necessarily used to describe entities in geometry. For example, they can represent states of a system. This is the more general interpretation I was addressing. Like here:
http://en.wikipedia.org/wiki/Vector_space
closed account (z05DSL3A)
[sarcasm]
It is all a matter of perspective, as is much in reality. But what do I mean when I say perspective? Do I mean it Metaphorically (in relation to cognitive topics), Mathematically,... Should I over qualify everything said so as not to cause a big debate on the meaning of the smallest word.
[/sarcasm]
It is all a matter of perspective, as is much in reality. But what do I mean when I say perspective? Do I mean it Metaphorically (in relation to cognitive topics), Mathematically,... Should I over qualify everything said so as not to cause a big debate on the meaning of the smallest word.
[/sarcasm]
@Grey Wolf
Your quote is quite realistic. A computer program is pedantic and rigorous description. But the programming language is described in human terms. That can be problematic sometimes.
Your quote is quite realistic. A computer program is pedantic and rigorous description. But the programming language is described in human terms. That can be problematic sometimes.
Still interesting answers you have here. It would be nice if someone would provide a link for a visual aid. It may be a video or java applet or flash content.
Yeah I think I may have contributed to over complicating this thread ;)
Technically they are even more difficult than they are being made out here :P
However, going back to the OP's question, in flat Euclidean space, once we have introduced coordinates (and hence an origin), vectors and points can effectively be considered the same as far as some simple computations are concerned.
However high you go there seems always to be another level of abstraction, but in most contexts none of this should matter...
@simeonz, fixing the radial coordinate of a spherical coordinate system does result in a sphere, which can be seen to be curved. However, one can also used all three spherical coordinates to represent Euclidean space, which is not curved. So my point earlier was the distinction between representing a flat space like Euclidean space with 'strange' coordinates, and a genuinely curved coordinate system like a sphere. However, this probably isn't relevant to this discussion...
| They're not as difficult as they're being made out to be in here |
Technically they are even more difficult than they are being made out here :P
However, going back to the OP's question, in flat Euclidean space, once we have introduced coordinates (and hence an origin), vectors and points can effectively be considered the same as far as some simple computations are concerned.
However high you go there seems always to be another level of abstraction, but in most contexts none of this should matter...
@simeonz, fixing the radial coordinate of a spherical coordinate system does result in a sphere, which can be seen to be curved. However, one can also used all three spherical coordinates to represent Euclidean space, which is not curved. So my point earlier was the distinction between representing a flat space like Euclidean space with 'strange' coordinates, and a genuinely curved coordinate system like a sphere. However, this probably isn't relevant to this discussion...
closed account (z05DSL3A)
A Point is a 0-dimensional mathematical object that can be described in n-diminutional space with n-coordinated. So really points aren't anything, they are more of a location; even Euclid defined them as "that which has no part"[1].
A Vector isdefined as described by
1, A mathematical entity that has both magnitude (which can be zero) and direction.
2, An element in vector space.
The second one is posable less helpful without going into what vector space is, which will go into vector addition....
Without geometry, life is pointless.
[1] http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html
A Vector is
1, A mathematical entity that has both magnitude (which can be zero) and direction.
2, An element in vector space.
The second one is posable less helpful without going into what vector space is, which will go into vector addition....
Without geometry, life is pointless.
[1] http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html
Last edited on
Topic archived. No new replies allowed.