Understanding Affine Geometry and Space: Origins, Definitions, and Applications

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In summary: Applicable Differential Geometry by Crampin and PiraniTensor Geometry by Dodson and Poston (A Springer yellow and white.)These two references give equivalent, but different-looking, definitions of affine spaces.A Course in Mathematics for Students of Physics - Bamberg & SternbergApplied Differential Geometry - BurkeA Course in Mathematics for Students of Physics - Bamberg & SternbergSince no manifold has an implicit origin then does this imply that all manifolds are affine spaces?No, an affine space can be obtained by forgetting the origin of a vector space, or by adding an aff
  • #1
pmb_phy
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Many times when I see the term Affine space used, the person using it seems to define it as a space with no origin or something akin to that. Its hard to find a definition of this term except the one that says an affine space is a space with is affinely connected where affinely connected is defined prier to this definition. MTW (page 242) use the the term affine geometry as that branch of mathematics which adds geodesics, parallel transport and curvature (shape) to a manifold. They then go on to say that that branch of mathematics which adds a metric is called Riemannian geometry.

I never could understand why people would think that an affine space is a space without an origin since the manifold on which the space is defined is simply a collection of points, anyone of which could be defined to be the origin of the manifold.

Could those who use the term "affine space" to mean a space without an origin please give me a reference to the source in which you found it defined like this? Thank you.

Best wishes

Pete
 
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  • #2
pmb_phy said:
Could those who use the term "affine space" to mean a space without an origin please give me a reference to the source in which you found it defined like this? Thank you.

Applicable Differential Geometry by Crampin and Pirani

Tensor Geometry by Dodson and Poston (A Springer yellow and white.)

These two references give equivalent, but different-looking, definitions of affine spaces.
 
  • #3
A Course in Mathematics for Students of Physics - Bamberg & Sternberg
Applied Differential Geometry - Burke
 
  • #4
robphy said:
A Course in Mathematics for Students of Physics - Bamberg & Sternberg
Applied Differential Geometry - Burke
Since no manifold has an implicit origin then does this imply that all manifolds are affine spaces?

Pete
 
  • #5
to me, an affine space is a flat space in which no specific point has been chosen as origin. hence a curved manifold with no chosen origin is not affine, and a flat space like R^n in which (0,...,0) is the obvious preferred origin has more structure than an affine space.

so an affine space can be obtained by forgetting the origin of a vector space, or by adding an affine structure to a manifold, but in my opinion only to certain manifolds which are in some sense flattenable.

so people starting from a vector space think of forgetting some structure, whereas people in differential geometry starting from a manifold think of adding some structure, to get an affine structure.

oh and people in algebraic geometry are perverse, and speak of k^n even with its origin, as "affine space", since they are contrasting it with projective space.i am an amateur here on the differential geometry side of things.
 
  • #6
i think affine space in diff geom, may be a space with a certain affine group of symmetries.

a space with a group acting may be called a homogeneous space, if no origin is chosen. after an origin is chosen it becoems a group. so one has the same distinction between groups and groups without origians.

if the groupo concerned is an affine group, then ropesumabkly one has affine space.

spaces with no origin but also with no group action are then not homogeneous and certainly then not affine.

i hoope this is approximately right.

so there is a hierarchy of spaces: vector spaces, which are groups of translations. affine spaces which are spoaces on which a vector space acts faithfully, and hence are vector spaces withut an origin chosen. lie groups, which are more general groups with a topological space structure. homogeneous spaces which are manifolds on which a lie group acts.

and general manifoklds on which no group acts.

but it seems to me an "affine structure" in diff geom is a bit more general than what i described as an affine space. i never quite grasped it.
 
  • #7
Thank you all for your helpful response. I would like to ask those who have a text to quote the text where it defines affine space. Perhaps this will help me understand this better. Thanks.

Pete
 
  • #9
https://www.amazon.com/gp/sitbv3/reader/104-1255686-8703936?ie=UTF8&p=S00K&asin=0521406498&tag=pfamazon01-20 (see pg 1-6)

https://www.amazon.com/gp/sitbv3/reader/104-1255686-8703936?ie=UTF8&p=S00U&asin=0521269296&tag=pfamazon01-20 (see pg 11-13)
 
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  • #10
pmb_phy said:
I never could understand why people would think that an affine space is a space without an origin since the manifold on which the space is defined is simply a collection of points, anyone of which could be defined to be the origin of the manifold.
The fact that any point 'could be an origin' is precisely the answer to your own question.

R^n can be viewed as a vector space, and the maps of vector spaces send (0,..,0) to 0. R^n can also be viewed as affine n-space, and this means we no longer treat (0,..,0) as special, and we care about properties that are preserved by maps sending x to x+v for some v.

You see, it is the *maps* of the object that are important, not the elements of the object. I mean, I could put any structure I like on an abstract set of points. But that doesn't mean I should.

If we just take a collection of points, like those on a manifold, then if we make it a vector space with X as an origin, and then as a vector space with Y as an origin, the two vector spaces are not equal. There may be a map from one to the other, but that map cannot be seen as a linear endomorphism of 'the' space with X as the origin to itself because it would have to send X to X. But as affine spaces, it is the same affine space - there is an affine map sending X to Y.
 
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  • #11
robphy said:
https://www.amazon.com/gp/sitbv3/reader/104-1255686-8703936?ie=UTF8&p=S00K&asin=0521406498&tag=pfamazon01-20 (see pg 1-6)

https://www.amazon.com/gp/sitbv3/reader/104-1255686-8703936?ie=UTF8&p=S00U&asin=0521269296&tag=pfamazon01-20 (see pg 11-13)
Thanks Rob

That kin of makes sense now. If I were able to afford only one of these books at a time then which one do you recommend I buy first?

Thanks

Pete
 
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  • #12
matt grime said:
The fact that any point 'could be an origin' is precisely the answer to your own question. etc
So let me get this straight: Spacetime is an affine space, Minkowski spacetime is not. Correct?

Pete
 
  • #13
mathwonk said:
i think affine space in diff geom, may be a space with a certain affine group of symmetries.

a space with a group acting may be called a homogeneous space, if no origin is chosen. after an origin is chosen it becoems a group. so one has the same distinction between groups and groups without origians.

if the groupo concerned is an affine group, then ropesumabkly one has affine space.

spaces with no origin but also with no group action are then not homogeneous and certainly then not affine.

i hoope this is approximately right.

so there is a hierarchy of spaces: vector spaces, which are groups of translations. affine spaces which are spoaces on which a vector space acts faithfully, and hence are vector spaces withut an origin chosen. lie groups, which are more general groups with a topological space structure. homogeneous spaces which are manifolds on which a lie group acts.

and general manifoklds on which no group acts.

but it seems to me an "affine structure" in diff geom is a bit more general than what i described as an affine space. i never quite grasped it.
From what I'm reading right now from those references kindly provided by Rob (thanks Rob!) it appears that you're somewhat correct. I can't say you're exactly correct because I have yet to absorb all your post and the textbook quotes above.

But I'm pretty sure that I'll be picking up one of those textbooks. :biggrin:

Pete
 
  • #14
pmb_phy said:
Thanks Rob

That kin of makes sense now. If I were able to afford only one of these books at a time then which one do you recommend I buy first?

Thanks

Pete

It really depends on just what you want to learn.

George Jones's references are good references as well.
Each reference has its own strengths.

In my experience, among these four, ...

Burke is probably the most readable... for physical intuition accompanied by a unique viewpoint of related mathematical structure... but probably not for doing calculations.

Bamberg & Sternberg is fascinating reading, developing some topics in introductory physics with rather deep (and possible obscure at first glance) mathematical formulations [like treating circuit theory with algebraic topology, optics with symplectic geometry, electromagnetism and thermodynamics with differential forms]. (Burke also treats some of these topics, with less calculation.)

Crampin and Pirani was helpful to me in understanding the relationship between the tensor notations of the mathematician and the physicist... helping me do and better understand calculations. It was also helpful to me when I was first introduced to geometrical formulations of mechanics. (In addition, I've been trying to understand some papers by Pirani et al on the interpretation of various curvature tensors and hierarchy of mathematical structures in GR.)

Dodson and Poston was helpful in understanding and visualizing structures involving tensors and differential geometry. Unlike the other books, this book focuses more on General Relativity.

It might be best to flip through these books [assuming you have access to a good bookstore or university library... or try using http://books.google.com" (as I did) to "look inside"] ... or else read the reviews on Amazon and elsewhere (like, say, AJP or other journals). Here are some non-Amazon reviews that came up on a google search [including "review" with the author names]:
http://blms.oxfordjournals.org/cgi/reprint/20/2/183-a.pdf (Crampin & Pirani)
http://www.ucolick.org/~burke/forms/bamberg.html (Burke's review of Bamberg & Sternberg)

You might find some author homepages helpful in deciding:
http://www.ucolick.org/~burke/home.html
http://www.math.harvard.edu/~shlomo/
http://www.maths.manchester.ac.uk/~kd/homepage/coursenotes.html
 
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  • #15
pmb_phy said:
So let me get this straight: Spacetime is an affine space, Minkowski spacetime is not. Correct?

Pete

I would say that Galilean and Minkowskian Spacetimes are affine spaces.
I would say that time in Galilean physics is an affine space.
I would say that the Euclidean plane is an affine space.

If I distinguish an event to call my origin in Galilean or Minkowskian Spacetime, then I would say that "the set of spacetime displacements from that event" would form a vector space.
I would say that the space of 4-momenta at an event in spacetime is a vector space.
(I think it is correct to regard a vector-space is a particular type of affine-space.)
 
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  • #16
pmb_phy said:
So let me get this straight: Spacetime is an affine space, Minkowski spacetime is not. Correct?

Pete

I have no idea, nor inclination to bother checking. Vector spaces have vector space endomorhisms, affine spaces have affine endomorphisms. It is the maps that matter, not the underlying set of points. R^n is both an affine space and a vector space depending on the context.
 
  • #17
robphy said:
I would say that Galilean and Minkowskian Spacetimes are affine spaces.
I would say that time in Galilean physics is an affine space.
I would say that the Euclidean plane is an affine space.

If I distinguish an event to call my origin in Galilean or Minkowskian Spacetime, then I would say that "the set of spacetime displacements from that event" would form a vector space.
I would say that the space of 4-momenta at an event in spacetime is a vector space.
(I think it is correct to regard a vector-space is a particular type of affine-space.)
As I read Schutz's textGeometrical Methods of Mathematical Physics I read on page71
A manifold Mwith a metric g is called Minkowski spacetime only if there exists a single coordinate system covering alll of M in which the components have the form n_uv.
With this in mind, is Minkowski spacetime an affine space, at least according to Schutz (who does not define the term)?

Pete
 
  • #18
i am a beginner here but i have the following opinion/questions:

minkowski space is an affine 4-space but is also equipped with a metric. I do not think this metric can detract from it being an afFINE SPACE, but am not sure.

i.e. to me minkowski space should be considered together with its metric, so its transformations should be not just aFFINE OneS BUT METRIC prESERVING ONES.
 
  • #19
mathwonk said:
i am a beginner here but i have the following opinion/questions:

minkowski space is an affine 4-space but is also equipped with a metric. I do not think this metric can detract from it being an afFINE SPACE, but am not sure.

i.e. to me minkowski space should be considered together with its metric, so its transformations should be not just aFFINE OneS BUT METRIC prESERVING ONES.
A metric associated with a space does not imply the existence of a point chosen to be origin. However once you start talking about coordinate transformations then it appears to me that you've adopted a coordinate system and perhaps an origin. This part I'm not sure about. Is there a origin in spherical coordinates? No.

Pete
 
  • #20
pmb_phy said:
As I read Schutz's textGeometrical Methods of Mathematical Physics I read on page71

With this in mind, is Minkowski spacetime an affine space, at least according to Schutz (who does not define the term)?

Pete
Hi Pete,
Sure it is an affine space.
Schutz simply means that we don't need different coordinate patches to cover everything if it is a Minkowski space, since it is all flat.
 
  • #21
pmb_phy said:
A metric associated with a space does not imply the existence of a point chosen to be origin. However once you start talking about coordinate transformations then it appears to me that you've adopted a coordinate system and perhaps an origin. This part I'm not sure about. Is there a origin in spherical coordinates? No.

Pete

Maybe there needs to be some clarification of terms.
Minkowski spacetime [which has no distinguished event to declare as the origin] is an affine space.
However, Minkowski spacetime has a Minkowski-metric tensor field, which assigns a Minkowski-metric tensor to the tangent vector-space associated with each point.
 
  • #22
pmb_phy said:
A metric associated with a space does not imply the existence of a point chosen to be origin. However once you start talking about coordinate transformations then it appears to me that you've adopted a coordinate system and perhaps an origin. This part I'm not sure about. Is there a origin in spherical coordinates? No.

Pete
Mathwonk was going in an unrelated direction -- he's just pointing out that "Minkowski space is affine" is too weak of a statement, because that forgets the metric tensor structure.
 
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  • #23
MeJennifer said:
Hi Pete,
Sure it is an affine space.
Schutz simply means that we don't need different coordinate patches to cover everything if it is a Minkowski space, since it is all flat.
I guess I can take that with Schutz to mean that while certain coordinate systems must exist it doesn't mean that they have been attached to the spacetime manifold. Is that how you see it?

Rob - The kind of book I'm looking for is that book that will help me with my forevering continuing studies of SR and GR.

Thank you all for your input

Pete
 
  • #24
pmb_phy said:
I guess I can take that with Schutz to mean that while certain coordinate systems must exist it doesn't mean that they have been attached to the spacetime manifold. Is that how you see it?
It means that if you can fully cover a space with only one coordinate patch with diag(-1, 1, 1, 1) it is a Minkowski space.

Contrast that to a curved space where you need more than one chart to fully cover it.

For instance, say you have a green apple that you cut in half and you discard one part. Then you can cover the remaining half with two charts, one representing the disk (the white flat part) using for instance polar coordinates and the other one (the green part) with for instance latitude and longitude coordinates.
 
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  • #25
mathwonk said:
i am a beginner here but i have the following opinion/questions:

minkowski space is an affine 4-space but is also equipped with a metric. I do not think this metric can detract from it being an afFINE SPACE, but am not sure.

i.e. to me minkowski space should be considered together with its metric, so its transformations should be not just aFFINE OneS BUT METRIC prESERVING ONES.

Yes.

The metric preserving affine transformations on Minkowski space are called Poincare transformations; the metric preserving linear transformations are called Lorentz transformations. A Poincare transformation relates two physical observers by some combination of spacetime translations, rotations of axes, and boosts (relative motions).

Unitary, irreducible representations of the Poincare group lead naturally to relativistic wave equations, like, for example, the Dirac equation. An overview of this route is a nice interplay between mathematics and physics.

The state space of an elementary physical system is a space for an irreducible representation of the Poincare group.

In quantum theory, physical systems are represented by states in state space. A change in observer results in a change of perspective that changes the state "seen", so a representation (action) of the Poincare on a quantum state space is needed. If the the physical system is, like an electron, elementary, then there should not be any invariant subspaces. Any inaviant subspace could lay claim to being a more elementary system.

The representation is unitary.

The probabilty that a radioactive element will decay is an intrinsic property of the element and does not depend on choice of obersver. Thus, the representation has to preserve quantum probabilities, so it has to be unitary.

Wave equations.

There is a physically natural way of defining the action of Poincare transformations on state spaces, but the resulting representation are not unitary and irreducible. A relativistic wave equation (e.g., the Dirac equation) projects onto an invariant subspace on which the Poincare transformation are represented by unitary operators.

(Actually, because of quantum phase factors, it's (spinor) representations of the universal cover of the Poincare group that are important.)
 
  • #26
MeJennifer said:
It means that if you can fully cover a space with only one coordinate patch with diag(-1, 1, 1, 1) it is a Minkowski space.

Contrast that to a curved space where you need more than one chart to fully cover it.
This is news to me. Is this a theorem? I see that it might take more than one patch for curved spaces but I never heard that it must have more than one patch.

Pete
 
  • #27
pmb_phy said:
This is news to me. Is this a theorem? I see that it might take more than one patch for curved spaces but I never heard that it must have more than one patch.

Pete

For example, it seems to me that open Friedman-Robertson-Walker universes which expand forever can be covered by one coordinate patch.
 
  • #28
George Jones said:
For example, it seems to me that open Friedman-Robertson-Walker universes which expand forever can be covered by one coordinate patch.
Thanks George. That's what I was looking for, i.e. an example.

If I didn't mention it before then I want to thank you for all of your input into this thread.:smile:

Best wishes

Pete
 
  • #29
MeJennifer said:
It means that if you can fully cover a space with only one coordinate patch with diag(-1, 1, 1, 1) it is a Minkowski space.

you can go even stronger with that: it *is* Minkowski space. Any 4-manifold M with one coordinate patch has a priori a diffeomorphism phi from itself to R^4. R^4 has a well-defined metric of signature (1,3) -- namely, the Minkowski spacetime structure. PUshing the metric backwards via phi, we can then define a metric of signature (1,3) on M so that phi will be an isometry of spacetimes. Which means that M is essentially Minkowski space.

In fact, M is essentially R^4: any structure that you can put on R^4 has a corresponding and equivaletnly acting one on M.
 
  • #30
The organization of geometries a la Klein

Some of the posters here are in my "ignore list", but the following caught my eye:

mathwonk said:
i am a beginner here but i have the following opinion/questions:

minkowski space is an affine 4-space but is also equipped with a metric. I do not think this metric can detract from it being an afFINE SPACE, but am not sure.

i.e. to me minkowski space should be considered together with its metric, so its transformations should be not just aFFINE OneS BUT METRIC prESERVING ONES.

That's exactly correct!

Such varieties of "homogeneous" geometries are all best understood via Kleinian geometry. Then we see clearly how projective < (1,3)-conformal < Lorentzian. (Meaning that by adding suitable structure to a projective space we can obtain a conformal geometry, then a metric geometry.) In particular, Minkowski spacetime is a particular example of a Lorentzian manifold, but it happens to have uniform curvature (zero curvature, even), so it also a homogeneous space, and by carefully "forgetting" some of the geometric structure of Minkowski geometry we can obtain progressively more general notions of geometry (conformal, affine...).

If we want to fit in alternatives such as 4-conformal, or if we want to introduce more notions of geometry, we'll have to consider partial orders, e.g. we also have affine < 4-conformal < Riemannian and affine < areal < Lorentzian.

Needless to say, Klein's key insight was that this partial order is governed by the subgroup relationships of various symmetry groups. Nowadays we'd say that the geometries arise as coset spaces of some group by the stabilizer of some point. (Since these geometries are homogeneous, all points are equivalent.)

Exercise: by excising a hyperplane "at infinity" from a projective space, we obtain something to which we can add just the right structure to make "affine geometry". What is that structure?

See for example Paul B. Yale, Geometry and Symmetry, Dover reprint, 1988.

The common generalization of Kleinian geometry and Riemannian geometry is Cartanian geometry, which allows us to introduce "curved manifolds" whose "isotropy group" is something more interesting than [itex]SO(n)[/itex] (Riemannian manifolds) or [itex]SO(1,n-1)[/itex] (Lorentzian manifolds), such as the unimodular group, or the projective group. The textbook by Richard Sharpe, Differential Geometry, Springer, 1997, offers an introduction to Kleinian and Cartanian geometries (but not the perenially popular physical applications to Weyl's notion of a "gauge theory"; I note however that Weyl and Cartan were both inspired by the then general theory of relativity, and they were both motivated in part by potential applications to "unified field theories"). In general, we will then deal with "connections" having both "curvature" and "torsion", and we can obtain for example notions of geometry in which "curvature" measures departure from de Sitter rather than Minkowski geometry.

(OK, that was a bit of trick question up above, but it's a good exercise nonetheless!)
 
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  • #31
Chris Hillman said:
Some of the posters here are in my "ignore list", ...
I'm sorry but I don't understand why this fact is important enough to be stated in this post. It does not help anyone with understanding what you wrote.

Pete
 
  • #32
The origin of a vector space has a special meaning. It's the (unique) additive identity operator. In other words, I can add the vector zero to any vector and not change it. Addition of vectors has all the nice and expected properties like associative, commutative...

However, the addition of elements of an affine space is not well defined. It makes sense to write v + w, for a pair of vectors. But it makes no sense to write P + Q for a pair of points in an affine space. By analogy, one can drive a mile east, then turn right and drive another mile. But what does it mean to "add" the location of the city of Boston to that of the city of Los Angeles?

To me, saying that an affine space has no origin is saying that there's no additive identity element. An observation that follows directly from the idea that one can't "add" affine points.

(The vector that represents the "difference" between the locations of those cities can be defined. Drive thataway to get from Boston to Los Angeles. But even if you represent points with ordered tuples of numbers and vectors as ordered tuples of numbers, they are not the same kind of mathematical objects. The "drive thataway" vector is not an element of the affine space of points.)
 
  • #33
Affine spaces are characterized by the fact that you have a notion of line, and a notion of parallelism of lines. These are the two main ingredients, and they allow you to take affine linear combinations of points.
 
  • #34
I'm enjoying this thread immensely and learning a lot. Nobody is yet on my ignore list. :smile:
sundried said:
Affine spaces are characterized by the fact that you have a notion of line, and a notion of parallelism of lines. These are the two main ingredients, and they allow you to take affine linear combinations of points.

I believe you are quite on target; parallel lines are the heart of affine geometry. Can you help me understand what you mean by "affine linear combinations of points"?
 
  • #35
With pictures it is easier to grasp what convex combinations are: Have a look at the YouTube video: WildTrig35: Affine geometry and barycentric coords
 
<h2>1. What is affine geometry?</h2><p>Affine geometry is a branch of mathematics that deals with the study of geometric objects and their properties under affine transformations. An affine transformation is a type of transformation that preserves parallelism and ratios of distances between points, but not necessarily angles or lengths.</p><h2>2. What are the origins of affine geometry?</h2><p>Affine geometry has its origins in ancient Greece, with the work of Euclid and his study of geometric figures and their properties. However, it was not until the 19th century that affine geometry was formally developed by mathematicians such as Jean-Victor Poncelet and Julius Plücker.</p><h2>3. What are the key definitions in affine geometry?</h2><p>Some key definitions in affine geometry include affine transformations, affine spaces, and affine maps. An affine transformation is a transformation that preserves parallelism and ratios of distances between points. An affine space is a set of points with a defined notion of parallelism and a set of affine transformations that can be applied to those points. An affine map is a function that preserves affine combinations of points.</p><h2>4. What are the applications of affine geometry?</h2><p>Affine geometry has many applications in fields such as computer graphics, computer vision, and robotics. It is also used in physics and engineering to model physical systems and in economics to study consumer preferences and demand.</p><h2>5. How is affine geometry different from Euclidean geometry?</h2><p>Affine geometry and Euclidean geometry are both branches of mathematics that deal with geometric objects, but they differ in their assumptions and definitions. Euclidean geometry assumes the existence of a metric (distance) and a notion of angle, while affine geometry does not. Additionally, affine transformations only preserve parallelism and ratios of distances, while Euclidean transformations preserve angles and lengths.</p>

1. What is affine geometry?

Affine geometry is a branch of mathematics that deals with the study of geometric objects and their properties under affine transformations. An affine transformation is a type of transformation that preserves parallelism and ratios of distances between points, but not necessarily angles or lengths.

2. What are the origins of affine geometry?

Affine geometry has its origins in ancient Greece, with the work of Euclid and his study of geometric figures and their properties. However, it was not until the 19th century that affine geometry was formally developed by mathematicians such as Jean-Victor Poncelet and Julius Plücker.

3. What are the key definitions in affine geometry?

Some key definitions in affine geometry include affine transformations, affine spaces, and affine maps. An affine transformation is a transformation that preserves parallelism and ratios of distances between points. An affine space is a set of points with a defined notion of parallelism and a set of affine transformations that can be applied to those points. An affine map is a function that preserves affine combinations of points.

4. What are the applications of affine geometry?

Affine geometry has many applications in fields such as computer graphics, computer vision, and robotics. It is also used in physics and engineering to model physical systems and in economics to study consumer preferences and demand.

5. How is affine geometry different from Euclidean geometry?

Affine geometry and Euclidean geometry are both branches of mathematics that deal with geometric objects, but they differ in their assumptions and definitions. Euclidean geometry assumes the existence of a metric (distance) and a notion of angle, while affine geometry does not. Additionally, affine transformations only preserve parallelism and ratios of distances, while Euclidean transformations preserve angles and lengths.

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