What is the Affine Hull in 2D and 3D for Different Numbers of Points?

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An affine combination of points allows for the creation of new points by combining existing ones without the restriction of non-negativity, unlike convex combinations. In 2D, the affine hull of two points is a line, while for three points it forms a plane, and for more than three points, it remains the entire plane if not collinear. In 3D, the affine hull of two points is a line, three points form a plane, and four or more points span the entire space. The discussion emphasizes the distinction between affine and convex hulls, with the former being the smallest affine set containing the points. Understanding these concepts is crucial for grasping the geometry of points in different dimensions.
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An affine combination of k points (x1, x2, ..., xk) is the sum of the form:

b1*x1 + b2*x2 + ... + bk*xk

with b1 + b2 + ... + bk = 1
where the condition that any of the b's do not have to be greater than or equal to zero, as it is required in the convex combination.

In 2D, what is the affine hull of two points? Three points? n>3 points?
In 3D, what is the affine hull of two points? Three points? n>3 points?

The professor said it was easy but I'm still not quite yet grasping the concept.

thanks guys
 
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Well, I can imagine two ways to proceed:

(1) Draw a picture.
(2) Work through algebra.
 
JasonJo said:
An affine combination of k points (x1, x2, ..., xk) is the sum of the form:

b1*x1 + b2*x2 + ... + bk*xk

with b1 + b2 + ... + bk = 1
where the condition that any of the b's do not have to be greater than or equal to zero, as it is required in the convex combination.

In 2D, what is the affine hull of two points? Three points? n>3 points?
In 3D, what is the affine hull of two points? Three points? n>3 points?

The professor said it was easy but I'm still not quite yet grasping the concept.

thanks guys

With a convex combination, such at you bi must be between 0 and 1, you get the area between the points.
What is the convex hull of two points? The convex hull of 3 points?

Now extend that outside the points, allowing the bi to be negative or greater than 1. Your professor was right- it is easy.
 
eh wow, i really feel like a dummy.

in 2D i got:
2 points - the affine hull is the line connecting the two points
3 points - the affine hull is an infinite parallelogram or plane through the 3 ponts
n>3 - I'm *guessing* it's the hyperplane that passes through all n points?

in 3D i got:
2 points - the line that passes through both points
3 points - no idea (i would guess the plane)
4 points - no idea ( i would guess a sphere)
n>4 points - no idea (no clue at all)
 
n>3 - I'm *guessing* it's the hyperplane that passes through all n points?
There are hyperplanes in 2-D??
 
yeah hyperplanes in 2D are called lines =] j/k but really ithink they are(isn't the def'n of a hyperplane as something that divides the region in 2 halvesn>3 points in 2D with the given non-convex criteria is the plane isn't it...that is if they are not all colinear

3 points in 3D is easy...think vectors draw them out and then think what you can get from those vectorpick up o'rourke computational geometry book in there I'm sure he has the proofs in there.
 
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yeah hyperplanes in 2D are called lines =] j/k but really ithink they are(isn't the def'n of a hyperplane as something that divides the region in 2 halves
You're right -- but I figured quibbling over the details would disrupt the spirit of my interjection!
 
neurocomp2003 said:
yeah hyperplanes in 2D are called lines =] j/k but really ithink they are(isn't the def'n of a hyperplane as something that divides the region in 2 halves


n>3 points in 2D with the given non-convex criteria is the plane isn't it...that is if they are not all colinear

3 points in 3D is easy...think vectors draw them out and then think what you can get from those vector


pick up o'rourke computational geometry book in there I'm sure he has the proofs in there.

that'sthe book I'm using right now, it's good but it's really advanced for me.
 
In 2D the affine hull of 3 or more points is the entire plane.

In 3D the affine hull of 4 or more points is the entire space.
 
  • #10
In 2D the affine hull of 3 or more points is the entire plane.
I feel I should point out, since it hasn't been mentioned yet, that this is for three generic points -- there are special cases where this would fail.
 
  • #11
also assuming nonconvex right? or rather the restriction stated in the first post?
 
  • #12
Oh I just noticed you did mention this exception.

Yah, I said the affine hull (as opposed to the convex hull).

Jason: if you want some intuition, the point of these things is:

The convex hull of S is the smallest convex set containing S.
The affine hull of S is the smallest affine set containing S.
 

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