# Affine Hull

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

Hurkyl
Staff Emeritus
Gold Member
Well, I can imagine two ways to proceed:

(1) Draw a picture.
(2) Work through algebra.

HallsofIvy
Homework Helper
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)

Hurkyl
Staff Emeritus
Gold Member
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 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.

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Hurkyl
Staff Emeritus
Gold Member
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.

HallsofIvy
Homework Helper
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.

Hurkyl
Staff Emeritus
Gold Member
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.

also assuming nonconvex right? or rather the restriction stated in the first post?

Hurkyl
Staff Emeritus