Affine Geometry

[dreamy walrus]

Given two points, let's say A(2,1,-1) and B(-3,0,2), how can i determine the equation of the bundle of planes passing through A and B?

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[carmine karma]

Hm... I'm thinking of how to make this in the most symmetric way possible.
First, find an affine transformation that brings A to the origin and B to, say, (0, 0, 1). Then make the equation of the plane as x cos(θ) + y sin(θ) = 0, where θ is an arbitrary parameter. Then apply the inverse transformation to this plane, then you'll get an equation of a plane that passes through A and B, and the parameter θ will just rotate the plane around the line AB.

[dreamy walrus]

Thank you, but i figured it out with a simpler solution

[hardy egret]

What did you do?

[dreamy walrus]

took a point which doesn't belong to the line AB, obtained the equation of a plane which contains these 3 points, then took another point which doesn't belong to the line and the previously obtained plane either, which helped me create the second plane. And now the bundle is written αplane1 + βplane2 = 0

but i think there was a simpler solution. i ll ask a teacher today to find out

[hardy egret]

Sorry the last line did not make all too much sense

[dreamy walrus]

You can obtain the equation of a plane if you have 3 points which are not colinear

[hardy egret]

No I get that part

[dreamy walrus]

by using the determinant formula

[hardy egret]

But I don't get the alpha plane1 + beta plane2 = 0

[dreamy walrus]

ok, then α*plane1 = plane2?

see you in 1 or 2 hours

you can search about bundle of planes and we can talk

maybe i didn t get something right

[hardy egret]

No I already have a solution in mind

I just wanted to know what you did

[dreamy walrus]

that is the equation of a bundle of planes

ok

[hardy egret]

It's more on a formalism level that I have a problem with that thing

pi1 and pi2 are equations, not numbers

So what's multiplying an equation by a scalar?

And when is an equation equal to a number?

[dreamy walrus]

try to test it on Geogebra and tell me too

i don't know either

i have something to do for 1-2 hours now and can't focus on this

brb

my partial exam at affine geometry is tomorrow and now i'm focusing on other things too

[hardy egret]

I mean if you are studying for an exam then just remember whatever your notes tell you

My opinion matters not

[dreamy walrus]

it's ok

[dreamy walrus]

what did you study?

mathematics related university?

[hardy egret]

A bit of everything

My main focus is data science now but I know a bit of every field in math

[dreamy walrus]

nice

[hardy egret]

So I guess mainly applied mathematics

[carmine karma]

Ohh, I see. Nice idea!
As for the coefficients of the bundle, maybe a good idea is to take them as cos(θ) and sin(θ)? That should account for all cases.

[dreamy walrus]

@carmine karma Ok, i will try to think about that. Thank you!