Can Any 2-Plane Be Mapped to Another Using Linear Maps and Translation?

[WWGD]

Hi, just curious as to whether we can map any 2-planep: ax+by+cz=d into any other

2-plane p': a'x+b'y+c'z=d' by using a linear map (plus a translation , maybe). I was thinking

that we could maybe first translate to the origin , for each plane, then , given the

angles ( t,r,s) with the respective x,y,z axes, we could rotate by (-t,-r,-s) to have

a plane z=constant , and do the same for c'. Would that work?

 

[micromass]

First translate each plane to the origin. Then take a basis for each plane and take a linear map which maps the basis vectors to each other.

 

[WWGD]

But how do you find a basis for a plane using only the equation ax+by+cz=0?

 

[micromass]

But how do you find a basis for a plane using only the equation ax+by+cz=0?

Well

$$(-b/a,1,0),(-c/a,0,1)$$

is a basis (if a is nonzero). If a is zero, then you must do something analogously with b and c.

 

[blue_raver22]

I am not an expert in mathematics or geometry, but I can provide some information based on my understanding of the subject. It is possible to map one 2-plane onto another using a linear map and a translation. This process involves translating each plane to the origin and then rotating it by a certain angle in order to achieve a z=constant plane. However, the success of this method depends on the specific values of the coefficients and angles involved. It may not work for all cases and may require further adjustments. Additionally, other methods such as using matrices or vector representations may also be used for mapping planes. It is important to carefully consider the mathematical principles and properties involved in order to accurately map one plane onto another.