Question on quad planes in XYZ coordinates

[Flak]

As we know, 3 points define a plane at XYZ, if you add another point then you create a 3D polygon.

 

My question is that if there is a way to align the new point to the others so you can create a quad plane.

 

The info:

 

3 points with XYZ coordinates (base plane)

1 point with XYZ coordinates wich have to be moved on X or Y or Z to be aligned to base plane and create a quad plane.

[Country Boy]

I'm afraid you will have to explain what you mean by a "quad plane"!

[Flak]

I'm afraid you will have to explain what you mean by a "quad plane"!

 

Quad Plane: a plane made of two triangles (wich is know as plane).

 

You can move a point on a plane to any position and you ever get a plane, let say a planar surface. However on a quad plane you move a point and you get a 3D surface, so the task would be move that point on X or Y or Z to get back a planar surface.

[Bignose]

I'm glad HallsofIvy asked, because it still isn't making sense. That is, can you explain in great detail what you mean by "quad plane"

 

Because, if you want to locate a point on a plane, once you've established that plane, you have the equation. ax + by + cz = d. Given three points, you can establish what a,b,c are. (see http://en.wikipedia.org/wiki/Plane_(mathematics) ). Then, any other point that solves that equation is also on the plane. Any point that doesn't solve that equation, isn't on the plane. It's a simple as that.

[uncool]

Quad Plane: a plane made of two triangles (wich is know as plane).

A plane is not made of two triangles; do you mean a surface which is made of two triangles? A plane is always flat.

 

You can move a point on a plane to any position and you ever get a plane, let say a planar surface. However on a quad plane you move a point and you get a 3D surface, so the task would be move that point on X or Y or Z to get back a planar surface.

[insane_alien]

maybe he means intersecting planes?

[Flak]

Ok is difficult to explain and this board do not allow uploading.

 

You have 3 points in an XYZ enviroment, that create a plane, lets call it 3-plane because it is made of 3 points. You can move any of the points at any place on XYZ and will still have a plane.

 

Now , you have 4 points on an XYZ enviroment, that can create a plane, lets call it quad plane because is made of 4 points. If you move any of the points in XYZ you will get a 3D surface. I indicated that it can create a plane because if the extra point is correctly aligned that will create a planar surface. That extra point can be moved on X or Y or Z to be aligned.

 

Then the info is XYZ values for 3 fixed points that already create a plane, and the value XYZ of the point that need to be aligned to the 3 fixed points to create a planar surface by change its X or Y or Z value.

[D H]

Are you asking how to compute the projection of a point in R³ onto a plane embedded in R³?

[Flak]

http://i220.photobucket.com/albums/dd193/Flak_SFN/quadplane.jpg

[insane_alien]

so you meant two interscting planes.

[Bignose]

so you meant two interscting planes.

 

That's what I got from the pictures, too. But, Flak, you have to very careful with your terminology there. 4 points cannot "create a plane" 4 points can all lie on a plane, but all 4 have to satisfy the equation of the plane. It takes 3 points to define a plane, and then all you have to do is check to see if the 4th also satisfies the equation that describes the plane the first 3 points are on. Otherwise, as your picture showed, it will take two different planes to describe all 4 points. I think that you really should drop calling it a "4-plane" or "quad plane" because that terminology really isn't correct unless you want to start talking about 4 dimensions. Even then, the terminology usually is to call it a hyperplane.

 

That wikipedia link I posted earlier, http://en.wikipedia.org/wiki/Plane_(mathematics) has a formula for calculating a distance from a point to a plane, so you can use that to see how far you have to move a point to get back aligned with the plane of the first three points.

[Sisyphus]

What I think you're asking is how do you place a point such that it lies on a preexisting plane. And the answer is that you place it such that it's x, y, and z coordinates satisfy the equation for that plane. The equation for any plane can be written in the form ax + by + cz = d, where a, b, c, and d are constants.

 

I think perhaps the source of confusion is that you're thinking that three points somehow are the plane, or "make up" the plane. This is incorrect. It is true that "three points define a plane" (at least, three points that are not all in a line), but what is meant when we say that is only that only one plane can pass through all of those points. There are an infinite number of points on any plane, however, and any three of them could just as well define that exact same plane.

[Flak]

Ok I agree, I used the term "quad plane" to explain that type of plane.

 

How the ecuation ax + by + cz = d is used?, let say in the plane the values are (just any value as example):

 

Point A

X = 23.5

Y = 10

Z = 20

 

Point B

X = 10.6

Y = 5

Z = 2

 

Point C

X = 35

Y = 20.2

Z = 14

 

Point D (the one I want to aling to the plane by moving it on X or Y or Z)

X = 4

Y = 46

Z = 52.4

[Xerxes]

It takes 3 points to define a plane,

Are you sure? I can define a plane, or rather the plane using only 2 coordinates. Any plane defined with more than 2 coordinates is what? A hyperplane, maybe? Dunno.

 

But I do know that, as commonly understood, a plane is a (flat) two-dimensional surface, wherein any point on the plane can be referenced by two (real or complex) numbers on the coordinate set.

 

PS, by edit: There are other non-planar surfaces where any point can also be referenced by two and only two numbers (this harks back to another thread) Anyone?

[insane_alien]

if you only have 2 points there can be infinite planes that would satisfy it. they would just be rotated on way or aother. thats like saying you define a line with one point and you don't need any points to define a point.

[Xerxes]

Huh? Lost me there.

[insane_alien]

okay, take three points and a board. there is only one oreintation you can put the board in where it can touch all three points. these three points define the plane. if you have 2 points only, the board can rotate around these two points easily and still touch them. the two points define a line but there can be infinite planes that have those two points.

[Xerxes]

okay, take three points and a board.

So your "board" is two dimensional, right? Are your "points" on the board, or not. I'm confused.

there is only one orientation you can put the board in where it can touch all three points. these three points define the plane.

Ugh! This makes no sense to me.

the board can rotate around these two points easily and still touch them. the two points define a line but there can be infinite planes that have those two points.

Even worse. Sorry, but what on earth are you talking about?

[insane_alien]

well, it was meant to be a tangible visualisation of it but obviously it didn't work.

maybe someone else can explain it better.

[Bignose]

Yes, I am sure. Xerxes, take a piece of paper in both hands. Hold with just your thumb and one finger, and pull it so it is taut, so that it is flat like a plane would be. Where you are holding it will be the two points. Now, notice that you can rotate your hands while keeping your thumbs in place, symbolizing that there are an infinite number of planes that can go through those two points -- your thumbs. This is just like using only two points to try to define a plane -- you can keep two points fixed, but there are still an infinite number of planes that can go through them. So, which plane you you want? You have to have a third point, that is not on the same line as the first two points, then you can completely specify a single plane. Again, look at that wikipedia link I've posted twice now.

 

Same thing Flak, if you look at the wikipedia link I posted, it tells you how to use three points to find the equation of the plane, and then how to calculate how far a point is from a given plane.

[Xerxes]

Ah, OK, we are at cross purpose, it seems. You are talking about a plane as a two-dimensional surface embedded in (or as a slice of) a 3-space. In which case I agree with you.

 

I was talking about the 2-plane as a "free" construction, i.e. with no reference to an embedding space, in which case any point on the plane is referenced by two numbers only.

[Flak]

Please can you explain me how to use that ecuation in the indicated question (align of the point to the plane)?

[Bignose]

Please can you explain me how to use that ecuation in the indicated question (align of the point to the plane)?

 

Flak, what are you specifically asking here? The wikipedia link shows how to calculate all the info you want, at least as far as I understand what you want. Are you unfamiliar with the notation and concepts used in the wikipedia article? Are you unfamiliar with vectors, and operations on vectors like dot product and cross products?

 

Or does the wikipedia entry not show you how to calculate what you want? In that case, you need to be more specific about what you want.

 

Please be more specific where your confusion is, so we can more directly help you. Thanks.