What is the Concept of Signed Distance in Relation to a Plane?

[Physicsissuef]

Homework Statement

\vec{PM_1}=d*\vec{n_0}

Here are the both pictures.

"[URL 0401.jpg"]Picture 1.[/URL]

"[URL 0502.jpg"]
Picture 2[/URL]

Homework Equations

d = \frac{\left | a x_1 + b y_1 + c z_1+d \right |}{\sqrt{a^2+b^2+c^2}}.

The Attempt at a Solution

Why in my book says that, if d > 0 then the coordinate M is of opposite site of the plane (relative to the (0,0,0)), and if it is d < 0 it is on same site of the plane (relative to (0,0,0)) when d is always positive?

 

[HallsofIvy]

Bad editing? I suspect they meant
d= \frac{ax_1+ by_1+ cz_1+ d}{\sqrt{a^2+ b^2+ c^3}}
without the absolute value! And then |d| is the distance.

 

[Physicsissuef]

if d > 0 then the coordinate M is of opposite site of the plane (relative to the (0,0,0)), and if it is d < 0 it is on same site of the plane (relative to (0,0,0))? Where is the logic?

 

[Physicsissuef]

and d up in the fraction should be D:

like this: <br /> d= \frac{ax_1+ by_1+ cz_1+ D}{\sqrt{a^2+ b^2+ c^3}}<br />

 

[VietDao29]

if d > 0 then the coordinate M is of opposite site of the plane (relative to the (0,0,0)), and if it is d < 0 it is on same site of the plane (relative to (0,0,0))? Where is the logic?

No, it's not like that.. It depends on whether D is negative or positive..

Say, I have a 3-variable function:

d(x, y, z) = \frac{ax + by + cz + D}{\sqrt{a ^ 2 + b ^ 2 + c ^ 2}}

where a, b, c, D are all coefficients of the plane function: ax + by + cz + D = 0, and (x, y, z) \in \mathbb{R} ^ 3 is the co-ordinate of a point in 3D space.

There should be a theorem in your books, which states:

Two points A(x₀, y₀, z₀), B(x₁, y₁, z₁) are on the same side of the plane: ax + by + cz + D = 0, if and only if:

d(x₀, y₀, z₀) d(x₁, y₁, z₁) > 0..

They are not on the same side (i.e, one is on one side, and the other one is on another side), if and only if:

d(x₀, y₀, z₀) d(x₁, y₁, z₁) < 0..

-------------------------

Back to your question:

The origin O has co-ordinate (0, 0, 0), so d(0, 0, 0) = \frac{D}{\sqrt{a ^ 2 + b ^ 2 + c ^ 2}}, which will take the sign of D, since the square root in the denominator is always positive, hence, does not affect the sign of d(0, 0, 0).

So, a random point with co-ordinate (x, y, z), and the origin are on the same side of the plane (whose function is: ax + by + cz + D = 0), if and only if:

D * d(x, y, z) > 0..

They lie on 2 different sides if and only if: D * d(x, y, z) < 0..

Is this clear? :)

 

[Physicsissuef]

d is vector? I don't think so, that d have co-ordinates, because
<br /> \vec{PM_1}=d*\vec{n_0}<br />
To be honest I don't understand what are you talking about.

 

[VietDao29]

d is vector? I don't think so, that d have co-ordinates, because
<br /> \vec{PM_1}=d*\vec{n_0}<br />
To be honest I don't understand what are you talking about.

d is not a vector, it's a function.. Just like those of f(x), f(x, y), and f(x, y, z)..

d(x, y, z) is just a 3-variable function, and it returns real numbers..

Do you know the general method to check whether 2 points are on the same sides of a plane, or on 2 different sides?

BTW, is that book in Russian?

 

[Physicsissuef]

No, I don't know the general method to check whether 2 points are on same sides of plane.

The book is in Macedonian, which have similar letter with Russian.

 

[HallsofIvy]

That question is different, and simpler, than determining the distance from a point to a plane!

The equation of a plane can be written A(x-x_0)+ B(y- y_0)+ C(z- z_0)= 0
Any point on the plane makes the left side of that 0, of course. Any point not on the plane make it either positive or negative. Two points are on the same side of the plane if they make the same sign: (x_1,y_1,z_1) and (x_2,y_2,z_2) are on the same side of the plane if and only if A(x_1-x_0)+ B(y_1- y_0)+ C(z_1- z_0) and A(x_2-x_0)+ B(y_2- y_0)+ C(z_2- z_0) have the same sign.

That is, of course, essentially the same as the numerator of your "distance" formula without the absolute value. Since the denominator is positive, the sign of fraction is the same as the sign of the numerator.

 

[Physicsissuef]

And what is numerator and denominator?

 

[Physicsissuef]

If
<br /> \vec{PM_1}=d*\vec{n_0}<br />
then d is not vector, but it is module. How is possible than that d<0 ?

 

[VietDao29]

And what is numerator and denominator?

In a fraction, a numerator is the number on top, the denominator is the one that lies at bottom..

Say, \frac{2}{5}, then 2 is the numerator, and 5 is the denominator. Or, generally, in a fraction, we have: \frac{\mbox{numerator}}{\mbox{denominator}}

If
<br /> \vec{PM_1}=d*\vec{n_0}<br />
then d is not vector, but it is module. How is possible than that d<0 ?

No, d is a scalar, in different words, a real number. It is positive when \vec{PM_1}, and \vec{n_0} point in the same direction; and negative, otherwise..

 

[Physicsissuef]

Ok, thanks. And where is the logic about d>0 and d<0 ?

 

[vurentjie]

a distance is always absolute and unsigned and positive, |d| is the distance of whatever point from a plane, the signed distance is the negative or positive values of d, so -d will be on one side of the plane and it will be |d| units from that plane, if +d then it is on other side but also |d| units from plane,

in other words -2 is exactly 2 units from 0, +2 is also exactly 2 units from 0, but is situated on the other side of zero 0.

that is probably the simplest way it could be explained.