How to define for which plane lies a given point with coordinates?

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Aknur 2023-4-4

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评论： Bjorn Gustavsson 2023-4-6

采纳的回答： Bjorn Gustavsson

Hello!

Kindly ask about how to find in which plane lies point A(x, y, z), example A(1.5, 1.5, 3.0)

and I have planes

planes(:,:,1) = [0 3 3; 0 0 3; 0 3 0; 0 0 0];

planes(:,:,2) = [0 0 3; 3 0 3; 0 0 0; 3 0 0];

planes(:,:,3) = [3 0 3; 3 3 3; 3 0 0; 3 3 0];

planes(:,:,4) = [3 3 3; 0 3 3; 3 3 0; 0 3 0];

planes(:,:,5) = [0 3 0; 3 3 0; 0 0 0; 3 0 0];

planes(:,:,6) = [0 3 3; 3 3 3; 0 0 3; 3 0 3];

location_plane = 6;

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Answer 1

Bjorn Gustavsson 2023-4-4

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This is easiest to do with a vector-algebra approach. Something like this:

1, the plane can be described as all points

such that

, where

is the normal to the plane and l is a scalar.

2, to use this, first use 3 of your 4 points to calculate the surface normal, (create 2 vectors that are not parallel, cross-multiply them and normalize that vector).

3, use the formula above with that normal-vector and all 4 points - to check that you get the same l for all four points.

4, test if

also is identical to l

5, start over at point 1 with the next four points.

HTH

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Bjorn Gustavsson 2023-4-5

在 MATLAB Online 中打开

No you dont find the normal using the dot-product. A dot-product produces a scalar, and you need a vector. In order to produce a vector you have to do something else. One way to do this is using the cross-product between 2 vectors - provided that they are not parallel you will get a vector that is perpendicular to both vectors.

For you to do this you might have great use of the 2 best arrow-drawing functions on the file exchange: arrow3 and arrow. They will let you draw the arrows you work with and that ought to help you understand what your results are.

So first you should do something like:

p_4 = [3 3 3; 0 3 3; 3 3 0; 0 3 0];
plot3(p_1(:,1),p_1(:,2),p_1(:,3),'r.','markersize',18)
hold on
grid on

That shows the four point in what is claimed to be the fourth plane. If it is a plane (check by rotating the axes around), you know that the vector from one point to another on that plane is perpendicular to the normal (still unknown) - that means that we can generate two vectors that are perp to

something like this:

v1 = p_4(2,:) - p_4(3,:);
v2 = p_4(1,:) - p_4(3,:);
arrow3([0 0 0],v1)
arrow3([0 0 0],v2)
axis auto
axis equal

Note that these are vectors "parallel to the plane" meaning that if you start at any point in the plane and move in the direction of v1 or v2 you will remain in the plane. Verify this by drawing an arrow from the fourth point in the direction of v1 or v2.

Then you calculate the normal vector by taking the cross-product:

n = cross(v1,v2);
n = n/norm(n);
arrow3([0 0 0],n)

Now you have the normal-vector of the plane. What remains is to determine l. You do that as explained above, and once you've done that you check if the given point A is also in that plane.

Aknur 2023-4-6

@Bjorn Gustavsson thank you for your answer and clarification. I did smth like this. But at the end answer which is number of plane does not correct. Could you please check . Thank you so mcuh for your comments

X0 = 1.5;

Y0 = 3;

Z0 = 0;

planes(:,:,1) = [0 3 3; 0 0 3; 0 3 0; 0 0 0; 0 0 0];

planes(:,:,2) = [0 0 3; 3 0 3; 0 0 0; 3 0 0; 0 0 0];

planes(:,:,3) = [3 0 3; 3 3 3; 3 0 0; 3 3 0; 3 0 0];

planes(:,:,4) = [3 3 3; 0 3 3; 3 3 0; 0 3 0; 0 3 3];

planes(:,:,5) = [0 3 0; 3 3 0; 0 0 0; 3 0 0; 0 0 0];

planes(:,:,6) = [0 3 3; 3 3 3; 0 0 3; 3 0 3; 0 0 3];

location_plane = 6;

for j=1:6 % j is number of plane

j

plane = planes(:,:,j);

p0 = plane(1,:); %p0 is top left point of plane

p1 = plane(2,:); %p1 is top right point of plane

p2 = plane(3,:); %p2 is bottom left point of plane

p3 = plane(4,:); %p3 is bottom right point of plane

V0 = plane(5,:); %point on the plane

% Pi is initial start point on the ray

Pi = [X0 Y0 Z0]; %initial start point

%Ri = [XBar YBar ZBar]; %direction vector with unit length

A = p0-p2; %calculate A and B then

B = p0-p3; %then to calculate Normal of each plane

n=cross(A,B); % Normal for each Plane

n

l0 = dot(p0,n);

l1 = dot(p1,n);

l2 = dot(p2,n);

l3 = dot(p3,n);

l4 = dot(Pi,n);

if l1 == l0 || l1 || l2 || l3

RS = planes(:,:,j);

else

RS ~= planes(:,:,j);

end

RS

plane_i = j;

j

end

Aknur 2023-4-6

@Bjorn Gustavsson

I got same l for all 4 points (l0, l1, l2, l3 = =27), and l4 = dot(Pi,n) equal 0,

how then I can define on which plane lies my point with

Bjorn Gustavsson 2023-4-6

You have to make the same check with your point A. If you get the same l for

as for one plane then A lies on that plane, if you get the same l for multiple planes then A lies in all those planes.

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