Position of a 3d Point

Discussion in 'Homework Help' started by zulfi100, Mar 17, 2015.

  1. zulfi100

    Thread Starter Member

    Jun 7, 2012
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    Hi,
    I want to determine position of a 3d point w.r.t. planes. Let's suppose the point is (10, 15, 20). In which plane this point would lie: xy, yz or xz. If its not in any of these planes where it would lie? Some body please guide me.

    Zulfi.
     
  2. WBahn

    Moderator

    Mar 31, 2012
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    What do you know about ANY point that lies in the xy plane? Does this point satisfy that condition?

    What do you know about ANY point that lies in the yz plane? Does this point satisfy that condition?

    What do you know about ANY point that lies in the xz plane? Does this point satisfy that condition?

    What do you mean "where would it lie"? It lies at the point (10,15,20). It also lies on an infinite number of lines. And it lies in an infinite number of planes that contain any given one of the infinite number lines it lies on.
     
  3. zulfi100

    Thread Starter Member

    Jun 7, 2012
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    Hi,
    Thanks for your reply.

    What do you know about ANY point that lies in the xy plane? Does this point satisfy that condition?
    Ans: z=0. No point does not satisfy this condition.
    What do you know about ANY point that lies in the yz plane? Does this point satisfy that condition?
    Ans: x=0. No point does not satisfy this condition.
    What do you know about ANY point that lies in the xz plane? Does this point satisfy that condition?
    Ans: y=0. No point does not satisfy this condition.

    Please give me one example of this.

    Please give me one example of this.

    Zulfi.
     
  4. WBahn

    Moderator

    Mar 31, 2012
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    The line defined by the parametric equations {x=15t, y=15t, z=20t}

    I'll give you three: the x=10 plane, the y=15 plane, and the z=20 plane.
     
  5. zulfi100

    Thread Starter Member

    Jun 7, 2012
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    Hi,
    Thanks for your reply.
    I cant understand how can it be 3. If you say that x=10 is one representation then it means that you are considering y=z=0 which is not correct according to my above answers.
    In my view, a 3d point (where x , y & z are not zero) its not a point but its a combination of 3-planes. It gives us a notion of space. It can represent some thing like my water tank, my fridge or my microwave oven.

    Zulfi.
     
  6. WBahn

    Moderator

    Mar 31, 2012
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    The plan x=10 allows ANY values for y and z. The only constraint on x, y, and z is that x must be 10. So that is a plane that is parallel to the yz plane but that is offset so that all points for which X=10 are within it.

    Those three planes are mutually perpendicular and intersect at the point (10,15,20).

    It doesn't matter what you view is. A point is a point. Claiming that it is a combination of three planes is largely meaningless. Pick up a book and consider the plane that is the front cover, the spine, and the bottom edge of the book. The intersect at the lower-left corner of the cover. Now fix that point in space by putting that corner of the book at a specific point on the desk. You now have three planes what "combine" to give you that point. Now twist the book however you want while keeping the corner at the same point on the desk. You know have three different planes that "combine" to give you that same point. Now open the book. You still have three planes that "combine" to give you that same point, only now they aren't all mutually perpendicular. Now consider that the surface of the desk defines a fourth plane and you can pick any three of the four (three from the book and one from the desk) to "combine" and give you that same point.

    And how does any of that represent a volume such as your water tank?
     
  7. zulfi100

    Thread Starter Member

    Jun 7, 2012
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    Hi,

    I got the concept:

    Although, I would admit that book example was way above my head, I would show you a simple example of room corner:

    xyz axis and 3d point4.png
    AD= 10cm, DC=15cm and DH=20 cm. ABFE is a plane perpendicular to x-axis, BFGC is the plane perpendicular to Y-axis, and EFGH is a plane perpendicular to Z-axis. F is the point at which the 3 planes intersect. Is this description correct?

    Volume is obtained by multiplying the 3 dimensions.

    Zulfi.
     
  8. WBahn

    Moderator

    Mar 31, 2012
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    Volume of what? You haven't defined an enclosed volume? You are ASSUMING that this point defines SIX planes. It doesn't define ANY planes!
     
  9. WBahn

    Moderator

    Mar 31, 2012
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    Can you see that there is a plane that contains the x-axis as well as the point in question? If so, can you see that the same applies to the y-axis and the z-axis? Therefore we have three plans that interest at the point in question? What is the "volume" that corresponds to this? (HINT: There isn't one!)
     
  10. MrAl

    Well-Known Member

    Jun 17, 2014
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    Hi,

    I think what he is looking for is the intersection of three planes where the three planes are explicitly predefined.
    Probably he was interested in the planes that are PARALLEL to the three basic planes where either x,y, or z is zero.
    So one plane would be parallel to the xy plane, one parallel to the xz plane, and one parallel to the yz plane. Then one 3d point would define those three planes.
    But this is more of an application of analytic geometry not part of basic geometry itself, because more generally a single point has an infinite number of planes passing through it. It's only after defining how we want to restrict our possible set of planes do we make any sense out of it.

    There are tests to find out if a given point is in a given plane, for example, and the result is either 'yes' or 'no'.
    There is also a calculation we could do to find the distance from a given point to a given plane, where that distance would be zero if the point lies in the plane.
     
  11. WBahn

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    Mar 31, 2012
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    I don't think he is aware of all of the assumptions that he is making and, instead, thinks that those assumptions are basic truths.
     
  12. MrAl

    Well-Known Member

    Jun 17, 2014
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    Hi,

    Yeah i agree fully, i think that's all the problem is here. Once the OP realizes that i think it will make more sense to them why the explanation is not so simple.
     
  13. zulfi100

    Thread Starter Member

    Jun 7, 2012
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    Hi,

    Thanks for your replies. It really removed the confusion.
    Actually my problem was solved when i was able to draw that point with the help of a room (cube) example. However i am not yet able to understand how can we have infinite representation of point. This would be a complex thing. 3d is similar to real world things and when we do measurements we usually specify 3 components and that points to a specific point and does not have infinite representations. When we say that points are not unique then we cant do any construction work in a real world environment.
    Thanks, i drew this cube (room) and i am able to understand that point F is the intersection point for the 3 planes.

    Actually in real world we use 3 quantities to represent volume similar to the 3 components of the point we just discussed and we use the unit cft (cubic feet). So it creates a confusion but you are right the 3 components of a point intersect each other at the defined cartesian coordinates but since its not a closed surface or polygon so a point's dimensions are not similar to volume.

    So now my problem is that how a point can have multiple representation in a 3d environment. Kindly explain me with room picture i provided above.

    Zulfi.
     
  14. WBahn

    Moderator

    Mar 31, 2012
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    The similar situation in 2D should suffice. Take a piece of paper and put a point on it with a pen. Now draw a horizontal line and a vertical line that each go through that point. You now have two lines that can be used as a means of specifying that point. But now erase those two lines and draw a line coming down and to the right that goes through that point and another that is going up and to the left through that point. You now have two different lines that can be used as a means of specifying that point. If you put the original lines back on you now have four lines, any pair of which can be used to specify that point, giving you a total of six possibilities. Add a fifth line and you have added four more possibilities for a total of ten.

    Take a rectangular room and use the floor as one plane and then build a wall from the center of each wall to the center of the opposite wall and use those as the other two planes. The intersection of those three places is the point on the floor in the center of the original room.

    Now take the original room but this time build two walls going from a corner of the room to the opposite corner. Where do these three planes intersect? At the point on the floor in the center of the original room.
     
  15. MrAl

    Well-Known Member

    Jun 17, 2014
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    Hi,

    You have to realize that, in a manner of speaking, mathematics is larger than the universe. By this i mean that we can calculate many things that will never be possible in the real universe we happen to live in. This means we have to have an application so that we can narrow down the math needed to solve the problem.
    You seem to want to specify a point using three planes, and you can do that, as long as the planes intersect either other and you restrict the planes to say being normal to each other. You also have to realize that some planes will intersect two other planes at more than one point if there is no restrictions on how we can place the planes relative to each other.
    In your 'room' view, the planes are all normal to each other so your point can be well defined.
     
  16. zulfi100

    Thread Starter Member

    Jun 7, 2012
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    Hi,
    Thanks for taking pains in explaining me this.
    In both these examples, the location of the point has not changed. It lies on the same plane in the centre of room's floor. How is the possible that a point can lie on infinite planes at the same time?
    If its an imaginary thing, its acceptable.
    Zulfi.
     
  17. WBahn

    Moderator

    Mar 31, 2012
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    How is it possible for the same point to lie on an infinite number of lines at the same time?
     
  18. MrAl

    Well-Known Member

    Jun 17, 2014
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    Hi,

    Take a piece of sheet steel, say 1 foot by 1 foot, maybe 1/16 inch thick, that's your plane. Drill maybe a 1/2 inch hole in the center, that's your point.
    Take a long 1/4 - 20 threaded rod, push it through the center and bolt it to the plate using washers and nuts, hang a small weight on one edge of the plate so that it hangs on the bolt at an angle, like 45 degrees.
    Here you have your plane and your point, and the plane is oriented in space at a 45 degree angle relative to the bolt shaft. Your point is in the middle of the plane.

    Now holding the other bolt end, rotate the shaft 360 degrees, one time. You just illustrated that point being at the center of an infinite number of planes even though we kept the plane at the same angle reglative to the bolt all that time. As the steel plate revolves, it represents a new plane for each and every angle, and we can divide 360 degrees into an infinite number of angle thus we have an infinite number of planes.

    So from the above you can see that when the steel is oriented at a zero degree angle it represents a different plane than when it is oriented at say a 10 degree angle (horizontal) so you can see that a plane is not a physical object (like the steel plate). The steel plate is always the same size but the plane is always different.

    You should look up the definition of a plane in 3d space because it is becoming clear that you just dont understand yet what a plane really is. Once you do this i think you'll have a better grasp of what is happening here. The plane can be described by a mathematical equation with constants, and if you vary the constants you change the plane and because of that it is not considered to be the same plane anymore, but a different plane in geometry.
    For example, if i give you a simple expression:
    y=2*a+b^2
    If we then make a=1 we have a number of solutions for y depending on what b is, but if we change a to a=21, then we have a different set of solutions. For the 'plane' in geometry it's the same thing. We have constants that we change and that changes the plane so it is no longer the same plane.

    For the plane we have:
    a*x+b*y+c*z+d=0

    If we start with a certain set of a,b,c, and d, we have ONE particular plane. If we change these constants, even one of them, we end up with a DIFFERENT plane. It is considered an entirely different plane once the constants change.

    Attached is a 3d drawing of two planes sharing the same point. One plane lines in the xy plane, and the other is tilted at an angle. You can see they have at least one point in common (the red dot, but probably others too). If we tilt the upper plane at a different angle, we would have another entirely different plane yet it would share the same 3d point.
     
    Last edited: Mar 22, 2015
  19. zulfi100

    Thread Starter Member

    Jun 7, 2012
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    Hi,
    Thanks for your reply and your great example and the accompanied picture. I like your example and then the figure or drawing completely shows that we can have different planes but the point can be same. This clearly solves my problem. However, it still have some question but please dont consider that i am disregarding your efforts.
    Your example has a point which has physically the same position. When we rotate the sheet, the plane i.e the sheet remains the same. This means that its x, y and z values also donot change. Similarly your picture shows two planes. But in the upper plane it looks that the x, y & z values are different? So what i conclude is that the physical position is same but x, y & z may change. Am i right?
    One another question is that tilting of sheet will change the angle but how its going to change the plane. The equation of plane does not contain any angle term. It has constants but again these constants do not depend upon angle. Please guide me.

    Thanks for helping me. You have solved my problem. I highly appreciate your efforts. You have explained me that same point can exist in multiple planes and this idea can be extended to infinite planes but now i have confusion with x, y, & z values and impact of angle on the equation of plane.

    Zulfi.
     
  20. MrAl

    Well-Known Member

    Jun 17, 2014
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    Hi,

    The x,y,z variable values just help to generate the planes.
    The a,b,c,d constants do in fact change the angle. With all these equal to zero except c=1, the plane lies flat on the xy plane.
    The equation for any plane again is:
    a*x+b*y+c*z+d=0
    and solving that for z we have:
    z=-(a*x+b*y+d)/c
    If we make d=0 and c=1 for convenience, we have:
    z=-(a*x+b*y)
    This can generate planes of that one type, and yet still if we make a=-1 and b=1, we get different angles than if a=1 and b=-1.
    For a=-1 and b=1 we get:
    z=x-y
    and for a=1 and b=-1 we get:
    z=y-x

    So throw a few numbers in for x and y in both those equations and see that the angle of the plane is different for each one.

    I think you may still be looking at a 'plane' as a physical object. It's not a physical object that can be moved around at will. A flat steel sheet is like one plane if you hold it very still, but if you move it up or down or rotate it in any way it becomes an entirely different plane. It's still a flat steel sheet, but it is now lies in a different plane. The action of moving the plane can be described by changing the constants a,b,c, or d, or maybe all four or just three or just one or two sometimes.
     
    Last edited: Mar 22, 2015
    zulfi100 likes this.
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