A very abstract problem about angle,centre,movement and space.

I am in my first year of degree in electrical information engineering and just started my university life for 2 months now.Let me just say that I am fresh from high school and still have no experience of real engineering projects or whatsoever.I am having trouble finding a solution to a problem that 2 researchers gave me and I have to present the solution on this coming monday.so here is the MAIN question.My BIG problem with this is that there absolutely no measurements given.The diagram is not drawn according to scale and it's just too abstract and not specific.
[IMG]

Can the arm(the one with 3 fingers)touch all points throughout the whole space?pplease prove your answer by including proposition,hypothesis,theorem,formulae,etc.

Before proving the MAIN question,first solve these problems as these can lead to proving the MAIN question.

1.It's the same thing but first we think simple.Think about a circle.if the radius is 2,and the length of the arms between the node(think of it as a centre) is 1,how can you prove that it is able to touch all points in the circle?
[IMG]

My teacher only gives me one example of stating a prove which is as written in the diagram,the arm won't move out of the circle as it always moves at a constant distance and therefore it is able to touch the points shown in the diagram(including downwards movement).

2.But what if we were to touch a random point?how do we determine the angle needed to rotate the arm so that it touches a certain point as shown in figure 2 below?one of the way is by changing the position of the node(the centre of the arm)but how much does it need to be repositioned?
[IMG]

3.After solving the circle,think of square.Can the arm touches all points in the square?What is the prove?
[IMG]

4.After solving the square,solve a triangle.
[IMG]

So this is what the 2 researches told me to do.I really need a complete solution to proving wether the arm stated in the MAIN question can or cannot touch all points in the space given.My hypothesis is that the arm can touch all the points but I do not know how to prove it as the 2 researches reject my logical explanation.They want me to prove this mathematically.

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Ankita Katdare

Hi Aruwin,

I will try to find a mathematical solution to your problem.
Meanwhile, go through this paper: #-Link-Snipped-#

Is it of any help?

PS: I just did a random search and posted the link, didn't go through it completely.
Ankita Katdare

@Aruwin: On what premises did you lead to the hypothesis that the arm can touch all the points?
Please share all details with everyone.
Aruwin

I just managed to think about the circle,though.If we assume that the arm length is equal to the radius of the big circle,then it is able to touch all points since it wont go out of the circle.In order to touch random points,what I think is needed is to change the centre of the arm and determine the angles of the centre of the arm and also the angle between the centre of the circle and the arm.I think for the main question,if we could play around with the angle and the length and the position of the centre,then it does manage to touch all points.However,I cant find any way to prove this.
ISHAN TOPRE

Yup Aruwin you are on a right track, However I would differ slightly in my opinion. The total arm length should be equal to diameter. But as you can see it is joined at two places, the other two links should be half of the radius. the joints should act as pivots.
ISHAN TOPRE

Now think of square. The link joining the center will rotate in an anticlockwise direction and the other link would have a relative motion with respect to the first link and the stationary square. Isnt it?

Now we can vary the speed of the link at the center. (Think of hands of a clock).
Aruwin

ishutopre
Yup Aruwin you are on a right track, However I would differ slightly in my opinion. The total arm length should be equal to diameter. But as you can see it is joined at two places, the other two links should be half of the radius. the joints should act as pivots.
equal to the diameter?I dont quite get that part.I hope u dont mind drawing it out so that i can see easier.
ISHAN TOPRE

You have yourself drawn it there. I will draw something for a square instead. 😀
Aruwin

but if the arm is equal to the diameter of the circle,wouldnt it go beyond the circumference??The arm i drew is actually the same length as the radius which is 2.
ISHAN TOPRE

Yup Gotcha. here. Apply the same analogy to circle.

Now keep the length of both the links half of the diameter. 😀

OK as in your second figure you are using just one joint, let us keep the length same as the diameter.
ISHAN TOPRE

Now you can make out a relation between the rotating link with the other link joining the periphery of circle and square, can you? 😀
Aruwin

thanks.But before going to the square we need to finish proving the circle first.I havent got to determine the angles and the centre positioning.
ISHAN TOPRE

OK, for a single point on periphery of circle, determine the angle between the two positions of link with respect to the speed of your center link. That is your answer.

The angle will vary according to the speed of your centerlink. 😀
Aruwin

ishutopre
OK, for a single point on periphery of circle, determine the angle between the two positions of link with respect to the speed of your center link. That is your answer.

The angle will vary according to the speed of your centerlink. 😀

urm..what is speed of centerlink?and can u give me a concrete example?formula maybe?previously my teacher asked me to write out an example which is to prove how can the arm touch coordinate (x,y)
ISHAN TOPRE

Well, you can decide it there. The problem is now between the length of links and the speed. Find out the angle subtended by the link for a radius "R" and speed N rpm. 😀 Keep the length of links for simplicity as R/2. You can keep them 3R/4 and R/4 also, the angle will vary according to it.
The drawing takes it as R/2 for each link.
Aruwin

But i dont think this has anything to do with speed at all.is there any other way because my teacher said this has nothing to do with finding speed
Aruwin

But i dont think this has anything to do with speed at all.is there any other way because my teacher said this has nothing to do with finding speed
CIVILPRINCESS

ishutopre
Well, you can decide it there. The problem is now between the length of links and the speed. Find out the angle subtended by the link for a radius "R" and speed N rpm. 😀 Keep the length of links for simplicity as R/2. You can keep them 3R/4 and R/4 also, the angle will vary according to it.
The drawing takes it as R/2 for each link.
i don't think its a matter of speed from what i've understood ishan. its about co-ordinating the arm inside the circle now right?
ISHAN TOPRE

How about on concentrating on angle instead? See this angle? Your teacher has given radius as 2 cm. 😀
ISHAN TOPRE

In this way, by simply putting any arbitrary value of angle, you can touch all points inside the circle too. 😀 you can even touch the center by keeping the angle between the two links as 0 deg, right? I think, there is no formula and it is only a matter of putting any value (common sense)
CIVILPRINCESS

yeah and we don't have a radius for the circle right?
Aruwin

ishutopre
In this way, by simply putting any arbitrary value of angle, you can touch all points inside the circle too. 😀 you can even touch the center by keeping the angle between the two links as 0 deg, right? I think, there is no formula and it is only a matter of putting any value (common sense)
yeP,that's exactly what i told the teacher but then he asked me to show atleast a theorem,not just saying common sense.He then asked me,how do you wanna prove that the arm touches lets say (-1,1)???
Aruwin

How do I answer that question?Whenever I say that we can simply put any number he will then ask me to prove it concretely by stating an example.So how do I say the arm touches coordinate (-1,1)?
Aruwin

so how are u gonna prove that the arm touches coordinate (1,-1)?what answer do i give to my teacher?
ISHAN TOPRE

Well try calculating angle for (0,0) as center and (-1,1) as other point, that would be very simple.
Give this angle as example to your teacher and state your ideas in words.
Also generalize it for points (x,y) instead of (-1,1) once you get it for that point.

You will get angle in terms of x,y. 😀
ISHAN TOPRE

Calculate the angle subtended by link joining the center with the horizontal with respect to the angle subtended by the two links( assume it as theta). and use simple co-ordinate geometry. 😀

Get the solution in terms of x,y and angle theta.
Aruwin

ishutopre
Calculate the angle subtended by link joining the center with the horizontal with respect to the angle subtended by the two links( assume it as theta). and use simple co-ordinate geometry. 😀

Get the solution in terms of x,y and angle theta.
can u give me a diagram so that i can see it clearly??right now i cant really imagine it in my mind. and what about the explanation for figure 3(square)?
ISHAN TOPRE

Perhaps this would be helpful. 😀
ISHAN TOPRE

Now can you find out the distance between 0,0 and x,y for a constant angle theta? That is your answer. 😀
Aruwin

ishutopre
Now can you find out the distance between 0,0 and x,y for a constant angle theta? That is your answer. 😀
ok,i think i can try that out,but what about square?how to prove that all points in the square can be touhed?for circle is easier to find the angle but what about square?
ISHAN TOPRE

It is simple, as you can find out the equation of trajectory for a point moving in a circle, find out the trajectory for a point moving in a square.

In a circle, the point moves in a circular manner, in a square, it would move in a linear fashion. Simple,

In circle, it would be equation of circle for a particular angle, for square it would be equation of a line with respect to a particular angle. 😀
Aruwin

ishutopre
It is simple, as you can find out the equation of trajectory for a point moving in a circle, find out the trajectory for a point moving in a square.

In a circle, the point moves in a circular manner, in a square, it would move in a linear fashion. Simple,

In circle, it would be equation of circle for a particular angle, for square it would be equation of a line with respect to a particular angle. 😀
Again,can u show me a diagram?i think i can do it if there"s a diagram.
ISHAN TOPRE

Here is the image. Here both the angles will vary with respect to each other to keep the trajectory straight and hence will cover the whole square.
Aruwin

ishutopre
Here is the image. Here both the angles will vary with respect to each other to keep the trajectory straight and hence will cover the whole square.

sorry,i still dont get it.what if we insert a circlr of radius 1 inside the square?and move the pivot (length 1)around so it touches all the sides n corners?
ISHAN TOPRE

Yeah you can do that too.
Aruwin

ishutopre
Yeah you can do that too.
If i use that that i just find the distance n then get the angle just like we did for the circle.is that possible?
ISHAN TOPRE

Yup that is possible. You got my point. 😀
Aruwin

ishutopre
Yup that is possible. You got my point. 😀
just finished presenting!!!OMG,I did it!!My 2 teachers were satisfied with my explanation for the circle and square.However,there is one question arise.this is the question,@
"How do you move the angle?"
Aruwin

I need to prove for a rectangle now and then a triangle I have to finish the MAIN question by this week.Please,someone help me.Here's the figure for the rectangle.
Aruwin

cam someone give me any info about any research that has been done about this?

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A very abstract problem about angle,centre,movement and space.

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