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Minimum distance between a disk in 3d space and a point above the disk 
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#1
Jan214, 11:28 AM

P: 202

Hi all,
How can I calculate the minimum distance between the perimeter of a disk in 3d space and a point above the disk? (the point can be inside or outside the area above the disk) I've been trying to work this out for a while, but i'm getting no where. For example, a point at (1,1,1) and a disk with center (0,0,0) and radius 0.5. The distance between the centers is: [tex]\sqrt{(10)^{2}+(10)^{2}+(10)^{2}} = \sqrt{3} \approx 1.73[/tex] But how can I work out the shortest distance from the point to a point on the perimeter of the disk? Thanks! 


#2
Jan214, 11:34 AM

P: 2,991

You could try vector addition:
Rcenter + Rmin + Rradius = 0 and so Rmin =  (Rcenter + Rradius) 


#3
Jan214, 11:45 AM

P: 202

Thanks! That seems obvious now you've mentioned it aha. I was trying to do it using trigonometry.



#4
Jan214, 12:05 PM

P: 202

Minimum distance between a disk in 3d space and a point above the disk
Although shouldn't it be...
Rmin = Rradius  Rcenter so in the example i gave Rmin = (0.5,0,0)  (1,1,1) Rmin = (0.5,1,1) so it has length 1.5 


#5
Jan214, 12:21 PM

P: 202

Also, I've just realised this only works when Rradius lies on the x axis. How would you find the vector Rradius (the vector between the disk center and min point on the perimeter) if it does not lie on the x axis?



#6
Jan314, 08:04 AM

P: 202

I think i've figured this out, it would help if someone could verify if i'm doing this correctly though.
There is a particle at [5,5,5] above a disk with center [0,0,0] and radius 3. The disk is not tilted: It lies on the x,y plane. To calculate the vector between the disk center, and the point on the perimeter of the disk that is closest to the particle: 1) calculate unit vector from center to particle unitvec=[5,5,5]/[5,5,5] = [0.5774, 0.5774, 0.5774] 2) set the z element to 0, and multiply by the radius vec1=3*[0.5774, 0.5774, 0] = [1.7321,1.7321,0] To calculate the vector between the particle and the point on the perimeter of the disk that is closest to the particle: 1) subtract the vectors [5,5,5][1.7321,1.7321,0] = [3.2679, 3.2679, 5.0000] The magnitude of this is the minimum distance. This is 6.8087 Am I making any mistakes here? 


#7
Jan314, 09:04 AM

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P: 15,167

Your step 2 is erroneous. Your vector [1.7321,1.7321,0] does not represent a point on the perimeter of the disk.



#8
Jan314, 09:54 AM

P: 82

Yeah I think the problem here is step two where you set the z coordinate to zero and assume that unitvec is still a unit vector. [0.5774, 0.5774, 0.5774] is a unit vector in 3d space, but [0.5774, 0.5774, 0] is not a unit vector on the x,y plane on which the disk lies. You can see for yourself if you evaluate (0.5774)^2+(0.5774)^2.



#9
Jan314, 10:26 AM

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P: 15,167

As I understand the problem, you need to find the distance between some point in 3D space and the perimeter of a disk (i.e., a circle). You do not have to find the point on circle that is closest to the point in question to solve this problem.
Break the problem into parts. Start with the 2D version of the problem, finding the distance between a point [x,y] and a circle of radius r centered at the origin. Hint: You do not need to solve for the coordinates of the point on the circle that is closest to the point [x,y] to determine the distance between this point and the circle. 


#10
Jan314, 11:33 AM

P: 202

so this will give a vector that points towards the position on the x,y plane that the particle is above, and it will have a magnitude of 1. So multiplying this by the radius will give the vector that points to the min point on the perimeter of the disk. 


#11
Jan314, 11:54 AM

Mentor
P: 15,167

You do not need to solve for that point. The problem is much easier to solve by not finding the location of that point.



#12
Jan314, 12:08 PM

P: 202

I need to find the closest distance to the particle. This is so that I can later work the solid angle subtended by the disk. Once I have worked out this distance I can use a similar method to work out the maximum distance, and then using these I can work out the solid angle subtended by the disk.
I cannot think of a way of finding the distance between a point and a circle without first finding the coordinates of the closest point on the circle. Unless I work out a general equation for the distance between coordinates on a circle and a point, and then find where the derivative of this equation equals zero to find the minimum. However this method seems more difficult. 


#13
Jan314, 12:14 PM

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P: 15,167

Think of the point [x,y] as being one point on a circle of some radius R with origin at the center.
What is the distance between two concentric circles, one of radius R and the other of radius r? 


#14
Jan314, 12:27 PM

P: 202

so if [x,y] is a point on the circle with radius R, and [a,b] is the location of the particle, then r will equal (a^2+b^2)^0.5 so the minimum distance between the circle and particle will be Rr I can't work out how to use this method so that it will work in 3 dimensions though 


#15
Jan314, 01:21 PM

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P: 15,167

What kind of triangle is this? (In other words, what is the angle between those two yellowgreen sides?) 


#16
Jan414, 05:45 AM

P: 202

There is a particle at [5,5,5] above a disk with center [0,0,0] and radius 3 R = 3 r = ( (5^2)+(5^2) +(0^2) )^0.5 = 7.071 distance between concentric circles = ((37.071)^2)^0.5 = 4.071 height (the dashed line) = 5 so R_{min} = ( (4.071^2) + (5^2) )^0.5 = 6.448 Is this correct? I hope it is because this method seems much easier, thanks! 


#17
Jan414, 06:03 AM

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P: 15,167

That is correct.



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