Hello everyone.

I've been trying to help my nephew with his math quiz from school but it is difficult enough to do in Moi that I can't solve it yet.

So we provided with 3 lines:
- 1 horizontal blue line with length of X√2
- 2 magenta lines with length of X

The task is to arrange them in specific ways so that: they form a continuous polylines with each endpoints are exactly on a perfect circle circumverence (red circle), like in the provided illustration (it's actually not an acurate drawing I made in Moi). The real task is to form a pentagon with 4 magenta lines (length = x) and 1 blue lines (length = x√2), but I believe we can ignore the other 2 since they'll be added later by mirroring the 2 existing magenta lines horizontaly.

It's really difficult to rotate both the Magenta Lines so that they meet the specified condition above!

A friend told me that the problem could easily done with 'constraints' feature, but I REALLY want to solve it just using Moi3D.

Please tell me what I have to do? Thank you very much in advance.

Assume without loss of generality that x = 1 unit. Scale for higher values of X.

(Currently researching the problem, and equations. Maybe numeric solution???)

Let 2*theta be the radial angle of the sqrt(2) chord.
Let 2*phi be the radial angle of the 1 unit chord.
2*theta + 8*phi = 360 degrees = 2*PI
theta = PI - 4*theta
Let r = radius of circle.
sin(theta) = sqrt(2)/2r = 1/(sqrt(2)*r)
sin(phi) = 0.5/r = 1/(2*r)

Recall Pythagorean theorem also:
https://byjus.com/maths/chord-of-circle/

Hi Bem... thank you very much for the reply and link. I will definitely study them since it's kinda cool.

However, I prefer 'non-mathematical' solution with involves no calculation at all. Instead, I would like to know if using only MOI, the problem can be solved.

For a better illustration: there are many many tutorial on youtube to model a DODECAHEDRON, which most of them require us to enter some 'predefined' numbers (be it length or angle) to successfully create the geometry. However, there is one tutorial here (https://www.youtube.com/watch?v=_XDg57dO94E&t=116s), he is a friend of mine, that successfully model the geometry inside Moi WITHOUT any number inputs!.. So, I would like to know this kind of method in solving the math problem I posted before.

Hope you all can understand what I mead, and pardon for my english.

Thank you very much in advance.

Hi Elang

So I am dropping math formulas, which may have required numerical methods anyway.
MoI has some intrinsic way of making curves match up...

Establish the base line of sqrt(2), by typing it in the command window when creating a horizontal line.
Create an edge pentagon, with base of sqrt(2).
The other 4 sides are too long, so the final circle is contained inside the pentagon.
Create a line of 1 unit along the right leg of the pentagon, and also a perpendicular bisector, which will pass through the eventual final circle center.
Mirror will create the left leg.
Rotating the right leg, will rotate the left leg, due to History.
Need to tie the upper two arms in somehow.
Many 3 point circles will pass through the two base points, and the top of the right leg.
...???...

- Brian

So the other piece of the puzzle is to use Orient LineLine to copy lower right leg to upper right arm.

I thought that a way to do it would be to position a circle of radius X centered at point A2. Then point B is on that circle and it just needs another geometric condition to intersect with the circle but I haven't found that other condition yet.



- Michael

So I have begun making many mistakes, so time to call it for the night.

The circle is approached by successive approximation.
The Radius of the circle is slightly larger that 0.928 * X. (And less that 0.9322 * X.)
The Angle of the base chord of sqrt(2), to the right lower leg, is slightly larger than 97.758235 degrees.

The unit chord to unit chord angle is (slightly larger I think), than 114.819736 degrees.

Place base chord of sqrt(2) units. Its end points are two points on many 3 point circles.
Place start of right leg chord segment of 1 unit, (with perpendicular bisector), at right end of base chord, at an angle of 97.758235 degrees. Its upper end point is the 3rd point on the candidate circle. Create said circle.

Using Orient LineLine, with copy checked, move a copy of right leg with perp bisector, with lower endpoint and upper end point, to upper end point, towards the upper center of the candidate circle.
The 1 unit is SLIGHTLY too long. Left hand two chords can be mirrored, but to not add anything to the proceedure.
So the candidate circle is slightly too long.

Rotate right leg chord segment and perp bisector, to slightly larger angle than 97.758235, and repeat proceedure.


https://imgbox.com/T04618V9

- Brian

I thought History might be helpful, (or not).

Maybe a NODE could be created, which performs the Rotation of unit chord, 3point circle, and OrientLineLine ???
Then something to check the ERROR of the Top Center intersection point versus the top center of the circle.
Make successive corrections to the angle.

- Brian

ps
I used MoI Dimension Angle to measure angles.
The upper unit chord angle is 114.844058, for the candidate circle.
Maybe the average with the other 114+ angle could be useful?

The perpendicular bisectors are not quite intersecting at the successive circle centers.

Does History work with Orient Line/Line?

- Brian

The average of the unit chord to unit chord angles is 114.831897.
But I doubt that the change with right leg chord rotation is linear.

Construct the two right hand chords to this angle, and rotate until the top centerline is reached, and check the two new 3point circle candidates...

- Brian

I may be wrong, but Orient Line/Line does not seem to snap to an intersect with a line?

- Brian

Hi Brian,

re:
> Does History work with Orient Line/Line?

it does if you use it with "Make copies" enabled and then select the generated object and use Edit > History > "Enable update".

- Michael

Maybe I over simplified it???

For an irregular polygon in a circle, the "Sum of all central angles is 2π".
The sum of the outer internal angles is 180 * (number of sides - 2). 180*(5-2) = 540 degrees.
So with the unknown being the rotation angle of the right unit chord,
the angle (~114.82 degrees) between two unit chords can be calculated, with a little simple math.

So devise a method to draw the right chord leg, and the right chord arm, with the changing rotation, (~97.75degrees), and snap the endpoint to top center, as the pair of chords is rotated.

https://math.stackexchange.com/questions/3184676/finding-the-interior-angles-of-an-irregular-polygon-inscribed-on-a-circle

- Brian

(114.82 * 3) + (97.76 * 2) = 539.98 = 540 degrees.
https://math.stackexchange.com/questions/3047152/if-the-total-number-of-angles-in-a-polygon-is-180n-2-why-are-there-just-36

Use radians instead of degrees...

3*phi + 2*theta = 3*PI with angles in radians

Simplifying, phi = PI - (2/3) * theta. (In radians)

phi is the inside angle, unit chord to unit chord.
theta is the inside angle between sqrt2 chord and connected unit chord.

Does History work with Dimension Angle. So that when an angle is increased by rotation, the Dimension Angle auto changes?
Is Dimension Angle Value available to some script or node? Does it have a radian display mode?
Is here a History manipulation Node?

I could not understand Burr's video. Is it a solution to the PentagonChord puzzle?

Thinking of MoI doing the Math, with a Snap to endpoint vs intersection or closed polyline, or ???, as the right leg is rotated, with dynamic angle update.

Or rotate pair of unit chords?
Or mirror unit chord pair, and check for closure to left endpoint of sqrt2 chord?
Rotation or mirror handle?

- Brian

Hi Elang,

I think it's impossible to have all sides touching the circle if one side is X√2 and other sides are x.
x are too short.
The answer would be a bird's house.

I scaled up by 10, for better visibility.
Base is now 10*sqrt(2).
Unit chords are now 10 units in length.

Initial MoI construction:
A right angle triangle, with angles 90, 45, 45, with the two perpendicular sides of length 10 units, has a hypotenuse of 10*sqrt(2).
Place the 10*sqrt(2) in the horizontal base position.
Place a future chord line segment of 10 units from the right end point, to the +x direction. It can be rotated to yield the right leg chord.
The inner angle theta is between 97.784758 degrees, and 97.7891905 degrees.
To use MoI's Rotate command, the angle is the supplement, (180 - theta) degrees. This rotation angle is about 82.2108095 degrees (plus a tiny amount).
((The rotation circle was pointed out by Michael.)

The angle between two scaled unit chords is Phi = 180 - (2/3) * theta.
Phi is (a little less than), approximately 114.8072063333 degrees.
To Rotate the right hand leg chord, to the right hand arm chord, use a Rotation angle of (360 - Phi), which equals (180 + (2/3) * theta).
So the angle to type in the Rotate box is (a little more than) 245.1927936667 degrees. (which is 114.x07... degrees on the other side of the angle.)

The independent variable is the angle theta, when rotating the right leg chord.
The three point circle, and the other 3 unit chords, and angle Phi, need to be updated during the rotation, and have a Snap, when the size of the circle, and the end points of the four 10*unit chords match up.

I need to learn more about History, and how to use it to do rotations, and 3 point circle, so that they will auto update, with theta angle change.

Or try to do a Node.




Alternate link:
https://imgbox.com/mfMcu1JY

The above image is a close approximation, but does not quite solve the problem.

- Brian

Manually doing trial Theta and Phi cord placement, to narrow down the values, takes too much time.

Play with trying to do History with 3 point circle, and History with two rotations, did not seem to work.

Hello all... Thank you very much for thoroughly replies.

I'm very sorry that I'm out of town for about a week (it's been 2 days till now), and I don't have access to Moi for this time being. So I'll keep reading your replies and will be continue to solve the task based on all of your replies immediately.

Cheers!

Dear BurrMan...

How do you define the circle radius/diameter at the very first place so it fits the categories?