Help diagramming Axioms and arbitrary angles

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Help diagramming Axioms and arbitrary angles

Postby Ori-JD » July 23rd, 2019, 6:51 am

So everyone knows how to fold a piece of paper according to diagrams - but how exactly does one go about actually drawing the diagrams accurately for axioms like these? Image
(example image from reference finder)

I have drawn full sets of diagrams in Adobe Illustrator before using common angles like 45 and 22.5 degrees, but I'm not sure how to approach these...
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Re: Help diagramming Axioms and arbitrary angles

Postby Kabuntan » July 23rd, 2019, 12:53 pm

The last three picture sets are simple bisectors. I don't know about Illustrator, but seeing how this is a basic function, your software should have a way to do this (CAD software can do it out of the box too).
For the first one, I have no idea right now.
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Re: Help diagramming Axioms and arbitrary angles

Postby origami_8 » July 23rd, 2019, 6:12 pm

So, second try. In the morning I wrote a lengthy reply and when I hit the submit button it told me to log in and all the text was gone (yes, it even sometimes happens to moderators). Unfortunately I then had to leave, but now I'm back home.

First of it depends on the program you are using and other ways may lead to the same result.

My approach to diagramming situations like in the first and last step would be to measure it on a paper sample and then transfer the measurements onto the drawing. Like lets say if I have a 9cm square and the line I want to draw starts 7cm from the left upper corner along the top edge, I'd take this measurement and calculate the percentage like (7/9)*100, what gives me around 77.78%
Then I'd draw a reference line as long as the upper edge of the square and resize it to the calculated value. The reference line should be placed along the upper edge so that one end touches the left corner.
Next on I'd repeat the same procedure on the lower edge, with measuring, calculating, drawing,...
Last step is to connect the ends of the two reference lines with the new fold line and delete the now unnecessary reference lines.

For steps two and three, if the program you use has no way to divide an angle by a set value, it can be done with a reference circle. The circle should be placed with the middle point touching the point where the two lines that you already have meet. Now draw a line that goes from where the edge of the circle meets the one line, to the point where it touches the other line. With most programs I know you can now just add a mid point to that line. (If that doesn't work, resize the line to 50%.) A line that goes through the mid point of the circle and this new point divides the angle. Afterwards the reference circle and line can be removed.
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Re: Help diagramming Axioms and arbitrary angles

Postby Baltorigamist » July 23rd, 2019, 7:33 pm

If you know the length of the exposed portion of the edge, you can use Haga’s Theorem in reverse. In the example above, since the raw edge touches the midpoint of the top, the bottom corner touches the left edge 1/3 of the way up. Some fairly simple algebra and trigonometry should allow you to calculate it for any situation.

To use the inverse of the theorem:
-Solve for the portion of the top edge that is uncovered (in the case above, 1/2).
-Double its reciprocal (to get 4) and subtract 1.
-The reciprocal (1/3) of the result (3) is where the corner touches the opposite edge.
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Re: Help diagramming Axioms and arbitrary angles

Postby origami_8 » July 23rd, 2019, 10:24 pm

You are right, it is one third! Well, that makes things way easier. This way you can make a line the length of one side of the square and place it from the third point through the halfway point on top. Now you make two reference lines connecting the end points of the line with the right corners of the square and find the mid point of these lines to get the line we are searching for in step one. No measurements, no calculations, brilliant.
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