Tutorial - 45 degree angles

by Goatmeal

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A Tutorial for Maintaining the Proper Thickness of 45º Angles. Cloning is HIGHLY recommended.
Info Created on 19th March 2011 . Last edited on 20th March 2011.
License Creative Commons
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This purpose of this font is to show how to approximate brick diagonals for making 45º angles.

While working on my "Alex Murphy Dings" font of OCP Logos,

Alex Murphy Dings

I, too, came upon the problem of trying to maintain the proper thickness while using 45º angles. This is where math — especially the Pythagorean Theorem — is your friend.

For a brick 1 unit long by 1 unit high, the diagonal length is 1.414 units long. So, to get a diagonal that is 1 unit in length (to match the brick height), the brick itself must be 0.7071 units long by 0.7071units high. Unfortunately, there is no 0.71 scale brick in FontStruct; everything is set to a 1-unit scale. However, we DO have a brick that has a diagonal length of 1.06 units, which is quite close to 1 unit!

I refer to them as diagonal units in the accompanying pictures:
1/4 Angle Brick = 0.35 --> "0.25" diagonal units
1/2 Angle Brick = 0.71 --> "0.50" diagonal units
3/4 Angle Brick = 1.06 --> "0.75" diagonal units
1 Full Brick = 1.414 --> "1.00" diagonal units

All you have to do is:
(1) Divide the Brick Height by 1.414 (the square root of 2)
(2) Round to the nearest 0.25; this gives you the "diagonal unit"
(3) Add the bricks listed above until they total the desired diagonal unit!

Here is a list of the first 10 Brick Heights and their corresponding Diagonal Units:
01 Brick High = 0.71 --> "0.75" Diagonal Units
02 Bricks High = 1.41 --> "1.50" Diagonal Units
03 Bricks High = 2.12 --> "2.00" Diagonal Units
04 Bricks High = 2.83 --> "2.75" Diagonal Units
05 Bricks High = 3.54 --> "3.50" Diagonal Units
06 Bricks High = 4.24 --> "4.25" Diagonal Units
07 Bricks High = 4.95 --> "5.00" Diagonal Units
08 Bricks High = 5.66 --> "5.75" Diagonal Units
09 Bricks High = 6.36 --> "6.25" Diagonal Units
10 Bricks High = 7.07 --> "7.00" Diagonal Units
And so on...

You can also do fractional bricks:
1.5 Bricks High = 1.06 --> "1.00" Diagonal Units

Of course, the bigger and more complex (and thus more) bricks you use in your FontStruct, the more accurate the thickness of your angle will become.
Comment by Goatmeal 20th March 2011
Comment by Goatmeal 20th March 2011
Thoroughly researched and very enlightling, FS User. Thanks.
Comment by thalamic 20th March 2011
Very intelligent and helpful.
Comment by Ganlo R. Ithsm (xenophilius) 20th March 2011
No wonder my thick and thinnesses are so hit-and-miss-arbitrary. I do everything by eye.
Comment by intaglio 20th March 2011
The love of geometry. Thank you for sharing this useful tutorial. I struggle with a similar problem at the angle of 30 or 60 degrees. Any suggestion? 10/10
Comment by Frodo7 20th March 2011
@everyone -- Thank you all for your kind words. I hope I've provided a possible solution for others struggling with this same problem.
Comment by Goatmeal 20th March 2011
@Frodo7 -- That's a terrific idea! I hadn't thought of those angles, but I'll perform some calculations to see if a similar pattern emerges for the 30º/60º angles as well.
Comment by Goatmeal 20th March 2011
@Frodo7 -- After performing some geometric calculations, I think the 30º/60º angles are far too difficult to come up with a similar solution. The bricks would have to be highly specialized and cannot be made through composites at this time.
Comment by Goatmeal 20th March 2011
thank you. this is so completely useful. With fontstruct it's easy to just subjectively eyeball things and when something's a tad off it's easy to not notice. This helps tighten things back down when "eyeballing" might fail.
Comment by fartheststar 10th September 2012
@fartheststar - You are most welcome. I'm certain there is an easier way to explain it, but I'm often prone to making things overly-complicated. ;^)
Comment by Goatmeal 11th September 2012
Hi Goatmeal. I really like your tutorial. It's very informative for Fontstructors alike. However I have found a few angles that could be improved, by using the advanced composites found in All Composites Saved In My Bricks 8 bricks and 9 bricks. Examples are below.
I understand that your tutorial uses current bricks for simplicity, but you may want to add the ones below as advanced options.
Comment by djnippa 26th October 2014
2 Bricks (Advanced)
Comment by djnippa 26th October 2014
3 Bricks (Advanced)
Comment by djnippa 26th October 2014
8 Bricks (advanced)
Comment by djnippa 26th October 2014
9 Bricks (Advanced)
Comment by djnippa 26th October 2014
@djnippa - Thank you for your contribution to this topic, as your collection (and expertise) of composites is invaluable.

As soon as I get the chance, I will update this topic to add the advanced composite bricks for even greater accuracy! :^)
Comment by Goatmeal 27th October 2014