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Calculating the length needed between two bridged circles to produce a given angle / slope

I have two circles which I bridge together. How can I calculate the length / distance required between the two circles to have the angle / slope be 13 degrees

Example:

I create and bridge 2 circles.

1st circle radius 3
2nd circle radius of 1.3

What formula would I need to use to calculate the length / distance required between the two circles to have the angle / slope be 13 degrees?

img1

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2 Answers 2

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From the mnemonic 'SOHCAHTOA', you know tan (theta) = opposite/adjacent ...
so opposite = adjacent * tan(theta) =

enter image description here

so, starting with the top profile, ('opposite' height = 1) .. we know
h = 0.85 * tan (13 degrees)
.. but the (Pythonic) entry fields expect angles in radians.

Just a note: The 0.85 comes from (3.0-1.3) / 2 = 0.85

So to scale the 1 height (here in Y) to the right length, you could SY the top figure, followed by typing:
0.85 * tan(13/360 * tau)
into the 'Y' field of the scale operator's F9 'Adjust Last Operation' panel, which yields about 0.196.

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    $\begingroup$ with a degree in math and 20 years of numerical programming, how have I never seen that mnemonic before? What are you using for your 3d-cad like annotations? $\endgroup$ Dec 31, 2021 at 16:20
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    $\begingroup$ @MartyFouts Ancient English school curriculum, taught by demobbed WWII vets with a Victorian attitude.. :) Looking back from here, it would have been more useful to take the trig. equivalences approach? 2. The shipped 'Measureit' add-on, which can also give you readable vertex-numbers, etc. $\endgroup$
    – Robin Betts
    Dec 31, 2021 at 16:31
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    $\begingroup$ @MartyFouts here's a video of it youtu.be/mnnc_EOSYlQ $\endgroup$
    – Rick T
    Dec 31, 2021 at 18:40
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    $\begingroup$ @Marty Just brought up on /..?n[ui]x/gm/, so I like the way they think. And they're not constantly dumping stuff on my machine, ffs, or nagging me, trying to corner me in to their monopolized ecosystems. $\endgroup$
    – Robin Betts
    Dec 31, 2021 at 20:22
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    $\begingroup$ @RickT Thanks for the edit :) $\endgroup$
    – Robin Betts
    Mar 6 at 8:38
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I'm no good at trigonometry, so I'll just use this online calculator.
It also gives you the right formula if you wanna try by hand.

  • Base is the difference between radius values (3 - 1.3 = 1.7)
  • Angle is 13 degrees

So the Height required is 0.392 units.

enter image description here

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    $\begingroup$ Slick! But the link is to the gif, not the calculator, and I can't seem to find the right one to pop in an edit for you :) $\endgroup$
    – Robin Betts
    Dec 31, 2021 at 11:44
  • $\begingroup$ thanks for the heads up, works now :) $\endgroup$ Dec 31, 2021 at 11:46
  • $\begingroup$ .. and it should be 0.85, not 1.7 .. you forgot to cut your cone in half :) .. (what a PITA I am ...) $\endgroup$
    – Robin Betts
    Dec 31, 2021 at 12:15
  • $\begingroup$ Hey Robin, thanks for checking, I'm no math wiz :D. Though the numbers work out fine when i actually model it... [confused] $\endgroup$ Dec 31, 2021 at 15:14
  • $\begingroup$ I might be up the creek myself / ... justification in my A. $\endgroup$
    – Robin Betts
    Dec 31, 2021 at 16:33

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