Hip/jack side cuts for octagon rafters

[JustCurious]

Hi everyone,

I've recently come across a partial reprint of an old Audel's Carpenter's Guide that depicts portions of a framing square marked with a table for cutting octagon rafters. The text around the figure in question is interesting; it says "no explanation is needed" since the table is to be used just like the standard tables are! That was probably true in 1923 when this little gem was published, but my brain is obviously addled by modern education methods and my complete lack of framing experience.

I assumed the "octagon rafters" the table refers to are those used in the construction of bay windows, turrets, gazebos, and the like, but the table values suggest the framing square maker, Eagle, had something entirely different in mind. I'll post a couple of the table values I can see under the 1" and 2" marks on the tongue then try to explain why I'm confused:


2" 1"
: :
15-5/6 16 SIDE CUT - OCT HIPS
6-1/2 6-7/12 SIDE CUT - OCT JACKS


The text explains that all cutting angles on the Eagle are constructed by using the 12" mark on one side and the table value on the other; the side with the 12" mark gives the cut angle. With the table values above for a rise of 1", the hip rafter side cut angle would seem to come to arctan (16/12), or 53.13 degrees. But, if the hip rafter was on a bay window, the side cut angle where it met the wall would be much closer to 22.5 degrees, would it not?

I can only assume the old framing square table values are correct for their intended purpose, but what that purpose is is not at all clear. Can anyone help the perplexed?

Thanks,

[Joe Bartok]

Just Curious:

Someone posted a similar question in the Timber Framer's Guild Forums.

Octagon Rafter Framing (http://tfguild.org/forums/ubbthreads.php?ubb=showthreaded&Number=3382&page=13)

One of the posters had an idea that (if I'm reading his post correctly) the tables were based on the tangent ... quote "not trig term but carpenters talk long lost" ... end quote. The OP never posted a pic so there was no final answer here.

I'm afraid my brain is also "addled by modern education methods", I have always calculated the framing angles for a complex roof. To me even an ordinary framing square is just a square - I never care about or use any of the tables. Nor am I familiar with other methods that have evolved over the centuries so I can't really explain your numbers. Sorry!

[JustCurious]

Thanks for pointing me back to the TFG forum. The OP was me, but I had given up monitoring the forum when I didn't get a response for about three weeks. I'll go back over the posts there to see if I can suss this out, then post back if I have a EUREKA moment.

Cheers,

P.S. Here's an image of the Eagle square octagon table I'm working with:

http://victoria.tc.ca/~ue191/images/EagleOctagon.jpg

[Joe Bartok]

I'll think about this some more. The tables for the Hip Rafter per foot of run and Jack Rafter length per one foot space from the Hip make perfect sense to me.

I'm as baffled by the other two tables as you are. The values don't fit any definition of a jack rafter or Hip rafter side cut that I'm aware of. I wonder what the instructions "use xxx" and "use vvv" on the end of the square mean? And the corresponding "x" and "v" marks on the square?

Edit ... never mind, these are just marks to find the numbers easily.

[Joe Bartok]

JustCurious, I posted your question in the JLC Rough Framing Forum (http://forums.jlconline.com/forums/showthread.php?t=40023).

There's a lot of knowledge and experience over there. Perhaps some of the older guys have worked with these tables or have a copy of the manual.

[Joe Bartok]

JustCurious ...

Check out the JLC Forum thread linked to in my last post. I was hoping that someone among the forum members would have knowledge of framing methods which have fallen out of fashion. And I was not disappointed.

Curtis Milton posted a diagram which made the mystery length on the framing square table click into place, at least to my satisfaction. I still don't understand why the value of the Hip Side Cut wouldn't be given directly like the Jack Rafter Side Cuts. To my thinking the numbers on the table as given seem cumbersome to work with. But what do I know? The carpenters using that square had to do without the advantages of cheap high tech portable scientific calculators. And modern education methods to addle their brains.

Anyway it looks like the mystery of where the numbers are coming from has been resolved. :cool:

[Topaz]

Hello,
I have just read through that fascinating exchange on the JLC Rough Framing Forum. The "missing link" appears to be Curtis Milton's diagram.
I don't have registration acces to that Forum, would it be possible for someone (and correct netiquette) to post that link here ?
May be ask Curtis first ?
Topaz
PS I just do not understand why Oscar1 was so obtuse (to use a geometrical description). I had the feeling information was being held back - or it was made to sound as if it was . . .

[Joe Bartok]

Hi Topaz:

I contributed to that thread and should add that, for most of that discussion, I was missing what angle was being developed. Ditto for the "side cut angle" the rafter tables posted in this thread are referencing.

The image of the regular octagonal footprint below is how I was thinking of the Hip rafter layout.

http://ca.geocities.com/web_sketches/framing_math_notes/octagon_rafter_arrangements/octagon_backed_hips.gif

But there are other means of assembling the rafters. The layout for the octagon square rafter table and the JLC thread is: four Hip rafters are set first, then the next four Hips are nested between the first four. This changes the side cut angle from how it would be developed or calculated for a roof as shown in my drawing above.

There are many ways of developing the Hip rafter side cut angles using geometry. If you want Curtis' method you should get in touch with him. Here is a link to the drawings he has posted for public access on the Internet:

Never Stop Learning (http://www.curtismilton.com/)

Curtis also posts over at the Timber Framers Guild forums.

......................................

Here is a link to my Geometry and Developments of Hip and Valley Roof Angles (http://ca.geocities.com/xpf51/PROJECTIONS/VALLEY_DEVELOPMENTS.html) page. It doesn't matter if we are talking about a section extracted from the Hip-Valley roof, or cutting a compound angle on the stick, the key to both the geometry and trigonometric formulations is always that tetrahedron composed of five right triangles. (Four triangles compose the tetrahedron ... developing the tetrahedron solves the fifth triangle, which is the Dihedral Angle/Saw Blade Bevel Angle. Here is a link to a slideshow of the Development of the Tetrahedron composed of Four Right Triangles (http://ca.geocities.com/net_geometry/tetrahedron_development_animation/tetrahedron_development_animation.html).)

A couple more links to my home pages ...
Timber Roof Framing and Joinery Angles (http://ca.geocities.com/xpf51/index.html)
Construction Math Formulas, Diagrams and Calculators (http://ca.geocities.com/web_sketches/)

Enjoy those pages while you can. I understand that Yahoo! is pulling the plug on its Geocities web hosting in the near future ...

[Don_P]

Say it ain't so!
I hope you put it all back up on another site Joe, there's a lot of good stuff there.

[Topaz]

Thank for the reply, Joe
There a lot of talk about tangents on Curtis's site but not many circles.
I always thought a tangent was a line that was perpendicular to the normal of a circle. So, do roofers have an "adapted" meaning for tangent ?

I am sure it has been asked before but I could not find the answer and this is as good a place as any to ask:
what drawing software do you use to create such clear graphics ?

I also really do hope that you find a way to salvage your site. "There's a lot of good stuff there" is a serious understatement !
Topaz

[Joe Bartok]

Thank for the reply, Joe
There a lot of talk about tangents on Curtis's site but not many circles.
I always thought a tangent was a line that was perpendicular to the normal of a circle. So, do roofers have an "adapted" meaning for tangent ?

I am sure it has been asked before but I could not find the answer and this is as good a place as any to ask:
what drawing software do you use to create such clear graphics ?

I also really do hope that you find a way to salvage your site. "There's a lot of good stuff there" is a serious understatement !
Topaz
Topaz, most of my diagrams were created as bitmaps using Windows 98 "Paint". The finished drawings are saved as a gif or a jpg.
.....................

This thread and the JLC thread discuss the application of the tangent in the context of the "Hip Rafter Side Cut Angle". I don't think you will need this angle for the garden toolbox you mentioned in your intro post and thread (http://www.construction-resource.com/forum/showthread.php?t=8167) (unless you intend reinforce the corners of the box with "Hips"?).

Before I explain the usage of the tangent ... here are some Wireframe Sketches and Developments of the Hip Rafter Side Cut Angle (http://ca.geocities.com/web_sketches/compound_angle_developments/hip_sidecut_angle_developments/hip_sidecut_angle_developments.html) so you can get an idea of the angle being talked about. This angle is located on the upper shoulders of unbacked Hips-Valleys. The tetrahedra being developed are extracted from the compound angle on the stick.

Scroll down my Geometry and Developments of Hip and Valley Angles (http://ca.geocities.com/xpf51/PROJECTIONS/VALLEY_DEVELOPMENTS.html) and you will find a different means of developing the Hip side cut angles. The angle labelled R4B is the side cut angle at the Hip rafter foot, or, since a Valley is an "upside down Hip", this same angle occurs at the Valley rafter peak. The angle labelled R4P is the side cut angle at the Hip rafter peak or Valley rafter foot (again because the Valley is an "upside down" or "inside out" Hip).

Moving on to the tangent: if I recall correctly, I made this over 16 Development of the Hip Side Cut Angle (http://joe.bartok.googlepages.com/hip_sidecut_lengths.jpg) based on Curtis Milton’s premise. However, the angle developed is for eight Hip rafters meeting one another and applies only to the angle at the Hip rafter peak (so the number is not the same as found on the ocatagon rafter square table). Apologies to one and all if there are errors in that diagram because I have misinterpreted what Curtis was saying (maybe he’ll see this thread and chime in … ?).

The moment you have been waiting for … this is a scan of my Developed Drawing of the Hip Rafter Side Cut Angles based on the Roof Lines (http://ca.geocities.com/net_geometry/image_files/hip_sidecut_trig_101271290.jpg). I show the circle of unit radius and as you can see there are indeed tangents involved in the solution … the perpendicular to the Hip run. Tangent having exactly the same meaning as in trigonometry and/or a tangent to a curve!

This same tangent line or perpendicular to the Hip run can be seen on the first page of the "General Hip/Valley Model", doc or pdf. This document may be found on my Log and Timber Framing Angles … Trigonometry Theory (http://ca.geocities.com/xpf51/LOG_AND_TIMBER_FRAMING_ANGLES/).

One more link: another Development of the Compound Angle related to the Hip-Valley Side Cut Angles (http://ca.geocities.com/net_geometry/image_files/hip_sidecut_compound_101271290.jpg) ... based on the roof lines but all the angles necessary to cut the compound angle are developed. Again we see the perpendicular to the Hip run, in other words, the tangent line.

O.K., that’s it for now. Sorry if it’s information overload, LOL … I don’t hold any information back. :-D

[Joe Bartok]

Say it ain't so!
I hope you put it all back up on another site Joe, there's a lot of good stuff there.
I'm afraid it is so. According to what I'm reading on the Internet the wave of the future is to remain competitive with social networking sites like "Twitter" or "Facebook" and Geocities just doesn't cut it any more.

I'm supposed to receive notification of how to save my online data in the near future. If Geocities idea of "saving" is to pay to host it on their sites ... then sorry, no. I suppose the cost could be financed with Google Ad Sense but it may all end up weighing more than it's worth.

A worse headache is that the site(s) grew "organically", so to speak. Some of the web pages are solutions to actual roofs, some arose in response to participation in forums, some pages are strictly theoretical math tutorials. One page was linked to another as they were uploaded over a period of several years and keeping the links intact if those pages are moved is going to be a nightmare.

So the future doesn't look good ...

[Topaz]

In the posts on this topic in JLC Rough Framing Forum
(topic 40023 - I am not allowed to post the URL)
the strange marks VVV and XXX are briefly discussed but then disappear.
Since the scale for Side Cut Octagon Jack Rafters ("use VVV") are considered to be correct, is it known what the tick-marks engraved at intervals on the 12th's scale of the body are for?
If so, then perhaps the similar X tick-marks can be interpreted and thus cast some light on the controversial Side Cut Octagon Hip Rafter scale.

Just an idea - and I am curious what the tick-marks are for.

Although I have read the JLC Rough Framing discussion several times I am still not sure what the conclusion was.
I understood that the Eagle square had a different idea about how the hips came together at the peak than the obvious one (8 identical hips coming togther at a point).
Would anyone who understands the answer (if there is one) risk an explanation ?

Greetings,
Topaz

[Joe Bartok]

In the posts on this topic in JLC Rough Framing Forum
(topic 40023 - I am not allowed to post the URL)
the strange marks VVV and XXX are briefly discussed but then disappear.
Since the scale for Side Cut Octagon Jack Rafters ("use VVV") are considered to be correct, is it known what the tick-marks engraved at intervals on the 12th's scale of the body are for?
If so, then perhaps the similar X tick-marks can be interpreted and thus cast some light on the controversial Side Cut Octagon Hip Rafter scale.

Just an idea - and I am curious what the tick-marks are for.

Although I have read the JLC Rough Framing discussion several times I am still not sure what the conclusion was.
I understood that the Eagle square had a different idea about how the hips came together at the peak than the obvious one (8 identical hips coming togther at a point).
Would anyone who understands the answer (if there is one) risk an explanation ?

Greetings,
Topaz
I don't feel there was a satisfactory conclusion either. This is the trouble with trying to work backwards from someone else's tables ... it's just about impossible to guess the premise they are working from. It was my hope when I started that JLC thread that one of the other guys had run across this square or the instruction manual and could shed some light on how to use the tables.

Curtis posted a couple more images (from the instruction manual, I think) in that thread, which I intended to attach to this post but it seems my attachment quota has been exceeded by ... holy cow! ... a whopping 3.51 MB?! So no can do ...

Using the math in this drawing (http://joe.bartok.googlepages.com/hip_sidecut_lengths.jpg) based on the drawing Curtis posted [download the "Octagonal Development" pdf file at his website, see Page #18], and making the Hip rafter length equal to 16 (in other words, fudging the numbers), I came up with the table in Post #48 (http://forums.jlconline.com/forums/showpost.php?p=340755&postcount=48) which is reasonably close to what is printed on the octagon framing square. But using the actual lengths that form his tangent plane ... 16 × 14.931856/18.557131 = 12.8742798 ... the framing square table says 14 11/12. I understand his method of developing the lengths and agree with it, but it's not giving the numbers on the framing square table. So I'm still missing something ...

This is a table of Hip Rafter Side Cut Angles at the Rafter Peak ... Post #64 (http://forums.jlconline.com/forums/showpost.php?p=341382&postcount=64) for eight rafters all meeting at the peak, based on the developmental geometry as I would do it [also see Curtis' pdf, Page #22]. This is how I would lay out the side cuts for the Hips on an octagonal gazebo (http://ca.geocities.com/web_sketches/gazebo_square_tail_hip_612/gazebo_square_tail_hip_612.html).

Unless someone can produce an instruction manual in its entirety, some of that octagon framing square such as the purpose of the "X"'s remains a mystery (to me, at any rate). This question has been posted in the TF Guild forum as well as here and JLC and to date no one has yet come forth with the original instruction booklet. :(

[Joe Bartok]

Check this out ...Octagon Framing: Then and Now? (http://www.thestuccocompany.com/construction/Octagon-Framing-Then-and-Now-8311-.htm)

This seems to be the Audel's Carpenters and Builders Guide (http://books.google.com/books?id=QVNXOgAACAAJ&source=gbs_navlinks_s) in question but there's no preview.

[Topaz]

It was that post (Octagon Framing - Then and Now) that got me interested in trying to get this going again; it seems that the Terry there has got a full version of the Audel book. I have registered on that forum and tried to post a question to him but I get an error plus a "please report to Webmaster" type message; which I have done.
I was hoping Terry's copy of Audel would explain the XXX and VVV, which at the moment I think could be vital.

Meanwhile I am studying Curtis's methods via a workshop ppt on his website w3.curtismilton.c** .
There is a post from Terry Branscombe on the TF Guild forum (which kind of started this whole thing). I wonder if that is the same Terry ?
Cheers,
Topaz
PS The forum will not let me post links so this is a rather incomplete post; sorry - but don't blame me :)

[Joe Bartok]

There is a post from Terry Branscombe on the TF Guild forum (which kind of started this whole thing). I wonder if that is the same Terry?

Likely is ... I didn't catch the name in the "Then and Now" post. "Just Curious" is the same person who posted in the TF Guild Forum.

This is a head scratcher, isn't it? Allowing for rounding errors, the length of the tangent line in the development is the length given on the framing square table. But what other number was it used in conjunction with?

Here's something else: on the framing square the Octagon Hip Length per Foot of Run is given as 13.92. Multiplying this by 4/3 to give the Hip length per 16 inches of run:

13.92 × 4/3 = 18.56... close enough to call it the Hip rafter length, which is the dimension the 14 11/12 should be used with to lay out the sidecut angle (see my graphic for a 5/12 octagon roof (http://joe.bartok.googlepages.com/hip_sidecut_lengths.jpg)).

But isn't the reason for the tables being on the square so a carpenter can just go ahead and use them? No further calculations should be necessary. Nor is a knowledge of geometry expected ... if a carpenter knew how to develop the angles he wouldn't need the tables in the first place ...

[Joe Bartok]

Topaz and other forum members, my drawing based on Curtis Milton’s development shows dimensions based on different Hip rafter lengths. Rather than cramming everything on one drawing I should have made three different diagrams based on the same fundamental geometry but with different scales or line lengths. I’m afraid my version is not properly annotated and is lacking in clarity, so I’m going to step back and explain the numbers one at a time. See if you agree or disagree:

Development and Trigonometric Scaling of Side Cut Angle at the Hip Rafter Peak (http://ca.geocities.com/net_geometry/image_files/hip_sidecut_trig_101271290.jpg) … where all the Hip rafters meet at the peak of the roof. The diagram isn’t for an octagonal roof nor does this matter, the logic to produce a formulation is the same for any and all roof slope and plan angle combinations.

Side Cut Angle at the Hip Rafter Peak = R4P
Hip Slope Angle = R1
Plan Angle = DD

From the development …
R4P = arctan (cos R1 ÷ tan DD)
or, using the commonly known names for the angles ...
Side Cut Angle at the Hip Rafter Peak = arctan (cos Hip Slope Angle ÷ tan Plan Angle )

………………..

Moving on to my drawing based on Curtis Milton’s Octagonal Roof Development (http://joe.bartok.googlepages.com/hip_sidecut_lengths.jpg)

For Hip rafters all meeting at the peak we use the formulation for the side cut angle given above. The Plan Angle is 67.5°. The calculation based on the lengths in the development is shown to the upper right hand of the image, or, using the trigonometric formulation:

Side Cut Angle at the Hip Rafter Peak …
R4P = arctan (cos 21.054201° ÷ tan 67.5°) = 21.134562°

………………..

The tables on the infamous octagonal framing square were presumed to be for four Hip rafters nested between four Hips already in place. This means that the Plan Angle is 45°. If you follow Curtis’ development at his site you’ll see that this is so, and furthermore if you scale the lengths, for a Common Run = 16, the Hip Run = 17.318275 and the Hip Length = 18.557131. For this Hip configuration the Plan Angle is 45°. Therefore the length of the Tangent Line (what I call the perpendicular to the Hip run in my diagrams) for a Hip Length of 18.557131 will also equal 17.318275. I’m giving this length as 14.931856 … ignore it for now, I’m going to go back to this shortly. From a Hip Length of 18.557131 and tangent line of 17.318275 …

Side Cut Angle at the Hip Rafter Peak …
arctan (17.318275 ÷ 18.557131) = 43.022238°

Or using a strictly trigonometric formulation, since the Hip Slope Angle is 21.054201° but the Plan Angle is 45° …

Side Cut Angle at the Hip Rafter Peak …
arctan (cos 21.054201° ÷ tan 45°) = 43.022238°

………………..

It was also suggested that the tables on the dreaded octagonal framing square used a run of 16. If this is the case then for a Hip Length of 16, the Hip Run, and since we’re still using a Plan Angle of 45°, the length of the Tangent Line or perpendicular to the Hip Run is …

16 × cos Hip Slope Angle
16 × cos 21.054201° = 14.931856

Using the measurements based on a Hip Length of 16 …

Side Cut Angle at the Hip Rafter Peak …
arctan (14.931856 ÷ 16) = 43.022238°

If you use the formula 16 × cos Hip Slope Angle you’ll find you can reproduce the values on the side cut tables with reasonable accuracy. Hopefully this clears up where the measurements on the diagram are coming from. But it still doesn’t explain the “X’s” and “V’s” on the framing square, nor why the tables give a Hip Rafter Length per foot of run since the side cuts seem to be based on a Hip rafter length of 16.

[Topaz]

I posted a request for info on the Usenet forum rec.woodworking.
They are great documentation collectors !
I am getting some interesting replies, with perhaps genuine copies of the manual for the Eagle square.
Look for thread "Audel Carpenters and Builders library".
Topaz

[Joe Bartok]

Topaz, you said you cannot post a link in this forum. I'm having issues myself of a different nature, I can't post attachments to the forum board nor send private messages. But I can post links so for the other members who wish to follow the discussion here it is:

Audel Carpenters and Builders Library at Rec.Woodworking (http://groups.google.ca/group/rec.woodworking/browse_thread/thread/210ad16c0c864264#)

[Topaz]

Joe, I came across that white-on-black fragment somewhere over the last few days, but I cannot remember where. Can you post a URL ?
I also found it unreliable because the hip configuration is missing (the one you can see is definitely not what the scale sets out). Do we need to see more of the fragment ?
May be that is the answer; the idea of using a 16 on the tongue has come up before and I wonder why it was "rejected" ?
It always try the formula on the shallowest-possible roof, so the side angles tend towards plan angles.

On another subject. I got to (theoretical) roof framing via pyramids on Joe Fusco's site - needed to solve (!) my tool box sides. (PS. I have cut the pieces and am now looking for a way to hold them together securely while I can drill for dowels, glue, nail; or some combination of those options.)
I now try to map everything into pyramids, using letters to mark the various points such as eave corners and the apex. This abstracts the problem away from the terminology. I did this with your tetrahedra and immediately it was much clearer for me. Now I am "following" Curtis's workshop it is agan a vital exercise, otherwise you are just left with "knitting" trying to decode all those intersecting lines :(

Are you interested in following this route and, if so, is this the best place to do it ?
Greetings,
Topaz

[Don_P]

I've asked Rich to check both your permissions to make it easier to post, pm, and add links. I wouldn't worry about the 70 year old copyright myself Joe although that is not up to me. Feel free to keep it going here if you guys want to.

[Joe Bartok]

Joe, I came across that white-on-black fragment somewhere over the last few days, but I cannot remember where. Can you post a URL ?
The urls where I found the images are …

http://forums.jlconline.com/forums/showpost.php?p=341142&postcount=56
http://forums.jlconline.com/forums/showpost.php?p=341198&postcount=57


This abstracts the problem away from the terminology. I did this with your tetrahedra and immediately it was much clearer for me. Now I am "following" Curtis's workshop it is agan a vital exercise, otherwise you are just left with "knitting" trying to decode all those intersecting lines.

Are you interested in following this route and, if so, is this the best place to do it ?

Some people understand the developmental geometry better if it’s expressed in terms of the roof lines. I’ve never been a fan of this method, it seems much easier to develop the tetrahedra in sequence. I also find that making the move from geometry to trigonometry or vector analysis is a piece of cake if one understands the tetrahedron.

Having said that, there’s nothing wrong with learning to develop the angles the way Curtis (and many other timber framers) presents them. You can never have enough math/geometry tools. More “information overload” … here is (some of) my version of developmental geometry based on the the roof lengths …

Developments of Hip-Valley Angles based on the Roof Lines (http://ca.geocities.com/net_geometry/developments_of_roof_lines/developments_of_roof_lines.html)
Tetrahedra and Triangles defining a Hip-Valley Roof (http://ca.geocities.com/web_drawings/hip_roof_triangles_tetrahedra/hv_tetrahedral_models.html)

Maybe you have already come across this while seaching the 'net, another excellent website that covers developmental geometry (you may find the crown molding sections interesting) …

SBE Builders Online Tools (http://www.sbebuilders.com/tools/)

[Joe Bartok]

I've asked Rich to check both your permissions to make it easier to post, pm, and add links. I wouldn't worry about the 70 year old copyright myself Joe although that is not up to me. Feel free to keep it going here if you guys want to.
Thanks Don, that will make life a lot easier.

I made Rich aware of my posting of the images just in case there is a problem in terms of copyright. (Or maybe you got that, I see you're a "Super Moderator"). LOL ... I had to use the "Report Post" option because I couldn't PM.

[Don_P]

Yup the big S on my chest means I got the report too :). I volunteered because I'm east coast and see alot of overnight trash earlier than most folks. I can delete spam and banish ne're do well's but can't change someone's member status... Although I suggested the trial period that is causing Topaz trouble. I think your trouble must be a computer glitch somewhere but still related to your member status. Anyway, we should be able to get you guys easier communications here directly.

[Joe Bartok]

Test post ... just wanted to try attaching images to the board (these are for the octagon framing square).

The following text and images have been moved from a previous post: "You can see in the first image where, for the side cut of an octagon Hip, it says to "use 16 on the end of the tongue and and 14 1/8 on the blade". For a 7/12 octagon roof, four Hips meeting at the peak, I get 16 × cos 28.32155° = 14.08478 tangent line length, so "over 16" makes sense. Too bad the drawing isn't clear on how the Hips meet one another at the peak."

[Topaz]

Joe, briefy about Paint.
Is Win98 Paint more powerful than XP Paint ?
What I think I miss is "groups".
Taking your octagon hip drawing, 8 hips all meeting at a point, I would want to draw one hip as accurately as I can; make the 7 (?) lines of the "master" hip a group; make 7 copies of the group and then rotate and drag the copies into position.
Can you do it like this - or equivalently - in Paint ?
Greetings,
Topaz
Sorry if this should be in Construction Software forum . . .

[Joe Bartok]

Topaz, there's no significant difference between the two versions of Paint that I'm aware of. Nor is there anything like a "Group" function (at least I've never found one). Since the bitmap shows the position of the cursor, to draw the octagon Hips I'd calculate and plot the vertices and center and then connect them with lines. Whenever possible I draw one quadrant of a polygon, then copy, paste and rotate it to save some time and effort.

Having said that: I'm way behind the times in terms of drawing software. The Windows bitmap is fine for the simple geometric drawings I need but it's not much good for decent construction drawings.

Try downloading a free version of Google SketchUp (http://sketchup.google.com/). A lot of guys on various framing/construction forums swear by this software. The tools are intuitive and easy to learn. There are also lots of SketchUp forums where you can get help if you get stuck. LOL ... it might be easy to learn but I still haven't got around to it.

[Topaz]

Joe, thanks for the Paint hint - I might see how bitmaps work-out.
Yes, SketchUp seems to be the way to go. I ordered a ". . . for Dummies" from Amazon.co.uk last night. Also set my version's default startup to be 2D - keep it simple when you start :)
Into the 21st century,
Topaz

[Topaz]

Joe, I'm going backwards.
After puzzling over octagon hip rafter side cuts I thought let's see how "quad" hip rafter side cuts are developed.
I reasoned this way. Take a real regular roof, with a main and the hip having the same pitch, 10/12 for example.
Choose one fixed ( run = 10, for example) and calculate the lengths of the common and hip rafters.
We now have a triangle in the plane of the main roof with sides of common, hip and eave, and
where the common rafter is perpendicular to the eave.

Isn't the hip rafter side cut angle the angle between the hip and common rafters in this plane? Or perhaps the compliment of it, depending upon the orientation of your protractor/square.
Most graphical developments I have seen make it much more complicated than this - am I missing something ?

For example, this link (which I am sure you know :)) sbebuilders.com CA_Roof_Framing_Development.pdf. On page 9 the hip rafter side cut angle is shown as arctan(hip rafter run / hip rafter length).
Or have I got it all wrong ?

BTW, I assume that the hip side cut angle is calculated before the backing angle, although this probably introduces a small error.
Regards, Topaz

[Joe Bartok]

Joe, I'm going backwards.
After puzzling over octagon hip rafter side cuts I thought let's see how "quad" hip rafter side cuts are developed.
I reasoned this way. Take a real regular roof, with a main and the hip having the same pitch, 10/12 for example.
Choose one fixed ( run = 10, for example) and calculate the lengths of the common and hip rafters.
We now have a triangle in the plane of the main roof with sides of common, hip and eave, and
where the common rafter is perpendicular to the eave.

Isn't the hip rafter side cut angle the angle between the hip and common rafters in this plane?
Topaz, can you post images or attachments to the board yet? It's difficult to judge from a text description so I could be wrong, but this sounds more like the Jack Rafter Side Cut Angle. The right angle in the triangle is between the Common Rafter Length and the Eave Line.

For example, this link (which I am sure you know :)) sbebuilders.com CA_Roof_Framing_Development.pdf. On page 9 the hip rafter side cut angle is shown as arctan(hip rafter run / hip rafter length).
Or have I got it all wrong ?
The instructions on Page #9 state that the length of line BC is the length of the perpendicular to line BE, where line BE is the Hip rafter run. The length of line AB equals the length of line BC. Since AB is drawn at right angles to Hip rafter length BD, this make the Hip Rafter Side Cut Angle arctan (Length of perpendicular to Hip Run/Hip Rafter Length) or arctan (AB/BD) and is in essence how both Curtis and I both develop this angle. (Lines AB and BD define a plane, and a plane passing through these two lines follows the slope of the unbacked shoulder of a Hip-Valley).

BTW, I assume that the hip side cut angle is calculated before the backing angle, although this probably introduces a small error.
Regards, Topaz
Generally speaking the tetrahedron of the "fundamental" or "primary" angles needs to be developed first, the angles in the tetrahedron being:

Plan Angle or Deck Angle
Common Slope Angle
Hip Slope Angle
Jack Rafter Side Cut Angle (or the complementary Sheathing Angle)
Backing Angle

Given these angles there is a surprising amount of latitude as to what order we choose to develop the remaining angles. Ditto for calculations using formulas.

When doing a layout on a Hip we can generally rip the backing angle directly with a bandsaw. On a Valley the side cut angle is indeed laid out first, then we rip the trough (backing angle) with a chain saw.

There has never been a problem with errors as far as the geometry or trig goes, the math is always dead on the money. The only errors, small or otherwise, have been self-inflicted human errors.

[Joe Bartok]

Topaz: here are a couple of examples I didn't have time to explain in detail yesterday; hopefully this will clarify the locations of the right angles and which side cut angles are being developed.

Development of Fundamental Roof Angles (http://ca.geocities.com/web_sketches/framing_math_notes/hip_valley_roof_ratios/hip_roof_development.html)... note that (as you said yesterday) the Common rafter length is perpendicular to the eave. The Jack Rafter Side Cut Angle develops on the plane of the roof, between the Common rafter length and the Hip rafter length. Since we've developed the triangle on the roof surface this drawing assumes a backed Hip rafter.

The next diagram has been linked to in prior posts. We don't need the Sheathing Angle and Backing Angle to find the Hip Rafter Side Cut Angles. All we need is the Plan Angle and the rise of the Common Rafter Triangle. This allows us to construct the Hip rafter length, and drawing the perpendicular to the Hip rafter run we have ...

Development of Hip Rafter Side Cut Angles (http://ca.geocities.com/net_geometry/image_files/hip_sidecut_trig_101271290.jpg) ... for the upper/lower shoulder of an unbacked Hip rafter.

Also, yesterday I misspoke in my response to ...
BTW, I assume that the hip side cut angle is calculated before the backing angle, although this probably introduces a small error.
Regards, Topaz
Generally speaking the tetrahedron of the "fundamental" or "primary" angles needs to be developed first, the angles in the tetrahedron being:

Plan Angle or Deck Angle
Common Slope Angle
Hip Slope Angle
Jack Rafter Side Cut Angle (or the complementary Sheathing Angle)
Backing Angle

Given these angles there is a surprising amount of latitude as to what order we choose to develop the remaining angles. Ditto for calculations using formulas.
It's convenient but not necessary to solve the "fundamental five" angles in order to develop or calculate other Hip-Valley roof angles. There may be a heirarchy or order in which the triangles are developed, or not!

You can begin with the funadamental Hip roof angles and develop subsequent set(s) of angles ... tetrahedra extracted from compound angles ... based on different groupings of pairs fundamental angles (for example: Developments of Purlin Angles (http://ca.geocities.com/xpf51/PROJECTIONS/developments_of_purlin_angles.pdf)... the second development is based on the Backing Angle and Sheathing Angle solved in the first development). Or, in theory it's possible to draw or calculate all the angles in a complex Hip-Valley roof, independently of one another, beginning with only the triangles of the Plan Angle and the Common Rafter Slope.

[Joe Carola]

Post was to short.

[Joe Bartok]

Post was to short.
Must be talking about my last two posts ... :D :twisted:

[Topaz]

Joe, Sorry to be so long in getting back - its summer and I spend most of my time up to my elbows turning soft fruit into something like jam.

a) I am not allowed to do fancy things yet, like post links, because I am a Newbie (<20 posts).

b) If we can step outside the maths for a moment, I imagine (in the good old days) you would mark the Hip Side Cut Angle on the top edge of the Hip Rafter with a protractor-like devicee (could be a framing square) and try to get your hand saw to follow the plane formed by the lines Hip Side Cut and Hip Plumb Cut.
Thus my question is the Hip Side Cut marked before or after backing is applied.

c) I cannot find an intuitive explanation of why the Hip Side Cut Angle is as is described on the famous Page #9. I don't doubt that mathematically it is correct, but why ?

d) I got my planes wrong. Before backing and sheathing, there are three planes at the hip: the main roof plane, the adjacent roof plane and the plane made by the unbacked top edge of the Hip Rafter. To try again: the Hip Side Cut Angle is the line formed by the intersection of the top-edge-of-hip plane and the vertical plane through the ridge. Ooph !

I feel I am living dangerously . . .
Topaz

[Don_P]

Try again Topaz,
You're either able to post links and pics or I've sent you to purgatory :)

[Joe Bartok]

b) If we can step outside the maths for a moment, I imagine (in the good old days) you would mark the Hip Side Cut Angle on the top edge of the Hip Rafter with a protractor-like devicee (could be a framing square) and try to get your hand saw to follow the plane formed by the lines Hip Side Cut and Hip Plumb Cut.
Thus my question is the Hip Side Cut marked before or after backing is applied.

Topaz, I have no idea how things were done in the old days. Most modern 1˝× framers don't generally back the Hip, just lay out the side cut and plumb line and cut with a hand saw (if the compound miter saw blade will tilt to the Plan angle we can forget about the side cut entirely). For Hips/Valleys where the width is significant, I wish I had a better answer than this (this is how we do it) ... When doing a layout on a Hip we can generally rip the backing angle directly with a bandsaw. On a Valley the side cut angle is indeed laid out first, then we rip the trough (backing angle) with a chain saw.
Which angle we lay out first depends on what we are cutting. I'll try some pictures:
The Hip rafters you see in the first images in this Photo Gallery (http://ca.geocities.com/xpf51/gallery/PHOTO_GALLERY.html) had the Backing angles laid out on the end grain, then ripped along their length on a bandsaw. No side cut angle layout was necessary.
The Valley rafters in these Cruciform Valley Roof images (http://joe.bartok.googlepages.com/logroofframing) had the side cut angles laid out first, the backing bevel was ripped afterward on the upper shoulders with a chain saw. The side cut angles remain on the lower shoulders and can be seen in the pictures of the Valley rafter/wall intersections.
c) I cannot find an intuitive explanation of why the Hip Side Cut Angle is as is described on the famous Page #9. I don't doubt that mathematically it is correct, but why ?And that, my friend, is my beef with developmental geometry. It's a useful mathematical tool, worth learning, but it doesn't really help the beginner visualize what's happening in the three dimensional world of the Hip/Valley roof. I've tutored a couple of people who took a class(es) at log building/framing conferences and then turned around and asked me what is in essence your question ... why lay out these lines in this order? The drawing on Page #9 of the pdf isn't intuitive, the lines have been moved from their "natural" positions.

The only good answer I have is to make a 3D bristol board model of the tetrahedra. (And yes, there's a bit of "which came first, the chicken or the egg" syndrome here ... to make the model we need to know a bit of developmental geometry). To begin with try making the tetrahedron at the top in these Developments of Hip-Valley Side Cut Angles (http://ca.geocities.com/net_geometry/image_files/hip_sidecut_compound_101271290.jpg). The angle labelled A5P is a saw blade bevel, don't worry about it for the time being.

Once you have a 3D model in hand you'll see how and why the Hip rafter length and the perpendicular to the Hip run work together to produce the Hip rafter side cut triangle.

I don't have any pics of the tetrahedra for the Hip side cut angles but here's something else for you: Images of 3D Model ... Interlocking Tetrahedra forming a Hip Roof (http://www.geocities.com/xpf51/backing_angle_3d_model/backing_angle_3d_model.html). Don't be intimidated by the title, this is simply the triangles of the "fundamental five" angles assembled to produce the tetrahedron extracted fron a Hip roof. There should be enough information in those images so you can piece together how to lay out (develop) the triangles in relation to one another (http://ca.geocities.com/web_sketches/framing_math_notes/hip_valley_roof_ratios/hip_roof_development.html).

[Joe Bartok]

I don't have any pics of the tetrahedra for the Hip side cut angles ...
Correction ... I do now.
d) I got my planes wrong. Before backing and sheathing, there are three planes at the hip: the main roof plane, the adjacent roof plane and the plane made by the unbacked top edge of the Hip Rafter.
Topaz, the attachments are a bit of a mess because I was in a hurry but should be good enough for our purposes ... try the making the two opposite hand tetrahedra as shown in the drawings. No complex instructions here, this is as easy as it gets. And that's why I like using tetrahedra, whether they are composed from the lines of the Hip roof or extracted from a compound angle: no muss, no fuss. The natural lines and planes of the triangles and the relationships between them is illustrated in a plain and simple manner ... the Pythagorean Theorem applied to three dimensions. The drawings and tetrahedra are also mathematical proofs of the Hip/Valley roof angles!

The triangles are scaled to a Hip run = 1; multiply all the lengths in the developments by any factor to produce a scale that's convenient for you to work to. Radius centers and arcs are shown so you can check your layout with a compass.
http://www.construction-resource.com/forum/attachment.php?attachmentid=1671&d=1246989306
http://www.construction-resource.com/forum/attachment.php?attachmentid=1672&d=1246989316
Cut out the developments, fold them and tape them along their edges of equal length (the three lengths we can check with a compass), then juxtapose them at the triangle of the Hip Slope Angle (labelled R1 on the drawings). To help you with the angle values and descriptions here's a link to my Online Roof Framing and Joinery Angle Calculator (http://ca.geocities.com/xpf51/HVFRAMING/framing_angles_trigonometry.html). (The angles in the developments are the calculator default values).

Now that you have a 3D model to study you can see the following ... before the Backing angles are cut (thereby morphing the Hip Side Cut Angles into the main and adjoining side Jack Rafter Side Cut/Sheathing Angles ... this is what happens with the cruciform roof Valleys (http://joe.bartok.googlepages.com/logroofframing)) there is only one plane following the upper/lower shoulder of the Hip rafter. This plane passes through the lines of the Hip length and the perpendicular to the Hip rafter run, and contains the two right triangles defining the side cut angles at the Hip rafter peak (or Valley rafter foot). And hopefully now you can see what lines, and why these particular lines, are being projected to the plane of the paper in the developments linked to in previous posts.

[Topaz]

Thanks Don, I am about to see whether I can crawl out of purgatory.

First, an attachment, being my canonical (not conical:)) Roof-Pyramid.
Oh, I am not allowed attachments yet . . .
So, I will try and put it on a website: link here (http://home.versateladsl.be/rsimpson/images/PyramRoof.jpg)

This is inspired by Joseph Fusco's excellant article on Dihedral Angles (http://www.josephfusco.org/Articles/Dihedral/Dihedral.html).

Joe, (and, hold on, I am going to try a quote now)

The triangles are scaled to a Hip run = 1; multiply all the lengths in the developments by any factor to produce a scale that's convenient for you to work to. Radius centers and arcs are shown so you can check your layout with a compass.
Cut out the developments, fold them and tape them along their edges of equal length (the three lengths we can check with a compass), then juxtapose them at the triangle of the Hip Slope Angle (labelled R1 on the drawings).

Thanks for very useful homework.
I use your calculators all the time to check things out, this is my favorite. (http://ca.geocities.com/web_sketches/hip_valley_dimensioning/framing_working_points.html)
Now a second quote just to see if I can

Now that you have a 3D model to study you can see the following ... before the Backing angles are cut (thereby morphing the Hip Side Cut Angles into the main and adjoining side Jack Rafter Side Cut/Sheathing Angles . . . there is only one plane following the upper/lower shoulder of the Hip rafter. This plane passes through the lines of the Hip length and the perpendicular to the Hip rafter run, and contains the two right triangles defining the side cut angles at the Hip rafter peak (or Valley rafter foot).
It has taken me about 45 mins to compose this reply so I am going to stop here and start looking for cardboard and glue.
Topaz

[Joe Bartok]

... I am going to stop here and start looking for cardboard and glue.
Good luck! Post a couple of pictures of the results if you can.

Here's a bit more on the construction of Hip/Valley roof and compound angle tetrahedra in general. Note how one line naturally follows the next and how all the radius centers are located on one triangle. There are links to more detailed information further down the page.

Slideshow ... Development of Tetrahedron composed of Right Triangles (http://ca.geocities.com/net_geometry/tetrahedron_development_animation/tetrahedron_development_animation.html)

Try the construction above with D = 55.00798° and A = 30.25644°. Compare the results to the 3D Model of the Interlocking Tetrahedra composing a Hip Roof (http://www.geocities.com/xpf51/backing_angle_3d_model/backing_angle_3d_model.html)

Gotta log off for the night so everyone can have some peace and quiet here ... 'til tomorrow ... :twisted:

[Topaz]

Joe, I have made the tetrahedra following your plans, using a 1:10 scale. i.e. the Hip Run is 10 cms long. I would have preferred a larger scale but that is the cardboard that I have available.

Examining these models from all angles , I have been trying to apply your explanation :???: . This bit I can understand:
This plane passes through the line(s) of the Hip length

but this bit I cannot follow:
and the perpendicular to the Hip rafter run,

do you think you could expand that second phrase a bit ?

BTW my models look exactly like those in your article hip_models pdf (http://ca.geocities.com/web_sketches/trig_notes/hip_models_and_developments.pdf) . That's right ?

Something else. This phrase hit me thereby morphing the Hip Side Cut Angles into the main and adjoining side Jack Rafter Side Cut/Sheathing Angles
When morphed, is the HSCA the Sheathing Angle at ridge/hip corner? If so, and we can describe development-wise the morph, this would give us the pre-backing HSCA.

If anyone else is listening, greetings to you too,
Topaz

[Joe Bartok]

BTW my models look exactly like those in your article hip_models pdf (http://ca.geocities.com/web_sketches/trig_notes/hip_models_and_developments.pdf) . That's right ?
The tetrahedra in that pdf would model the Plan or Deck Angle, Common Slope Angle, Hip Slope Angle and Jack Rafter Side Cut/Sheathing Angle. Alas, no Hip Rafter Side Cut Angle in that document.

I don't understand what went astray (just guessing, maybe the developments were cut to the wrong lines ... the triangles of the Common Slope Angles?). I think I'll have time to make the Hip Rafter Side Cut Angle tetrahedra tonight and try to have pictures posted tomorrow morning.

[Topaz]

Joe, hold everything. Very sorry, I made the wrong tetra's - the hip model ones - I didn't realise the significance of the blue cirles. I feel dumber than normal :oops: (I did wonder why the common lengths were not on the sketches !)
I will redo my homework, reading the question carefully this time.
Mortified, Topaz
PS It will be intersting to compare both pairs though . . .

[Joe Bartok]

It will be intersting to compare both pairs though . . .
And here they are. I had to make one model from a heavier weight of cardboard and the lighting sucks but I think we can live with this ... Study of the Side Cut Angles at the unbacked Hip Rafter Peak and the Jack Rafter Side Cut Angles (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_peak_sidecut_3d_model.html). I was going to suggest in any event that you make both pairs of models to more fully understand the relationships between the angles.

In other words, looking at this image ... Comparison of juxtaposed tetrahedra: Hip Side Cut Angles and Jack Rafter Side Cut Angles (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_tetrahedra_overview.jpg) ... if the juxtaposed Hip Rafter Side Cut Angle models on the left were made of wood, and we beveled the 10/12 side and 7/12 side at their respective backing angles (31.63186° and 16.79517°) along the joint (the line of the Hip length) we would have the juxtaposed Jack Rafter Side Cut Angle models on the right.
... I made the wrong tetra's ... Mortified
Don't worry about it. Mistakes are a part of the learning process, speaking as a guy who's made lots of them. When I first started out I used to throw out about three or four pieces of paper for every one worth keeping. It takes time to learn but you'll find the more models you make the easier becomes to grasp the relationships between the different lines, planes and angles. Rather than "information overload" you'll find that you almost develop a sixth sense of how to approach "new" problems you are unfamiliar with.

As you continue to explore the world of complex Hip/Valley roof geometry and math you'll find there are lots more tetrahedra ... each one connected to several others. For example, the Hip side cut angles discussed so far occur at the Hip rafter peak or Valley rafter foot. What about the similar (but not necessarily equal!) side cuts at the Hip rafter foot or Valley rafter peak? Or the side cuts at the unbacked Hip/Valley rafter foot to accommodate square hung fascia? An unbacked Valley peak intersecting a principal purlin?

[Topaz]

Great work Joe ! Will take some serious study.
If I examine your photos, for example here (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_tetrahedra_overview.jpg), then the 10/12 tetra with the long edge is on the left.
On mine it is on the right. I suspect this means I have folded them inside-out. I did that for the first one, then "corrected" it because it did not follow as I read the plan, although now they are not the same orientation as the standard hip models I made by mistake. Perhaps I have Valley Rafter Side Cut Angles :)
If your blue-circled triangle is the base then the sides are folded downwards it seems. Any comment on this ? Perhaps worth warning others !

I have been visualizing the HRSCA plane like a playing card balancing on the Hip Rafter (with the HR thickness reduced to a line). I could not fix its perpendicular. I am also visualizing the HR Side Cut Angle itself as the slope of the arris formed by the intersection of the HRSCA plane and the vertical plane through the ridge. Am I on target there ?
Thanks for your patience, Topaz

[Joe Bartok]

... If I examine your photos, for example here (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_tetrahedra_overview.jpg), then the 10/12 tetra with the long edge is on the left.
On mine it is on the right. I suspect this means I have folded them inside-out ...
This is fine provided you make all your other models inside out or opposite hand. After all, who's to say which side of the roof is the "main slope" (10/12 in this case) or the "adjoining slope" (7/12 for this study)? My assignment of names was arbitrary and it's just as valid to say that the main slope is 7/12 and the adjoining slope is 10/12.
... Perhaps I have Valley Rafter Side Cut Angles ...
Yes, since the corner angle between eaves in plan view is 90°. This isn't necessarily true for a completely irregular roof, say, 8.5/12 and 10/12 slopes at a 120° corner angle. (Stay tuned ... a model of this will be posted in the near future ...)

The best way to compare side cuts for Hip vs. Valley is this: when you've constructed your side cut angle model, flip it upside down. Think of a Valley as an inside-out or upside down Hip, with the trough line sloping down from the level plane of the plan angles. This will locate your side cut angles in the correct positions.
... I have been visualizing the HRSCA plane like a playing card balancing on the Hip Rafter (with the HR thickness reduced to a line) ...
I'm following this sentence but not sure I understand the rest. The plane of your "balanced playing card" must also follow the Hip Slope Angle (25.54245° for our model ... there's a note to this effect in my online photo essay). Where does this plane intersect the level plane of the plan view? Along the line of the perpendiculars to the Hip run!

One leg of the triangle is the Hip length, the other is the perpendicular to the Hip run. Compare these two images ...

Lines of the Hip Sidecut Triangle Hypotenuses (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_roof_sidecut_lines.jpg)
Tetrahedron modeling the Sidecut Angles at the Hip Rafter Peak (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_sidecut_model_shoulder.jpg)

[Joe Bartok]

... Stay tuned ... a model of this will be posted in the near future ...

Attached is a picture of the development for the model of the irregular roof Hip Rafter Side Cut Angles. If the attachment isn’t legible there’s a larger image here … Development of Irregular Hip Side Cut Angles (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/irregular_hip_sidecuts_dev.jpg).

If you don’t know where all the angles are coming from don’t worry about it for now. When you have made this Hip roof Side Cut Angle model, turn it upside down so the face of the side cut angles (the R4 angles) is sloping downward from the level plane of the Plan Angles. There’s the side cut angles for a Valley roof correctly positioned at the peak and foot of the Valley rafter.

The scaling is trigonometric with the Hip run = 1. There are several ways to check the value of the rise on a model of this type …

The Hip Slope Angle for this roof = 33.29403°, the Hip run = 1
Rise = 1 × tan 33.29403° = .65673

10/12 Side Slope Angle = 39.80557°
10/12 Side Common Run = .78807
Rise = .78807 × tan 39.80557° = .65673

8.5/12 Side Slope Angle = 35.31121°
8.5/12 Side Common Run = .92715
Rise = .92715 × tan 35.31121° = .65673

We can also take advantage of the R5 angles…

Checking the 10/12 Side:
.61558 × tan 22.01197° + .78807 × tan 27.36383° = .65673

Checking the 8.5/12 Side:
.37470 × tan 13.82450° + .92715 × tan 31.33650° = .65673

The other image is the model folded into it’s 3d shape, showing the face of the R4 or Hip (or Valley, as the case may be) Side Cut Angles. It’s a corner post with the corners following the Plan Angles and the top cut at the Hip Slope Angle. Btw, the post for a real hexagonal gazebo would of course have square corners rather than following the plan angles like this model. So the angles would be different … but that’s another story … or is it?

[Joe Bartok]

A couple more pics of the "post" model of the irregular roof side cut angles. One view is from the 10/12 side, the other from 8.5/12 the side.

In case you are wondering what the "R5" angles are for, these are the angles defining the intersection of an unbacked shoulder (usually the lower shoulder) of a Hip-Valley with the plumb plane of a wall. For example, the angles of this housing mortise (http://joe.bartok.googlepages.com/vv10_valley_wall_housing.jpg/vv10_valley_wall_housing-full.jpg) to accommodate a Valley rafter foot would be 90° ± R5.

[Joe Bartok]

Topaz, here is another "photo essay" ... LOL, like we really need another one.

The first set of images is a study of the "post" type model of the Hip Rafter Side Cut Angles for a 10/12 and 7/12 roof intersecting at a 90° corner angle in plan view. This is the same roof we made the tetrahedral Hip side cut and jack rafter side cut (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_tetrahedra_overview.jpg) models for, with the same trigomometric scale: Hip run = 1.

A Hip-Valley roof with a 90° corner angle has two pairs of equal side cut angles that make text descriptions of the geometry difficult. It's much easier to visualize the geometry with the 3D models. Try making an opposite hand models as well in order to observe how the side cut angles change location with mirror image (or reversing the "main and "adjoining" roof slopes) and whether we are making a Hip or Valley rafter.

The irregular Hip roof model images that I posted to the forum yesterday have also been added to this page.

Study of the Hip-Valley Rafter Side Cut Angles ... "Post" Type Models (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_sc_post_3d_models.html)

These last few "p:)sts" haven't really been about developmental geometry, but at this stage, in my opinion, it's best to have a variety of models (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_r490_all_models.jpg) at hand to look at. I'm going to be offline for a few days but when I'm back we can step back and look at developing the side cut angles again ... if you are still with us after wading through all this information, that is ... :)

[Joe Bartok]

Topaz, I have added a few more images (two of them are attached, I think you'll find them interesting) and text to the tutorial ...

Study of the Hip-Valley Rafter Side Cut Angles ... "Post" Type Models (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_sc_post_3d_models.html)

Now that you have the "post" type model as a template try constructing the juxtaposed tetrahedral models for the irregular Hip roof (8.5/12 and 10/12 slopes at a 120° corner angle) to satisfy yourself that all four side cut angles are different on the tetrahedral model.

The online photo essay doesn't cover the solid geometry of roofs such as the regular octagonal gazebo that started this thread. Such a roof will have main and adjoining side cut angles equal to one another at the foot of the Hip rafter. The side cut angles at the peak of the Hip rafter will have different values from the angles at the foot, but will also be equal for the main and adjoining sides of the roof. It's time to come full circle ... see if you can make all three models for an octagonal roof: Tetrahedral and Post Type Hip-Valley Rafter Side Cut Angle models, and the tetrahedra extracted from the Hip roof describing the Jack Rafter Side Cut and Sheathing Angles.

[Joe Bartok]

Topaz, when you've made your models try sketching on them like in the photo. Compare your model or the circled tetrahedra in my photo to the wireframe drawings on Pages #1 and #2 of General Hip/Valley Model.pdf (http://ca.geocities.com/xpf51/LTFAPDF/GENERAL_HIP_VALLEY_MODEL.pdf). Do the diagrams in the pdf make sense?

[Joe Bartok]

Something else. This phrase hit me
When morphed, is the HSCA the Sheathing Angle at ridge/hip corner? If so, and we can describe development-wise the morph, this would give us the pre-backing HSCA.
I think this answers your question quoted above ...

Two more model photos ... compare circled areas on these pictures to the sketch of the tetrahedron to the upper right on Page #7 of the General Hip/Valley Model.pdf (http://ca.geocities.com/xpf51/LTFAPDF/GENERAL_HIP_VALLEY_MODEL.pdf).

Also check out the development to the upper right in the Exploded Views of Triangles and Tetrahedra extracted from the upper shoulder of a Hip Rafter (http://joe.bartok.googlepages.com/hip_c5_r4p_r3.jpg). Cutting the upper shoulder of the Hip rafter at the Backing Angle (C5) changes the Side Cut Angle at unbacked Hip Rafter Peak (R4P) to the Jack Rafter Side Cut Angle (P2). Which is also what we are seeing when we look at the other Hip/Valley roof models side by side (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_r490_all_models.jpg).

Btw, if a picture is worth a thousand words how do models of three dimensional solid geometry rate? :) In other words, do the models help with visualizing the angle relationships?

[Joe Bartok]

Having made 3D models and gained insight as to how and which intersecting planes and lines produce the Hip/Valley Side Cut it's time to go back the developmental geometry.

Scroll down this Geometry and Developments of Hip and Valley Roof Angles (http://ca.geocities.com/xpf51/PROJECTIONS/VALLEY_DEVELOPMENTS.html) page and find the developments for angles R4P, the Side Cut Angles at the Hip Rafter Peak (or Valley Foot), and R4B, the Side Cut Angles at the Hip Rafter Foot (or Valley Peak).

Like the solid geometry models, in both drawings the Hip run = 1. The intersections of the perpendicular to the Hip run with the eave lines forms the Hip Side Cut Angle Rise. Note that these lines are the tangents and cotangents of the Main Plan Angle (DD) and Adjoining Plan Angle (D). The Hip length (1/cos R1), which forms the Hip Side Cut Angle Run, is rotated to the line of the Hip run. The tangent and cotangent lines are moved ahead (without changing their length!!!) to the endpoint of the line of the Hip length.

This process develops exactly the same rise and run for the Hip Rafter Side Cut Angles that we see on the tetrahedral models. Using a unit Hip run and scaling the line lengths in terms of the trig functions of the angles also gives us formulas for the side cut angles.

http://ca.geocities.com/xpf51/PROJECTIONS/r4p.jpg
http://ca.geocities.com/xpf51/PROJECTIONS/r4b.jpg

[Topaz]

Wow ! I can't keep up with that. But I have not been idle.
I threw away models and formula's for a moment and tried to go back to first principles : the roof. (Are you still listening?)
See this primitive diagram (attachment. How do I get that reference here?).

Imagine for the moment that this is a flat roof so that Hip run = Hip length. (And I know the "Crown"(?) rafter is missing, and for simplicity the Hip only butts to the ridge.) Also we have given some real width to the Hip Rafter, which results in the triangle ABS.

In this configuration (flat roof) the Hip Rafter Side Cut Angle (HRSCA) is the angle BAS and can be shown to be equal to the (main) Plan Angle.
As we now raise our flat roof, giving it some pitch, AD will increase, as will BC, but not by the same amount: thus AS increases. Since the width of the Hip Rafter has not changed, the HRSCA is changing.

With this insight I think I can show that the HRSCA is arctan(cos (Hip Rafter Pitch Angle) / tan(Plan Angle)).

PS Now, I'm sure this is just the same old thing expressed another way :)

[Topaz]

I would like to go back to the famous page 9 (http://www.sbebuilders.com/pdf/CA_Roof_Framing_Development.pdf) again. See your re-phrasing:
The instructions on Page #9 state that the length of line BC is the length of the perpendicular to line BE, where line BE is the Hip rafter run. The length of line AB equals the length of line BC. Since AB is drawn at right angles to Hip rafter length BD, this make the Hip Rafter Side Cut Angle arctan (Length of perpendicular to Hip Run/Hip Rafter Length) or arctan (AB/BD) and is in essence how both Curtis and I both develop this angle. (Lines AB and BD define a plane, and a plane passing through these two lines follows the slope of the unbacked shoulder of a Hip-Valley).

Unless there is a strange coincidence in the pitches chosen for their example, if the length of BC is always equal to BE why draw BC? I would construct (A)D perpendicular to BD, and then simply arc the length of BE onto that perpendicular to give A.
Perhaps BC is required later, but at page 9 ?
Topaz

[Joe Bartok]

Wow ! I can't keep up with that. But I have not been idle.
I threw away models and formula's for a moment and tried to go back to first principles : the roof. (Are you still listening?)
See this primitive diagram (attachment. How do I get that reference here?).

Imagine for the moment that this is a flat roof so that Hip run = Hip length. (And I know the "Crown"(?) rafter is missing, and for simplicity the Hip only butts to the ridge.) Also we have given some real width to the Hip Rafter, which results in the triangle ABS.

In this configuration (flat roof) the Hip Rafter Side Cut Angle (HRSCA) is the angle BAS and can be shown to be equal to the (main) Plan Angle.
As we now raise our flat roof, giving it some pitch, AD will increase, as will BC, but not by the same amount: thus AS increases. Since the width of the Hip Rafter has not changed, the HRSCA is changing.

With this insight I think I can show that the HRSCA is arctan(cos (Hip Rafter Pitch Angle) / tan(Plan Angle)).

PS Now, I'm sure this is just the same old thing expressed another way :)
That's it. For a zero slope roof AS = Hip Run, BS = perpendicular to the Hip Run. As the slope increases AS = Hip Length.

Setting the Hip run = 1 ...
Rise of Hip Side Cut Angle (at Hip Rafter Peak) = perpendicular to the Hip Run = 1 ÷ tan Plan Angle
Run of Hip Side Cut Angle (at Hip Rafter Peak) = Hip length = 1 ÷ cos Hip Pitch Angle

Solving for tangent of the Hip Side Cut Angle, or rise/run, leads to the formulation:
tan Hip Side Cut Angle = cos Hip Pitch Angle ÷ tan Plan Angle.
I would like to go back to the famous page 9 (http://www.sbebuilders.com/pdf/CA_Roof_Framing_Development.pdf)again. See your re-phrasing:

Unless there is a strange coincidence in the pitches chosen for their example, if the length of BC is always equal to BE why draw BC? I would construct (A)D perpendicular to BD, and then simply arc the length of BE onto that perpendicular to give A.
Perhaps BC is required later, but at page 9 ?
Topaz
The "coincidence" or equality is a result of using a 45° Plan Angle. Generally, BC doesn't equal the Hip rafter run BE. Try making the drawing with a different angle in plan. This is why I try to avoid "special cases" ... 90° corner angles, 45° plan angles, equal roof slopes or equal plan angles. These situations often produce geometry and formulas that apply only to the specific roof conditions one started with.

Line BC in the pdf corresponds to line BS in your attachment ... the perpendicular to the Hip rafter run (line BE). Line BD in the pdf corresponds to line AS in your attachment ... the Hip length. Line BC is "arced" (see Pages #3, #4, #5 and #6), creating the line AB at right angles to BD, thus forming the same triangle you are showing in your drawing; AB = Hip Side Cut Angle Rise, BD = Hip Side Cut Angle Run.

[Topaz]

Joe, I am still on about Hip Rafter Side Cut Angles (HRSCA's).
I still have to study your last post in detail (should be able to put a link here), meanwhile:
I have made a model with an extended roof, a vertical ridge plane and where the roof can hinge about the hip itself (see first attachment 243).
This allows me to tilt the roof up by the Hip Backing Angle and to watch what happens to the Ridge Line with respect to the ridge plane:
it sort of swings out and up as I would expect (see second attachment 244 - my eraser is acting as a crowbar).
I am sure there must be a tetrahedron in here somewhere which defines the HRSCA in terms of the Backing Angle and the Hip Pitch Angle and/or the (complement of the) Sheathing Angle.
I just cannot see it - too few brain cells left.
Can you nail it (sic) ?

[Joe Bartok]

I am sure there must be a tetrahedron in here somewhere which defines the HRSCA in terms of the Backing Angle and the Hip Pitch Angle and/or the (complement of the) Sheathing Angle.

This is the small tetrahedron circled in the two photos of the models attached to Post #52 (http://www.construction-resource.com/forum/showpost.php?p=48102&postcount=52). (There is a tetrahedron in there ... only two of the component triangles are visible on the surfaces of the models).

The same tetrahedron is developed to the upper right in the Exploded Views of Triangles and Tetrahedra extracted from the upper shoulder of a Hip Rafter (http://joe.bartok.googlepages.com/hip_c5_r4p_r3.jpg). The triangle of the Hip Rafter Side Cut Angle (R4P) rotated through the Backing Angle (C5) produces the Jack Rafter Side Cut Angle (P2). This is the simplest means that I have been able to find to describe the geometric transformation from one angle to the other. The Hip Slope Angle (R1) doesn't play a role in this particular development.

Nice work on the models. How the different triangles relate to one another may seem confusing now but keep experimenting, it will all come together.

[Joe Bartok]

These photos portray the same tetrahedron shown circled in Post Type Model of the Hip-Valley Side Cut Angles (http://www.construction-resource.com/forum/attachment.php?attachmentid=1692&d=1247583669), and the same tetrahedron is developed to the upper right in the Exploded Views of Triangles and Tetrahedra extracted from the upper shoulder of a Hip Rafter (http://joe.bartok.googlepages.com/hip_c5_r4p_r3.jpg). Since this tetrahedron extracted from the post type model is way too small to work with the model in the photo is scaled to allow it to be nested atop the tetrahedron extracted from the 7/12 side of the Hip Roof. All angles and proportions remain the same. The development is scaled to a Hip run = 1.

When morphed, is the HSCA the Sheathing Angle at ridge/hip corner? If so, and we can describe development-wise the morph, this would give us the pre-backing HSCA.

In my opinion this tetrahedron best illustrates the geometry between the Hip Side Cut Angle at the Hip Peak (R4P) and the Jack Rafter Side Cut Angle (P2). The side cut angle trigonometric relationships resulting from the tetrahedron geometry …

tan R4P = tan P2 ÷ cos C5
tan P2 = tan R4P × cos C5

or, using the longer more commonly known names for the angles …

tan Hip Side Cut Angle at the Hip Peak = tan Jack Rafter Side Cut Angle ÷ cos Backing Angle
tan Jack Rafter Side Cut Angle = tan Hip Side Cut Angle at the Hip Peak × cos Backing Angle

Geometrically: rotating the Jack Rafter Side Cut Angle through the Backing Angle (division) produces the Hip Side Cut Angle at the Hip Peak, conversely, rotating the Hip Side Cut Angle at the Hip Peak through the Backing Angle (multiplication) produces the Jack Rafter Side Cut Angle.


EDIT: I've uploaded one more photo comparing the "post type" Hip Side Cut Angle model, and the tetrahedron extracted from the upper shoulder of the Hip rafter sitting atop the 7/12 Side Jack Rafter Side Cut Triangle. The two models are not the same ... note the plumb line on the post model as opposed to the right angle on the nested model. The two juxtaposed models on the right represent enlarged tetrahedra extracted from the post type model on the left.

[Joe Bartok]

And that’s enough of that particular model, time to move on …

I am sure there must be a tetrahedron in here somewhere which defines the HRSCA in terms of the Backing Angle and the Hip Pitch Angle and/or the (complement of the) Sheathing Angle ... Can you nail it (sic) ?
A different group of angles so this requires a different tetrahedron, for the time being here’s the formula:

tan Hip Side Cut Angle at the Hip Peak = tan Backing Angle ÷ tan Hip Slope Angle

I’ll post a solid geometry model of this relationship in the near future. Or try to ... I have a sketch of the theoretical geometry but as drawn it's a mind bender, I'll see if I can reduce it to simpler elements.

[Joe Bartok]

And that’s enough of that particular model, time to move on …
Ooops! Spoke too soon. We are not yet done with the tetrahedron extracted from the upper shoulder of the Hip rafter, in fact we need to add another tetrahedron to it!

The attached images depict the development of the juxtaposed tetrahedra, the location on the post type model the tetrahedra are extracted from, and a view of the triangle of the Backing Angle. There is no particular scale for this development and model, I used a protractor to lay out the angles. A wireframe sketch of this model can be found to the upper right on Page #3 of General Hip/Valley Model.pdf (http://ca.geocities.com/xpf51/LTFAPDF/GENERAL_HIP_VALLEY_MODEL.pdf), exploded views are on Page #5.

The tetrahedron is mounted on a triangle which, when viewed from its far face (see photo in next post), defines the Hip Slope Angle. This triangle also represents a plumb section plane through the post type model, the section being taken through the Hip run.

[Joe Bartok]

Three more views of the tetrahedra from differing perspectives.

I am sure there must be a tetrahedron in here somewhere which defines the HRSCA in terms of the Backing Angle and the Hip Pitch Angle and/or the (complement of the) Sheathing Angle.
I just cannot see it - too few brain cells left.
Can you nail it (sic) ?
These are the tetrahedra which define most of the angles you specified ... alas, the "(complement of the) Sheathing Angle" or Jack Rafter Side Cut Angle isn't visible on the faces of this model, it's buried between the abutting faces of the two tetrahedra.

Here again is the formula relating three of the angles ...

tan Hip Side Cut Angle at the Hip Peak = tan Backing Angle ÷ tan Hip Slope Angle

... and as usual there are all kinds of other formulas connecting the other angles.

Likely the reason you are having a hard time seeing this tetrahedron is that the Backing Angle (in fact the entire tetrahedron!) has a rather unusual orientation (have a look again at the picture in my previous post (http://www.construction-resource.com/forum/attachment.php?attachmentid=1709&d=1248289952)).

Are you thoroughly confused yet? :)


EDIT: The images in my last few posts have been collected on one web page ...
Tetrahedra relating the angles on the upper shoulder of a Hip Rafter (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_upper_shoulder_3d_models.html)

[Topaz]

Joe, I have been a week away in France. You have given me a lot to study again !
I have been "in the field". See these attachments of a valley to post configuration:
goto attachments, pity can't put links here

Just to give me a road-map, which tetrahedron shows the rotation of the (complement of) the sheathing angle by the backing angle, using the hip rafter as hinge ? Are two tetra's required because a single one has no natural right-angle corners (in roof plane: ridge line, hip rafter and a line joining them is not a right triangle)?

Added by edit: You have a page which gives many trig. relations between Hawkindale angles. I cannot find this page back; can you provide a link? On that page you refer, AFAICR, to angles CV and CL. What are these?

[Joe Bartok]

Just to give me a road-map, which tetrahedron shows the rotation of the (complement of) the sheathing angle by the backing angle, using the hip rafter as hinge ? Are two tetra's required because a single one has no natural right-angle corners (in roof plane: ridge line, hip rafter and a line joining them is not a right triangle)?
Tetrahedra relating the angles on the upper shoulder of a Hip Rafter (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_upper_shoulder_3d_models.html) "shows the rotation of the (complement of) the sheathing angle by the backing angle, using the hip rafter as hinge". One is the component tetrahedra Exploded Views of Triangles and Tetrahedra extracted from the upper shoulder of a Hip Rafter (http://joe.bartok.googlepages.com/hip_c5_r4p_r3.jpg); the development and construction is shown first on the "Tetrahedra relating the angles on the upper shoulder of a Hip Rafter (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_upper_shoulder_3d_models.html)" page. The component tetrahedron is simply composed of the fundamental roof angles, but rotated on its axis. The triangle of the Jack Rafter Side Cut Angle (P2) and the complementary Sheathing Angle (90° – P2) is sandwiched between these two tetrahedra.

Added by edit: You have a page which gives many trig. relations between Hawkindale angles. I cannot find this page back; can you provide a link? On that page you refer, AFAICR, to angles CV and CL. What are these?
I'm not sure which page you are looking for. Material related to geometry is on my Geometry and Developments of Hip-Valley Roof Angles (http://ca.geocities.com/xpf51/PROJECTIONS/VALLEY_DEVELOPMENTS.html). Trigonometric relations arising from the tetrahedra are at my Table of Hip and Valley Roof Framing and Joinery Angle Formulas (http://ca.geocities.com/web_sketches/angle_formula_catalogue/angle_formula_catalogue.html).

Many of the angles on my web pages, both developed and expressed as formulas, are not Hawkindales. I used Hawkindales where I could, my thinking being that the formulas would be more familiar to timber framers. Not to mention steel detailers ... try Googling "Martindale Angles". Unfortunately Hawkindales don't all "fit" or express all possible relations in a complex Hip roof, and in many cases I modified the names (a case in point being our current Hip Side Cut Angles ... B appended after R4 indicates the angle is at the foot of the Hip rafter, P indicates the angle occurs at the peak. Ditto for the R5 and A5 angles.) For many angles I simply made up a name.

And I should clarify my statement above: "Hawkindales don't all 'fit' or express all possible relations in a complex Hip roof" ... neither do my formulas or geometry! Hip-Valley math has been an ongoing journey that never seems to end. While the basics are sorted out (indeed, the knowledge has been around for centuries), every now and again a problem comes up that requires a new way of looking at things ... more head scratching, more drawings, more formulas ... :taz:

[Joe Bartok]

Just to give me a road-map, which tetrahedron shows the rotation of the (complement of) the sheathing angle by the backing angle, using the hip rafter as hinge ? Are two tetra's required because a single one has no natural right-angle corners (in roof plane: ridge line, hip rafter and a line joining them is not a right triangle)?
Topaz, my response yesterday was somewhat lacking in clarity. The Jack Rafter Side Cut-Sheathing triangle is common to two tetrahedra, which are nested or juxtaposed at this triangle. Look again at this picture of the model, this is in essence a section view through the Hip rafter ...

Juxtaposed Tetrahedra viewed from the triangle of the Backing Triangle (http://www.geocities.com/xpf51/hip_peak_sidecut_3d_model/hip_us_c5.jpg) ... the triangle of the Backing Angle is the "hinge", not the Hip Slope triangle.

(1) Rotating the Jack Rafter Side Cut-Sheathing triangle through the Backing Angle produces the Hip Rafter Side Cut Angle at the Hip peak.

(2) Rotating the Jack Rafter Side Cut-Sheathing triangle through the complement of the Backing Angle produces the Hip Slope Angle.