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When I have an elliptical stairway to layout on my shop floor, I will draw the outer ellipse with a jig that I can adjust to any semi-minor and semi major axis that I need. I then draw the inside ellipse and then divide both ellipses into equal runs. One thing I have always noticed is when I extend these riser lines, they are always tangent to a circle as shown. This circle is of course tangent to the semi-minor and semi-major axis.
Can anyone prove the relationship of either of the ellipses to the radius of the circle that these extended lines are tangent to.
Joe: This would be most interesting with your graphics.
Ted: Ken: and all: Any grey matter left to churn after Kens last problem?
Replies
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So I have to go skiing for a week starting tomorrow..Sugarbush Area..Thankfully!
Should be a hundred posts to read when I return...
to near the stream,
aj
Looks interesting Stan...good luck boys...
*Enjoy aj!fv
*Stan,I think this problem/puzzle goes into the "non-trivial" category. But it also goes into the "very intriguing" category, at least for the math-types...Maybe AJ will get a "vision" while he's on the slopes, and provide an answer when he returns...Stan, I'm playing the skeptic, and not assuming that what appears to be a circle in your real-life layouts is actually a circle. I made that "part 1" of your question.Below is a sketch which illustrates the problem, simplified to only three steps in the staircase. I wanted to get a labeled sketch out early to make it easier to talk about.Joe F, maybe you can provide a better sketch. I'm not posting my .DWG file because even though it looks pretty accurate, it isn't. I just eyeballed the lengths of the elliptical arcs. I don't know any easy way to make AutoCAD find points equally spaced along the elliptical arc, and I'm too lazy to compute their coordinates mathematically.
*Ted: The perimeter of the outside ellipse is divided into equal divisions, or 'runs'. The inside ellipse is divided into equal runs as well. This assures that the inclined laminate projected from these points will be a straight laminate. Notice how the intersections of the riser with the outside stringer start square, become increasingly acute, then square up. The inside stringer starts square, becomes increasingly obtuse, then squares up. These make a very unique looking stairway. Obviously each tread is a different pattern. Again, I just have always noticed this relationship of these extended riser lines on my shop floor during layout.
*Stan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe: Thanks for the input. AS expected, a nice graph to view. I must make it clear that I am assuming the lines point to a circle. I can only draw one situation at a time and it is very time consuming. I have not tried any extreme cases. Something else I must point out is the outside stringer is a true ellipse. The inside stringer is drawn with a set distance perpindicular to the tangents of the outside ellipse. I do this to keep the stair width the same. This in reality makes the inside stringer NOT a true ellipse, but rather close to one. I am just looking for some excellent input from you, Ted, Ken, AJ, anyone else who has some insight on this.
*Stan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe,Any chance you labeled the "true ellipse" and the "plotted ellipse" backward? I believe the true ellipse should be on the outside.Stan, the picture below shows a problem that would occur if you use an offset in an extreme case.
*Ted,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Stan,Ted and Joe are doing a good job of sorting this out, so I'm just going to kick back and watch, and get ready for Superbowl. My first thought when I read your problem was, "what are concentric ellipses, and if they do exist, and can be defined, is that what you have here in your problem?"As I said, Joe and Ted are sorting this out for us.It's POSSIBLE that the circle that you mention actually does exist, regardless of whether or not the inside curve is a "true ellipse", and that there is an exact relationship between the radius of the circle, and the other variables involved, but I'm not prepared to comment on that aspect of the problem at this time.
*Joe,I guess I don't know what you mean by plotted ellipse. I thought your plotted ellipse was supposed to represent the one Stan constructs for the inside edge of his stair layout. How is your plotted ellipse constructed?
*Ted,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe: I had 167.22 inch radius figured for the 126 inch arc at 110.32 inch rise. My figures are in post #148 of " circular stair problem #3".
*Joe,Thanks, that cleared up the plotted ellipse for me. Now I think Stan needs to clear up something for us. In an earlier post he stated: "The inside stringer is drawn with a set distance perpindicular to the tangents of the outside ellipse."The nose edges of the stairs are not perpindicular to the tangent to the outer ellipse, so they will have slightly varying lengths.Your plotted ellipse makes all of the noses equal in length (30" in your last sketch), so I think we should be using the 30" offset if we want to duplicate what Stan is doing. By "offset" I mean the kind AutoCAD creates, which is along the perpindiculars to the tangents.I'd like Stan to reaffirm that he's measuring in along a perpindicular rather than along the noses of the steps.
*Ken: My situation has an inside stringer equal distant like Joe so well plotted. My understanding of ellipses is that it is impossible to have two ellipses equal distant from each other.Ted: On your exteme case graph, I would plot the inside stringer on lines perpendicular to the tangents of the outside ellipse. However, the inside stringer would be divided up equally and then the riser lines projected on through these points, continuing on to this "questionable circle" they seem to be tangent to. On my elliptical stairs to date, I just have noticed that these lines seemed to point to a circle that has a radius that is directly related to one of the ellipses.
*Stan,When you construct the oval for the inner stringer, do you measure in a fixed length along the riser line, or along a line which is perpindicular to a tangent on the outer ellipse?
*Ted: I measure a fixed length perpendicular to the tangent of the outside ellipse.
*Ted: Heres an elliptical stairway taken at an angle to show the ellptical stringers. This inside stringer is equal distant on lines perpendicular to the outside stringer.
*Stan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe,I'm glad to see you're continuing interest in the circular handrail thread ( Circular Stair Problem #3). It certainly is not a dead issue.Why not post your new chart, and information, over there, instead of in this new thread?Many of us, including Ted, Stan, AJ, and obviously yourself, have more information that we'd like to post in that thread. I for one, intend to build at least one more model to study the behavior of the plywood for tighter radii, in an effort to understand the twisting and bending factors that are involved, as well as the formula itself. I know that Ted has some additional information to post over there also.Since you've introduced the concept of radian measure into your new chart, I'd like to see you begin by giving an explanation of radian measure, a concept that the average poster has little understanding of.But once again, in that thread, not this one. Let's keep the 2 threads separate.Ken
*Here is a sketch of an ellipse which may have greater eccentricity than Stan usually encounters.The inner green oval was drawn as an offset from the outer red ellipse. This is in accordance with Stan's statement: "Ted: I measure a fixed length perpendicular to the tangent of the outside ellipse."It appears that the blue lines are not forming a circle. This is not a proof, just guiding evidence.Note: The 10 arcs of "equal length" in the sketch are not exactly equal, having an error of about 3%. I don't think correcting this error would be enough to noticeably change the appearance of the blue lines.
*Ted,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe,I did use the offset command in AutoCAD to produce the inner oval because that's the way Stan says he's constructing it. The line segments representing the risers are not equal in length, because they are not orthogonal (perpindicular to a tangent) to the outer ellipse. Stan says he measured inward a fixed distance along a line perpindicular to a tangent (which is the same thing the AutoCAD offset command does).I've attached my drawing file.
*Ted,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Ted: Your graph clearly shows that the risers extensions are not tangent to a circle. Evidently the stairs I have layed out were such that they had riser extensions very close to a circle. It must just have been been the right ratios to make the lines tangent to a circle. My last one that I drew out had the outer ellipse with a semi-minor axis of 78 inches-the semi-major axis was 118 inches. The inside stringer was plotted 48 inches away from the outside stringer-and perpendicular to it. Thanks for your input.
*Joe,Exactly my thoughts. In a previous post I asked Stan pretty plainly to reaffirm that he was measuring along the perpindiculars, and he said he was. It's an interesting problem either way...
*Stan,Perhaps extending those riser lines using your layout technique creates an ellipse, but not a circle? And maybe as the ellipse you use for the stairs becomes closer to a circle, the object the riser lines are outlining becomes closer to a circle?I'm just speculat'n. I drew a true ellipse inside a true ellipse, then divided them into equal arc lengths, and it appeared that the extended risers may have been outlining a circle then, even though the ellipses were very "narrow". Maybe I'll find it and post it.
*Joe,Exactly my thoughts. In a previous post I asked Stan pretty plainly to reaffirm that he was measuring along the perpindiculars, and he said he was. It's an interesting problem either way...
*Stan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
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View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe,That creates a nice looking oval, but it isn't an ellipse.You can see that easily if you make the major and minor axis equal. You should get a circle, right? But instead the curve you get is a parabola. A parabola is a conic section unrelated to an ellipse.But it does give you an oval with a nice shape.
*Here's a sketch showing the curve when the major axis is the same length as the minor axis. The yellow curve formed is a parabola.
*Ted,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe: I have used that method for easing differently inclined stringers together. Ted: I am hoping there is a definate relationship that can describe what these riser extensions are tangent to. I do not have the drawing program to try different ratios quickly. It takes me a half hour to draw just one different scenerio.
*Ted,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Stan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Stan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe: I can tell you have "been there-done that"That picture is exactly what I have done. Of course, a thin laminate works nice also. I will check out that drawing program. I presently have Design cad.
*Yep.And you can see a parabola when you turn on a water fountain, and you can see a hyperbola in the shadow made above a lamp which is against a wall.An ellipse is related to a hyperbola but not a parabola. If you have a good imagination, a pair of hyperbolas is an inside-out ellipse. Given two points, the difference of the distances from the points on the hyperbola is constant, whereas the sum of the distances from the points on the ellipse is constant.Shhhh!!Can you hear the sound of people clicking on their "back" buttons? :)
*Ted,Thanks for the interesting and accurate information. As we all know, pictures can be deceiving.
*Ted: Good description. I feel this little problem really has some kind of complex answer. If these lines aren't tangent to a circle, then I feel they are tangent to some kind of predictable graph. This is more a curiousity than anything for me. Ever since I layed out my first elliptical, this has always been in the back of my mind. The last elliptical I did was a year ago, and I didn't take the time to check this out on my full scale shop floor layout.
*Ted,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Any two fixed points and a constant for the distance defines a hyperbola. And yes, those two points are referred to as the foci for that hyperbola.
*Stan,Your layout starts with a "nice" geometric shape (the ellipse) whose properties are easily described. Then you form that inner oval using a perpindicular offset. That's where things get ugly.Since you're offsetting from a curve, the points are not simply translated (slid around), but are moved a fixed amount in varying directions. The result (as you mentioned earlier) is not an ellipse, nor any other special curve. I can write a nasty looking equation to describe the points along your inner stringer. But do you stop there? Noooooooo...... You then divide the arcs into N equal lengths. To figure the arc lengths for the outer stringer we'll have to use an elliptic integral (and probably for the inner oval too).Then we have to describe the riser lines in terms of those lengths and the equations for the curves, and try to figure out if they're all tangent to the same circle.I'd rather be fishin'...I may pursue this using an ellipse to approximate the perpindicular offset you use just to see if something nice shows up. I share your curiosity in the puzzle...
*Ted: I was afraid that my non-elliptical inside stringer would cause problems. I can see you are capable to pursue it to the nth degree, but it sounds like a lot of work. Please don't for my sake. I am curious, but not that much. I appreciate the effort you and Joe have put in this already. I figured it would pull your chains.Joe: Thanks for the link to the drawing program. I have not been able to download it completely yet. Someone seems to call during the 2 hour download, and my phone line shuts off to the computer. I am going to try late tonight.
*Stan,
View Image © 1999-2000"The first step towards vice is to shroud innocent actions in mystery, and whoever likes to conceal something sooner or later has reason to conceal it." Aristotle
*Joe: Yea. My last attempt at the free download, I was at 99%. My Real Download won't play it yet.
*Stan,There are programs you can run on your computer to download files in a more reliable way. These programs can resume downloading a file in the event that you lose your connection. You can also intentionally pause the downloading (so someone can make a phone call, for example) and then resume later. A few servers can't resume sending a file, but most of them can.I use one called MyGetRight for downloading larger files. It's freeware, and you can download it here (1.1 Megabytes). After installing it and running it, just use Ctrl/Alt/click on a link and MyGetRight will start downloading the file.It's a small program and not very intrusive.
*Ted: Thanks for the download program. I just installed it and will try to download Joes drawing program tonight.
*Just trying to see if there is any milk left in this thread before it goes to the archives.
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When I have an elliptical stairway to layout on my shop floor, I will draw the outer ellipse with a jig that I can adjust to any semi-minor and semi major axis that I need. I then draw the inside ellipse and then divide both ellipses into equal runs. One thing I have always noticed is when I extend these riser lines, they are always tangent to a circle as shown. This circle is of course tangent to the semi-minor and semi-major axis.
Can anyone prove the relationship of either of the ellipses to the radius of the circle that these extended lines are tangent to.
Joe: This would be most interesting with your graphics.
Ted: Ken: and all: Any grey matter left to churn after Kens last problem?