Re: Cylindrical Dial
Hello Roger, You wrote : My suggestion to calculate the hour lines is to treat the cylinder like a polygon with a series of flat vertical faces. Calculate the lines for each face the usual way as a series of vertical declining dials. Lay out the design for each facet and draw a smooth curve through the mid points for each face to approximate the hour angles. Because a cylinder can be described mathematically it is rather easy to calculate all the wanted points for a dial on the outside of a cylinder. The cylinder also can be unfolded to a flat plane so a drawing can be made. It depends on the dimensions of the tower and the length of the gnomon what the result will be. In the picture you see such a dial on a vertical cylinder with radius 250 units with a perpendicular gnomon of 25 units, pointing south, for latitude 45 degrees, reading local suntime. The hourlines are curves, not straight. Best wishes, Fer. Fer J. de Vries [EMAIL PROTECTED] http://www.iae.nl/users/ferdv/ Eindhoven, Netherlands lat. 51:30 N long. 5:30 E Attachment converted: Macintosh HD:cyl1.gif (TIFF/JVWR) (FFAF)
Re: Cylindrical Dial
Excellent! We now have two elegant solutions proposed: Tony Moss's orthographic projection and Fear de Varies's mathematics. I need to study both to understand and use them. Tony, your sketch is worth 10,000 words. I see from the sketch how things are laid out and points projected to different planes. But I am having some trouble with the words as you seem to be starting at the solution (P1) and working back to find the location of the sun. I would thing you need to start with the time and time angle and work towards the point on the cylinder from there. I have modified you sketch to try to do that. I hope this file is small enough to get through the filter. The explanation of this sketch is as follows. Define a new plane perpendicular to the style. On this equatorial plane, the style which is parallel to the earths axis, projects as a point and the trig on as a circle. The sun moves around this plane as the time angle (t), 15 degrees per hour. For any time angle, project the intersection with the circle back to the edge view and down to the plan view to establish time angles on the elliptical projection of the trigon. As you have done, project the sun's ray through the axis and onto the cylinder at (P1) and then up to establish (P2). As you proposed then move the trigon along the axis of the style to (Pn) to repeat for other points on the hour line. From this point of view shouldn't the trigon touch the edge of the cylinder so the various projections for (Pn) form a cone around the style axis? You are correct, the orthographic projection technique works but it is a tedious procedure to establish sufficient points to fix the hour lines. Maybe I should have another look at the mathematical solution. Roger Bailey Walking Shadow Designs N 51 At 07:49 PM 3/4/00 +, Tony Moss wrote: Roger Bailey Contributed SNIP Calculate the lines for each face the usual way as a series of vertical declining dials. Lay out the design for each facet and draw a smooth curve through the mid points for each face to approximate the hour angles. I am sure there are more elegant solutions but the mathematics is beyond me. Your challenge remains. I couldn't begin to contemplate a numerical approach to laying out such a dial but to do so by simple orthographic projection is fairly straightforward, though perhaps somewhat tedious. One picture is worth 10 000 words so I'm taking the liberty of including a small GIF (43k) showing how this can be achieved. Tony Moss Attachment Converted: c:\eudora\attach\PlotCylr.gif Attachment converted: Macintosh HD:Cyl-RTB.gif (GIFf/JVWR) (00010117)
Re: Cylindrical Dial
Roger Bailey contributed: SNIP Tony, your sketch is worth 10,000 words. I see from the sketch how things are laid out and points projected to different planes. But I am having some trouble with the words as you seem to be starting at the solution (P1) and working back to find the location of the sun. I would thing you need to start with the time and time angle and work towards the point on the cylinder from there. SNIP To avoid making the diagram too complex I left out the way in which the perpendicular view of the inclined trigon is achieved from a true circular view. Your re-draw supplies what I left out Roger. For those who are similarly puzzled I've re-worked the original diagram leaving out some of the previous detail but showing how it is done in drawing board methodology although I must admit that I cheated and simply 'squashed' a pre-drawn trigon electronically in Adobe illustrator sent herwith to my JPEG sub-list as a GIF. (on request to others as it won't pass Daniel's file size filter)
Re: Cylindrical Dial
At 07:29 PM 3/1/00 +0100, Alain MORY wrote: Hi, all diallists ! I'm asked to realise a vertical sundial on a circular tower. I don't know exactly (it's a truism !) how to proceed. This sundial will probably be built on stone, but we can't curve the stone. Then I think that it will be possible to realise like a multiface dial. What does the sundial list think about such a challenge ? Completely crazy or just a half ? Bonjour Alain, Your question is a good challenge. The answer is not easy. There is a good example of this type of dial in the town of Anduze in southern France (Gard). The Cadrans Solaires Francais Catalogues describes the setting as a tour de l'horloge. The cylindrical tower now has a clock and a sundial but its original purpose was a watch tower or barbican, a remnant of the fortifications around the old town. The dial demonstrates some of the features of a dial laid out on a cylinder. As Dan Wenger mentioned, it is useful for only a few hours as the cylindrical plane of projection curves away from the gnomon. The Anduze dial is essentially a noon mark, showing hour marks for only one hour on either side of noon. It has declination lines with zodiac marks so it shows the seasons well. It also shows that the hour lines are curves on the cylindrical surface. The curvature increases with the time from noon as the shadow moves around the cylinder. A simple model will demonstrate these effects. Stick a pin in a paper tube at an angle equal to your co-latitude. Hold it in the sun and rotate it to show different time angles. You will find that the shadow of the gnomon pin is curved and the dial is only useful for about two hours before and after noon. My suggestion to calculate the hour lines is to treat the cylinder like a polygon with a series of flat vertical faces. Calculate the lines for each face the usual way as a series of vertical declining dials. Lay out the design for each facet and draw a smooth curve through the mid points for each face to approximate the hour angles. I am sure there are more elegant solutions but the mathematics is beyond me. Your challenge remains. The alternative but practical way to lay out the dial is to do it on site. Install the gnomon, facing south at a slope to the vertical equal to your co-latitude. At local noon the shadow will be vertical. For each time increment from noon, mark the shadow of the gnomon on the cylinder surface for each hour line. This may take some time as a cloud will pass just as the time gets critical. Roger Bailey Walking Shadow Designs N 51 W 115
Re: Cylindrical Dial
All, I think a multiface dial would be the most practical way to go. However, what kind of shadows would be cast by a sphere (say double the size of the cylinder) impaled on the tower? ++ron - Original Message - From: Roger Bailey [EMAIL PROTECTED] To: Alain MORY [EMAIL PROTECTED]; Alexei Pace [EMAIL PROTECTED] Cc: sundial@rrz.uni-koeln.de Sent: Saturday, March 04, 2000 9:14 AM Subject: Re: Cylindrical Dial At 07:29 PM 3/1/00 +0100, Alain MORY wrote: Hi, all diallists ! I'm asked to realise a vertical sundial on a circular tower. I don't know exactly (it's a truism !) how to proceed. This sundial will probably be built on stone, but we can't curve the stone. Then I think that it will be possible to realise like a multiface dial. What does the sundial list think about such a challenge ? Completely crazy or just a half ? Bonjour Alain, Your question is a good challenge. The answer is not easy. There is a good example of this type of dial in the town of Anduze in southern France (Gard). The Cadrans Solaires Francais Catalogues describes the setting as a tour de l'horloge. The cylindrical tower now has a clock and a sundial but its original purpose was a watch tower or barbican, a remnant of the fortifications around the old town. The dial demonstrates some of the features of a dial laid out on a cylinder. As Dan Wenger mentioned, it is useful for only a few hours as the cylindrical plane of projection curves away from the gnomon. The Anduze dial is essentially a noon mark, showing hour marks for only one hour on either side of noon. It has declination lines with zodiac marks so it shows the seasons well. It also shows that the hour lines are curves on the cylindrical surface. The curvature increases with the time from noon as the shadow moves around the cylinder. A simple model will demonstrate these effects. Stick a pin in a paper tube at an angle equal to your co-latitude. Hold it in the sun and rotate it to show different time angles. You will find that the shadow of the gnomon pin is curved and the dial is only useful for about two hours before and after noon. My suggestion to calculate the hour lines is to treat the cylinder like a polygon with a series of flat vertical faces. Calculate the lines for each face the usual way as a series of vertical declining dials. Lay out the design for each facet and draw a smooth curve through the mid points for each face to approximate the hour angles. I am sure there are more elegant solutions but the mathematics is beyond me. Your challenge remains. The alternative but practical way to lay out the dial is to do it on site. Install the gnomon, facing south at a slope to the vertical equal to your co-latitude. At local noon the shadow will be vertical. For each time increment from noon, mark the shadow of the gnomon on the cylinder surface for each hour line. This may take some time as a cloud will pass just as the time gets critical. Roger Bailey Walking Shadow Designs N 51 W 115
