RE: Walking Shadow Riddle
Thanks to all who speculated about the walking shadow riddle. I think Dave Bell's answer was probably the best because he actually proposed a shape of the curved path (a half ellipse) at a mid latitude location. And it seemed to make sense, although my math isn't good enough to prove him right or wrong. But there was no real consensus or agreement if I read your letters correctly. If you want to reconsider the riddle in simpler terms, consider this average situation, avoiding the extremes of the equator or the poles: The trek begins at sunrise on the Equinox at Greenwich England at 51 degrees North. What would the curved path look like on a map of England? (Assume England is perfectly flat and has no trees, bodies of water, streets or buildings. And forget about all minor distracting details like head nodus diameter, apparent diameter of sun, refraction, altitude, footstep irregularities, etc.) John From: Dave Bell [mailto:db...@thebells.net] Sent: Tuesday, September 06, 2011 6:43 PM To: 'John Carmichael'; 'Sundial List' Subject: RE: Walking Shadow Riddle Interesting question!! I couldn't get my head around it, either, so tried plotting it in Excel. Grabbing a table of Solar Azimuth for Tucson on today's date, I estimated a 5 mph rate. Since the Sun rises just barely South of East, the path starts out heading North and mostly West. By noon, our trekker has reached 25 miles West and 12 miles North of his starting point. By sunset, he has walked back East to a point 25 miles North of his start. The path looks like half of an ellipse with an East-West major axis of roughly 50 miles, and a North-South minor axis of slightly over 25 miles. December 21st looks a lot different, and I'm having trouble buying it! Dave _ From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of John Carmichael Sent: Tuesday, September 06, 2011 11:08 AM To: 'Sundial List' Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I'm sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: walk towards your shadow all day, starting at sunrise and stopping at sunset at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can't figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He'll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he'll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he's at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
RE: Walking Shadow Riddle
If Meeus has a solution then he has different ones for each latitude and day of the year...and it will include the ridiculous graph of a day at the North Pole. There, the sunrises on about March 21 and sets after 183 24 hour periods on about September 21! I'd love to see Meeus' graph of those circles. The other end of the spectrum would be the approximately straight line walked by the guy at the equator on an equinox, walking west and then returning. Does he include these? Fritz From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of Gent, R.H. van (Rob) Sent: Wednesday, September 07, 2011 1:21 AM To: 'sundial@uni-koeln.de' Subject: FW: Walking Shadow Riddle Although I do not want to spoil the fun for those who want to figure it out by themselves, the general problem was solved in detail by Jean Meeus in Mathematical Astronomy Morsels III (2004), pp. 341-346 [Pursuing the Sun]. Here the walker always faces the sun while it is above the horizon but the answer is of course symmetrical to that of a walker always following his shadow. rvg From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of John Carmichael Sent: dinsdag 6 september 2011 20:08 To: 'Sundial List' Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I'm sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: walk towards your shadow all day, starting at sunrise and stopping at sunset at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can't figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He'll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he'll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he's at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
RE: Walking Shadow Riddle
Hi, You will probably find this link useful http://www.jgiesen.de/pursuit/index.html rvg From: Fritz Stumpges [mailto:fritz_stump...@fluidcomponents.com] Sent: 07 September 2011 17:25 To: Gent, R.H. van (Rob); sundial@uni-koeln.de Subject: RE: Walking Shadow Riddle If Meeus has a solution then he has different ones for each latitude and day of the year...and it will include the ridiculous graph of a day at the North Pole. There, the sunrises on about March 21 and sets after 183 24 hour periods on about September 21! I'd love to see Meeus' graph of those circles. The other end of the spectrum would be the approximately straight line walked by the guy at the equator on an equinox, walking west and then returning. Does he include these? Fritz From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of Gent, R.H. van (Rob) Sent: Wednesday, September 07, 2011 1:21 AM To: 'sundial@uni-koeln.de' Subject: FW: Walking Shadow Riddle Although I do not want to spoil the fun for those who want to figure it out by themselves, the general problem was solved in detail by Jean Meeus in Mathematical Astronomy Morsels III (2004), pp. 341-346 [Pursuing the Sun]. Here the walker always faces the sun while it is above the horizon but the answer is of course symmetrical to that of a walker always following his shadow. rvg From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of John Carmichael Sent: dinsdag 6 september 2011 20:08 To: 'Sundial List' Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I'm sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: walk towards your shadow all day, starting at sunrise and stopping at sunset at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can't figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He'll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he'll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he's at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
RE: Walking Shadow Riddle
Hey, he is pretty good with his numbers! Fritz -Original Message- From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of Gent, R.H. van (Rob) Sent: Wednesday, September 07, 2011 8:43 AM To: sundial@uni-koeln.de Subject: RE: Walking Shadow Riddle Hi, You will probably find this link useful http://www.jgiesen.de/pursuit/index.html rvg From: Fritz Stumpges [mailto:fritz_stump...@fluidcomponents.com] Sent: 07 September 2011 17:25 To: Gent, R.H. van (Rob); sundial@uni-koeln.de Subject: RE: Walking Shadow Riddle If Meeus has a solution then he has different ones for each latitude and day of the year...and it will include the ridiculous graph of a day at the North Pole. There, the sunrises on about March 21 and sets after 183 24 hour periods on about September 21! I'd love to see Meeus' graph of those circles. The other end of the spectrum would be the approximately straight line walked by the guy at the equator on an equinox, walking west and then returning. Does he include these? Fritz From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of Gent, R.H. van (Rob) Sent: Wednesday, September 07, 2011 1:21 AM To: 'sundial@uni-koeln.de' Subject: FW: Walking Shadow Riddle Although I do not want to spoil the fun for those who want to figure it out by themselves, the general problem was solved in detail by Jean Meeus in Mathematical Astronomy Morsels III (2004), pp. 341-346 [Pursuing the Sun]. Here the walker always faces the sun while it is above the horizon but the answer is of course symmetrical to that of a walker always following his shadow. rvg From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of John Carmichael Sent: dinsdag 6 september 2011 20:08 To: 'Sundial List' Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I'm sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: walk towards your shadow all day, starting at sunrise and stopping at sunset at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can't figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He'll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he'll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he's at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Walking Shadow Riddle
Hi Rob, A wonderful site. Thanks for the link. best wishes, Peter On 7/09/2011 11:43, Gent, R.H. van (Rob) wrote: Hi, You will probably find this link useful http://www.jgiesen.de/pursuit/index.html rvg From: Fritz Stumpges [mailto:fritz_stump...@fluidcomponents.com] Sent: 07 September 2011 17:25 To: Gent, R.H. van (Rob); sundial@uni-koeln.de Subject: RE: Walking Shadow Riddle If Meeus has a solution then he has different ones for each latitude and day of the year...and it will include the ridiculous graph of a day at the North Pole. There, the sunrises on about March 21 and sets after 183 24 hour periods on about September 21! I'd love to see Meeus' graph of those circles. The other end of the spectrum would be the approximately straight line walked by the guy at the equator on an equinox, walking west and then returning. Does he include these? Fritz From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of Gent, R.H. van (Rob) Sent: Wednesday, September 07, 2011 1:21 AM To: 'sundial@uni-koeln.de' Subject: FW: Walking Shadow Riddle Although I do not want to spoil the fun for those who want to figure it out by themselves, the general problem was solved in detail by Jean Meeus in Mathematical Astronomy Morsels III (2004), pp. 341-346 [Pursuing the Sun]. Here the walker always faces the sun while it is above the horizon but the answer is of course symmetrical to that of a walker always following his shadow. rvg From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of John Carmichael Sent: dinsdag 6 september 2011 20:08 To: 'Sundial List' Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I'm sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: walk towards your shadow all day, starting at sunrise and stopping at sunset at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can't figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He'll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he'll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he's at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial -- -- Peter Mayer Politics Department The University of Adelaide, AUSTRALIA 5005 Ph : +61 8 8303 5606 Fax : +61 8 8303 3443 e-mail: peter.ma...@adelaide.edu.au CRICOS Provider Number 00123M --- This email message is intended only for the addressee(s) and contains information that may be confidential and/or copyright. If you are not the intended recipient please notify the sender by reply email and immediately delete this email. Use, disclosure or reproduction of this email by anyone other than the intended recipient(s) is strictly prohibited. No representation is made that this email or any attachments are free of viruses. Virus scanning is recommended and is the responsibility of the recipient. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Walking Shadow Riddle
My guess is he ends up where he started, i.e., the wise man is telling him, in the usual pseudo-wise ways of pseudo-wise movies, that he already has the treasure but just doesn't realize it. On Tue, Sep 6, 2011 at 2:08 PM, John Carmichael jlcarmich...@comcast.netwrote: A Riddle: ** ** I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I’m sure this relates to sundials and mapping, but the answer eludes me. ** ** One of the characters was told by the wise man to: “walk towards your shadow all day, starting at sunrise and stopping at sunset” at which point the walker would discover the location of a treasure. ** ** So I asked myself, *what would the path of the trek look like on a map?*But I can’t figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He’ll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he’ll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he’s at the North Pole? ** ** John C. ** ** ** ** --- https://lists.uni-koeln.de/mailman/listinfo/sundial --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Walking Shadow Riddle
Here's my guess: If, as you suggest, you start at dawn on the equinox and on the equator you will walk east for six hours before noon and west for six hours before sunset, finishing at the point where you started. If you start at the north pole at the equinox you will move in a spiral, crossing each longitude once every 24 hours. John Lynes From: John Carmichael jlcarmich...@comcast.net To: 'Sundial List' sundial@uni-koeln.de Sent: Tuesday, 6 September 2011, 19:08 Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I’m sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: “walk towards your shadow all day, starting at sunrise and stopping at sunset” at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can’t figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He’ll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he’ll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he’s at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial --- https://lists.uni-koeln.de/mailman/listinfo/sundial
RE: Walking Shadow Riddle
Interesting question!! I couldn't get my head around it, either, so tried plotting it in Excel. Grabbing a table of Solar Azimuth for Tucson on today's date, I estimated a 5 mph rate. Since the Sun rises just barely South of East, the path starts out heading North and mostly West. By noon, our trekker has reached 25 miles West and 12 miles North of his starting point. By sunset, he has walked back East to a point 25 miles North of his start. The path looks like half of an ellipse with an East-West major axis of roughly 50 miles, and a North-South minor axis of slightly over 25 miles. December 21st looks a lot different, and I'm having trouble buying it! Dave _ From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of John Carmichael Sent: Tuesday, September 06, 2011 11:08 AM To: 'Sundial List' Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I'm sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: walk towards your shadow all day, starting at sunrise and stopping at sunset at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can't figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He'll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he'll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he's at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
RE: Walking Shadow Riddle
I believe that Brad had it right in terms of the philosophical intent of the question poser. But we must plunge ahead into the actual mechanical outcome. Dave, I believe you have approximately the right shape path for a walker with an apparent sun path that crosses his meridian south of his zenith. Continuing with that setup and the walker being on the equator to begin his walk at sunrise, I question whether the walker ever makes it back to the same line of longitude from whence he started. By walking away from the sun he has ended up extending the time from 6 hours between sunrise and solar noon to 6 hours plus. After solar noon, he is retreating from the sun and shortening the time to sunset to less than 6 hours. Hence, he winds up at a location north of his starting point, but longitudinally west of his starting longitude. In John's first setup with the walker on the equator on the equinox, the sun passes through the zenith so the walker would only be treading on the equator, but using the same argument above, heading west until solar noon would take 6 plus hours and the time spent heading back east would be under 6 hours so a constant speed walker would fall short of returning to his starting point. Larry Bohlayer From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of Dave Bell Sent: Tuesday, September 06, 2011 9:43 PM To: 'John Carmichael'; 'Sundial List' Subject: RE: Walking Shadow Riddle Interesting question!! I couldn't get my head around it, either, so tried plotting it in Excel. Grabbing a table of Solar Azimuth for Tucson on today's date, I estimated a 5 mph rate. Since the Sun rises just barely South of East, the path starts out heading North and mostly West. By noon, our trekker has reached 25 miles West and 12 miles North of his starting point. By sunset, he has walked back East to a point 25 miles North of his start. The path looks like half of an ellipse with an East-West major axis of roughly 50 miles, and a North-South minor axis of slightly over 25 miles. December 21st looks a lot different, and I'm having trouble buying it! Dave _ From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of John Carmichael Sent: Tuesday, September 06, 2011 11:08 AM To: 'Sundial List' Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I'm sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: walk towards your shadow all day, starting at sunrise and stopping at sunset at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can't figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He'll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he'll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he's at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial
Re: Walking Shadow Riddle
My intuitive solution was similar. You end up south of your original position as there is no north going component to bring you back to the start during daylight. The curve varies with latitude and declination but the results is the same. In spite of what Dorothy said in the Wizard of Oz, There is no place like home, go south to discovery the treasure. John, I guess you can say Been there, done that. Regards, Roger Bailey Walking Shadow Designs Life's but a Walking Shadow Been there, done that. From: Dave Bell Sent: Tuesday, September 06, 2011 6:42 PM To: 'John Carmichael' ; 'Sundial List' Subject: RE: Walking Shadow Riddle Interesting question!! I couldn't get my head around it, either, so tried plotting it in Excel. Grabbing a table of Solar Azimuth for Tucson on today's date, I estimated a 5 mph rate. Since the Sun rises just barely South of East, the path starts out heading North and mostly West. By noon, our trekker has reached 25 miles West and 12 miles North of his starting point. By sunset, he has walked back East to a point 25 miles North of his start. The path looks like half of an ellipse with an East-West major axis of roughly 50 miles, and a North-South minor axis of slightly over 25 miles. December 21st looks a lot different, and I'm having trouble buying it! Dave From: sundial-boun...@uni-koeln.de [mailto:sundial-boun...@uni-koeln.de] On Behalf Of John Carmichael Sent: Tuesday, September 06, 2011 11:08 AM To: 'Sundial List' Subject: Walking Shadow Riddle A Riddle: I was watching a dumb movie last weekend and there was a bit of dialogue that caught my attention. I'm sure this relates to sundials and mapping, but the answer eludes me. One of the characters was told by the wise man to: walk towards your shadow all day, starting at sunrise and stopping at sunset at which point the walker would discover the location of a treasure. So I asked myself, what would the path of the trek look like on a map? But I can't figure it out. This is as far as I get in my thinking: I started considering an example with these conditions.- The walk begins at dawn on the equinox, and the man is on the equator. And the walk ends at sunset. So we know that the walk will last twelve hours. If the average speed of a walking man is 5 km/hr. , it is 12 hours from sunrise to sunset; then we know that he will walk 60 km. He'll start walking towards the west at dawn and his path will turn towards the north in the morning as the sun heads south. After Solar Noon, his path will turn towards the east he'll end up facing due east at sunset. And we know that the path will be a curve since his shadow will always be changing direction. But what would the curve look like on a map? Would it be a hyperbola? How would the curve change if he walks on the summer solstice instead? What if he's at the North Pole? John C. --- https://lists.uni-koeln.de/mailman/listinfo/sundial --- https://lists.uni-koeln.de/mailman/listinfo/sundial
