南投 猴探井 日晷 A surprise dial at HouTanJing, NanTou, Taiwan

中文讀者請至 http://blog.xuite.net/nycl.chiu/blog/21498077

There was someone posting a picture of horizontal sundial in his web page saying that it was right on the grass behind the visitor center of BaGuaShan scenery park, ZhangHau. From the picture I was sure that the sundial was designed by an expert. There were hour lines as well as calendar lines. The nodus was the finger tip of a boy sitting on the high end of a seesaw while other three boys were on the low end touching the ground. Such a beautiful design for sundial is what I must see in person.
Today is the winter solstice day and the sky is blue. A perfect day to observe a sundial with calendar lines. My wife and I got up early and drove down to BaGuaShan park. Not far from the entrance of the park, about one km, we passed a natural park, which concerns mostly migratory birds who use that area as a rest stop in their journey. Making a right turn along the road, we found a parking lot. The lot keeper pointed out the way to the visitor center to us and said it would take only 5 minute walk to get there, so we parked our car. The way to the visitor center was uphill a little bid. It took 15 minutes for us to walk there.
There was an open space besides the visitor center building. From there one can view the whole ZhangHau city below, so we went over that way trying to reach the grass ground behind the building. The open space turned out to be an open theater but there was no grass ground behind the building. Of course, no sundial either. Astonished, we backed up and got in the visitor center looking for help.
There were five service persons on duty but none of them knew where the sundial was. Even all of them denied that a sundial existed in visitor center area. We were very depressed. Suddenly a service lady (volunteer) said: “I know. The sundial is at HouTanJing, not here. There is another visitor center of BaGuaShan scenery park at HouTanJing.” HouTanJing? I have never heard of that place. The lady unfolded a park map and point out the location of HouTanJing. The park area goes north-south; Zhang Hua is on the north edge and HouTanJing is on the lower quarter. “The road in front of us is road 137. You take south and follow the road. After few turns you’ll see a water supply. Make a left turn there and you‘ll be on road 139. Driving all way down, you’ll see the sign of HouTanChing.” (Ching should pronounce same as Jing. They tends to mean the same word in Chinese.) In my mind, the park had an area around a hundred acres only. But that was the impression I got more than two decades ago. The new map before me showed totally different things. I asked carefully: “How far is that?” “Half hour of driving if you know the road.” I took another look at the map. Yes, she must right. HouTanJing was about 25 km from where we were standing and only 5 km west from NanTou city.
When we arrived HouTanJing, we saw a small house there. It was a visitor center, but not that of BaGuaShan scenery park. We went in the house and found nobody inside. There was one service man in his sixties outside by the vending machine and he was our only hope. Knowing what we were looking for, he said:”The sundial is just over there, only a few steps down the stairs. Today you have to stand on the ‘12/22’ mark on the calendar line.” Wau! I was surprised. Although he did not make it right, it was not easy for an ordinary countryman to knew that today the shadow of the kid’s finger would fall along the ‘12/22’ calendar line. I was so anxious to see the sundial that I did not give him one more minute to introduce it to me. I ran to the stairs with the hope that there was a grass ground in front of them. But I did not have to walk down further, the sundial could be seen right from there. It was a sundial, an analemmatic dial! I was shocked again.


We didn’t mind that it was not the dial we were looking for. But where is that horizontal sundial? We checked web sites again after we came home. It turned out that the information we got the first time was wrong. The sundial is in the visitor center of the natural park where we passed at the beginning before we parked our car in the lot!

垂直、偏向垂直 和地平日晷的小時線 Hour Lines for Vertical, Declining Vertical and Horizontal Sundials

假設有個赤道日晷面(F1)，其南面的小時線分布如Fig. 1，每小時15度，其晷針垂直於紙面。如果將此圖用平行於晷針的光投影至一平面( F2)，其形狀將視兩平面相互之間的角度而定。
Fig. 1 shows the hour lines on the face (F1) of an equatorial dundial. Those lines are separated equally by 15 degrees. The gnomon, which is not shown here, should be perpendicular to the screen. If some parallel light is cast through this diagram onto a plane (F2), the shape of the new diagram formed by the shadows those lines varies according to the angle between F1 and F2.
若F2垂直，F1單純向南傾斜一角度M，則F2上之投影結果必然左右寬度不變，上下長度變長。以各自晷心為原點，在F1上某一點P1(x1,y1)投至F2上成為P2(x2,y2)點；其中x2 = x1, y2 = y1/cos M。就是Fig. 1 中所有的點(線)相對於6-18線的距離都同時向下移動，新的距離是原來距離的1/cos M倍，以形成Fig. 2。
If F2 is kept vertical and F1 is simply declined due south with M degree, then the resultant new diagram must be longer in height but without any change in width. Taking each dial center as its own origin, point P1 (x1,y1) on F1 generates point P2(x2,y2) on F2 with x2 = x1, y2 = y1/cos M. That means all points in Fig. 1 move downwards, referred to 6-18 line, to their new distances, which equal to the old distances multiplied by 1/cos M, to yield the shadow diagram, such as Fig. 2.
Fig. 2 是個M = 40 的例子。因赤道日晷傾斜40度，所以此圖即是北緯40度適用的垂直日晷小時線圖。晷針必與投影光線平行，故與水平面成50 (= 90 - M)度。
Fig. 2 is a result of M = 40 degrees. It is also the hour line diagram for a vertical sundial face on latitude 40N. Its gnomon ought be in the direction of the casting light, so it has an angle of 50 ( = 90 - M) degrees to F2 in this example.

如果將垂直面F2逆時鐘旋轉D度，D在此稱為偏向角，形成偏向垂直面F3，此時F1上的線條會在F3面上產生新圖樣，如Fig. 3。其小時線的角度分配決定於緯度和偏向角。
If the vertical plane F2 is rotated D degrees counterclockwise, (this rotation is called declination), plane F3 results. Those shadow lines on F2 are now on F3 and may have the hour line pattern as shown in Fig. 3. The pattern depends on latitude and declination.


Fig. 4說明了P2和P3的關係，其中藍色線是投影光線，距水平面角度 φ。P1和P2的關係已如前述。 P1 和赤道日晷原點(晷心)的關係是:x1 = sin ha, y1 = cos ha, ha = 時角 = (12 - hour) × 15。故可得垂直偏向日晷的時角h12公式
x3 = x1 ÷ cos D
y3 = y2 - x2 × tan D × tan φ
= y1 ÷ cos φ - x1 × tan D × tan φ
即
tan h12 = x3 ÷ y3
= cot φ ÷ {cot ha × cos D ÷ sin φ - sin D}


The mathematical relations between P2 and P3 are depicted if Fig. 4, where the blue line represents the cast light with an angle of φ from horizon. Those between P2 and P1 are given above. With x1 = sin ha, y1 = cos ha, and ha = hour angle for equatorial dial = (12 - hour) × 15, we have the formula for the hour angle h12 for declining vertical dial as
x3 = x1 ÷ cos D
y3 = y2 - x2 × tan D × tan φ
= y1 ÷ cos φ - x1 × tan D × tan φ
tan h12 = x3 ÷ y3 = cot φ ÷ {cot ha × cos D ÷ sin φ - sin D}

如果將垂直面F2向下轉成地平，形成平面F4，P1在此新平面F4上投至P4(x4,y4)。則 x4 = x1, y4 = y1 / sin φ，φ為緯度。將Fig. 1赤道日晷小時線各端點的(x1,y1)換成(x4,y4)，再由這些新的端點連線至晷心，即得地平日晷之小時線。
If the vertical plane is reclined so to give a horizontal plane F4, the shadow of P1 on F1 casts on F4 at P4(x4,y4) and x4 = x1, y4 = y1 / sin φ. That is
x4 = sin ha
y4 = cos ha / sin φ
Connect P4 points obtained with proper ha to the dial center with straight lines. These are the hour lines for horizontal dial at latitude φ.