Stereographic Projection, Chaucer and the Astrolabe
By Laura Jamieson and Maria Montero
What do Stereographic Projection, Climates, and Geoffrey Chaucer have in common?
They are all, in one way or another, related to the Astrolabe. But what exactly is an astrolabe? What is stereographic projection? What is a climate and what does Chaucer have to do with them? This project attempts to provide the basic answers to these questions through the use of pictures, proofs, and commentary. To organize all this information, we have divided the project into four main parts:
Part 1 What is an astrolabe? (This explains what a climate is too!)
What is an astrolabe?
Of all astronomical instruments, The Astrolabe was the most widely used throughout the middle ages. It was developed in ancient Greece for measuring the altitude of a heavenly body, but gained popularity all over the world until the sextant replaced it in the 18th century. It provided a means to mark the time of day and the seasons of the year as well as finding the angle of the sun, moon, stars and planets with respect to the horizon or the zenith. It was used for calculating the heights of buildings and mountains and for surveying the land. The Astrolabe was also used as a way of reading ones horoscope and a way for navigators to locate their longitudinal and latitudinal coordinates. Basically, it was a pocket watch, compass, fortune-teller, crude sextant, and theodolite all in a convenient little palm-sized case! The most common form of the astrolabe was the planispheric astrolabe, which was also the easiest to use. This marvelous invention, often made of brass, was approximately 5 to 10 inches in diameter and included a horse, a rete, climates or plates, the mater, the alidade, vane and pin.
Parts of the Astrolabe
As stated above, the astrolabe was made up of a horse, rete, some climates, the mater, the alidade, a vane, and a pin. The Mater was the main frame of the astrolabe and on it sat the climates, rete and horse. It was disk shaped and often had a hook or loop at one end so that the astrolabe could hang on a chain. The climates, or plates, which sat on top of the mater, were discs that had lines etched onto them. A person using an astrolabe would use different climates depending on their location on the earth. The rete sat above the climates and had specific pointers on it that would point out particular lines on the climate below it. The rete was free to rotate about a central pin that ran through the center of the mater, climates, and rete and was held in place above the rete by a horse, which was so named because it was usually made in the shape of a horses head. In addition to the climates, the mater itself would often have latitudinal lines etched onto its top side and on the backside all the information an observer would need concerning the degree of altitude, signs of the zodiac, months of the year, and days of the month were etched onto the mater. The information of a particular celestial body would be identified by the alidade (also called the double rule). The alidade sat directly below the mater and was basically a slide-rule with sighting holes at each end so that a particular body could be focused on. The alidade allowed the observer to calculate the altitude of the object seen within the sighting holes as well as discern the zodiac sign for the object by rotating beneath the mater and identifying the relevant information etched on the backside of the mater. Although the individual parts of the Astrolabe are simple, put together, the components of the Astrolabe create a highly versatile and indispensable instrument.
The Climates of the Astrolabe
One of the most important components of the Astrolabe are its climates. As mentioned above, the particular climate used by an observer depends on the location of the observer, and different climates are used in different locations. A typical climate would be a disc, most likely made of brass with a diameter only slightly smaller than that of the mater. It would be thin and several climates could be stacked on top of one another within the mater, and lines would be etched onto its surface. You may be wondering what these lines are. These lines, depending on the particular climate being looked at, may be representations of lines of latitude (such as the equator or tropic of Capricorn or tropic of cancer etc.) or they may be representation of almucantars, lines of azimuth, or hour angles. Almucantars are, like lines of latitude, "imaginary" rings on the earth but unlike latitude lines almucantars are concentric around the zenith of the observer and are parallel to the horizon. Thus, depending on your position in the world, almucantar lines would change and thus you would need to use a different climate. Also found on a climate may be etchings representing lines of azimuth. Lines of azimuth are imaginary lines that run from an observers zenith to the horizon, intersecting the almucantars at right angles. The final etchings that may be found on the plates are the Hour Angles. Hour Angles divide the twenty-four hour day into sections of a circle, assuming that both day and night are made up of twelve hours each. These lines run from the north celestial pole to the equator but when projected onto a climate, only the portions below the horizon are represented. An important question to ask is "How do we turn these imaginary lines on the earth into real lines on a climate and how do we represent them accurately?" The answer to this question is Stereographic Projection.
What is a stereographic projection?
There are many different ways of rendering three-dimensional objects into two dimensions. Different kinds of projections are able to represent realistically things like size, areas, distances, and perspective. One particular kind of projection used for representing spheres and circles on spheres in two dimensions (ie on the climates of an astrolabe, or on some maps of the earth or celestial sphere) is stereographic projection. Stere(o) hails from the Greek "stereos" for "solid", while graphic comes from the Greek "graphicus" meaning "formed by writing, drawing, or engraving". Also, a projection is defined by Websters Dictionary as "a systematic presentation of intersecting coordinate lines on a flat surface upon which features from the curved surface of the earth or the celestial sphere may be mapped". Thus stereographic projection involves the rendering of solid, or three-dimensional objects onto a two dimensional page.
What makes stereographic projection so apropos for spheres? Well, stereographic projection has two important characteristics that differentiate it from other kinds of projections:
This means that circles on a sphere (i.e. latitudes on the Earth) are represented as circles on a plane and the angles between lines are retained when the lines are projected. This is how the climates of the astrolabe are created. The lines of latitude, almucantar, azimuth, and hour angles are represented stereographically onto a plane (usually taken is the equatorial plane) and the climates are merely a scaled-down representation of this.
How does the projection work?
In regular perspective a point in three dimensions, say P = (x, y, z) is mapped to two-dimensions on the (x,y) plane with the following coordinates: P* = ( ax/a-z , ay/ a-z) where a is the distance position along the z axis where the eye sits (i.e. where one is looking from).
If we let the eye sit at (0, 0, 1) and take a sphere with radius 1 centered at ( 0, 0 , 0), and the x-y plane, then the x-y plane will divide the sphere into two hemispheres. Notice that the point (0,0,1), where the eye sits, would be mapped to the point (0/0, 0/0) and thus would be represented at ¥ and all points near this would be mapped very far away.
So, let us now take any sphere and mark off the poles on the sphere, which are diametrically opposite to one another. Next we identify the plane "E" which divides the sphere into two equal hemispheres and whose normal runs through both the poles. Now we take any point P on this sphere (except the poles which we know map to infinity) and we want to project this point P onto the plane "E" we defined above. We draw a line that contains a pole, the point P itself, and the plane E and where this line intersects with E we call P*.
So in our example above with the unit sphere centered at the origin, if we took the point P = (x, y, z) on the sphere and then take the south pole which has coordinate (0, 0 ,-1) we can parameterize the line that runs through both these points as:
Pole + t(P - Pole) = (0,0,-1) + t( x,y,z - 0,0,-1)So, when this line intersects the x-y plane (ie z =0) and t = 1/z + 1and so using this formula for t, our point is mapped to:
P* = (x', y', 0) where x' = tx = x/ z + 1 and y' = ty = y / z + 1To map circles then we would simply take points on the circle and apply the same procedures.
Proof that stereographic projection preserves circles
As mentioned above, stereographic projection has two important characteristics. One being that stereographic projection preserves angles and the other being that stereographic projection preserves circles. We now include a proof of this fact done in illustrations as well as an algebraic proof.
Note that a unit sphere with center at origin has the equation:
X2 + Y2 + Z2 = 1 ( 1)Also, A given plane will have the equation:
AX + BY + CZ + D = 0 (2) Where A,B,C,D are constantsSo, if we have a point on the sphere which satisfy (1) we have:
P = ( X, Y, Z)The projection of P onto the x-y plane will be given by:
P* = (x ,y,0)The equation of a pole is:
Pole = (0, 0 , 1)So, one way of representing the line that runs through all of P, P*, and Pole is:
P - Pole = t( P* - Pole) -> ( X, Y, Z - 1) = t( x, y, -1) (3)From Equation (3) we can see that:
X = tx, Y =ty, Z = 1 - t (4)We define a circle Q on the sphere in this way:
Q = x2 + y2Substituting the values from (4) we see
Q = (X / t )2 + (Y / t)2Also, substituting the value of Z from (4) into equation (5) we get
Q + 1 = 2 / t (6)And...
Z = Q -1 (7)Now we want to know what all this will look like on the plane:
AX + BY + CZ + D = 0We substitute our values for X, Y. Z that we got from (4) and (7)
Atx + Bty + C (Q -1) + D = 0We replace t by what we know from (6) and get
2Ax + 2By + C (Q -1) + D = 0Simplifying gives:
2Ax + 2By + C (Q - 1) + D( Q + 1) = 0Substituting our original Q gives us
(C + D)(x2 + y2) + 2Ax + 2By - C + D = 0 (8)Since the coefficient of both x2 and y2 are the same, then equation 8 is the equation of a circle and so the projection of the circle Q onto the plane is a circle.
The geometric proof relies on the fact that if a plane intersects a cone at an angle q with respect to the axis of the cone, and this intersection produces a circle, then if a plane was to intersect the cone at an angle 180o - q that intersection would also produce a circle. If we can then show that the stereographic projection of a circle on a sphere makes an angle of 180o - q with respect to the axis of a cone, then we will know that the projection is also a circle.
The pictorial proof begins with certain definitions of particular kinds of cones.
The first kind of cone is the circular cone, which is formed with a point and a circle. The circle is the base of the cone and the point is the vertex. If the axis of the cone (the line joining the center of the circle to the vertex is at right angles with the plane of the circle then the cone is said to be "right circular" and if not then it is "oblique circular".
The second definition is for elliptical cones, which are similar to circular cones except that instead of the base of the cone being a circle, it is an ellipse.
The next important point needed to begin the proof is that the intersection of any oblique circular cone is an ellipse
and thus any oblique circular cone can be represented as an elliptical cone. Why? Well, take an oblique circular cone Q, with circle b and vertex V, and the origin be the center of the circle
Note that the cone Q is symmetric around the YZ plane:
Place Points T,C, V and S as follows: All these lie on the YZ plane, and pick S such that SV=VT and C is the midpoint of ST.
Any plane passing through ST and perpendicular to YZ will intersect Q in an ellipse, therefore Q is an elliptical cone with axis VC according to Definition 2
How does this relate to our projection from a sphere?
With these important ideas clearly established, the proof begins with a sphere situated such that the origin, O, is at the North Pole. On the surface of the sphere is a circle called g. If light is shone from the origin, the rays of light and the circle g form a circular cone and from the above proof this circular cone can be considered an ellipse.
Let the rays of light from the origin through the circle g by a circular cone. We know from above that this cone can be considered elliptical
g is symmetric about the XY plane
So, our original circle intersects the elliptical cone made by the light at an angle q with respect to OC.
And no matter where you cut, as long as the angle is q, the result is always a circle:
Since the circle is symmetric about the XY plane, we can see that when the sphere intersects the XY plane
we will have a circle with a chord that corresponds to the diameter of our original circle (AB).
We can find the midpoint of the arc AB and call this C.
We see that if O is the origin, OC bisects the < AOB and is the axis of the cone.
As can be seen, the diameter of the circle is AB and C is the midpoint of the arc AB and so OC bisects the < AOB.
Another plane that also produces a circle is the plane that is 180o-q:
If we can show that the projection plane is at an angle 180o-q to OC then the projection will be a circle.
At this point we use the identity that inscribed angles opening to the same arc are equal, and the proof can be simplified into two dimensions:
Draw a tangent at C, and it will be parallel to AB:
And therefore corresponding angles are equal, so < OCP = q.
Now we draw S as the point diametrically opposite to the origin, making OS a diameter and P is the intersection of the tangent line at C and the tangent line at S.
From the picture we see that < OCS is an inscribed angle in a semi-circle and is therefore equal to 90o
but q = < OCP = < PCS + 90o
PS and CP are both tangent to the circle coming from the same external point P, so PC = PS and therefore < PCS = < PSC
f is the angle between the tangent line at point S and the line OC extended out of the circle. From the pictures we can see that q = < PCS + 90o or since f = 90o - < PSC then q = 180o - f.
The projection plane can be shown to be f which is 180o - q and so the projection is a circle.
Thus we have seen what stereographic projection is, how it works and some proofs of its very important property of preserving circles, now we can use stereographic projection to create some of our own Astrolabe climates!
As previously described, the climates of an astrolabe contain lines that represent latitudes, almucantars, lines of azimuth, or hour angles. Below are representations of these four types of imaginary lines as inspired by illustrations from J.D. North's article entitled "The Astrolabe" from Scientific American.
Lines of Latitude:
The easiest and probably most important lines of latitude are the Equator, the tropic of Cancer and the tropic of Capricorn. These three circles are depicted and the plane of projection is taken to be the equatorial plane. Lines connect the circles to the South Pole and their intersection with the plane produce the stereographic projection. The image below is a representation of the earth and below that, what a climate may have looked like depicting the Equator, Tropic of Cancer and Tropic of Capricorn.
For this particular representation, we take the observer's zenith to be 40o north of the equator and the horizon to be the last ring in the picture. The projection of two rings are specifically shown in red and blue and the total projection as it would have appeared on a climate is below.
Lines of Equal Azimuth:
Here we take angles at 90o (blue arc) and 60o (red arc) and focus on these. Lines of azimuth were only represented above the line of horizon and so these azimuth lines are cut off by the horizon ring. The climate projection is again show below the represented earth.
Hour angles we drawn only beneath the horizon line although they extended all way to the North Pole. These lines mark the hours of the day: twelve for day and twelve for night... we only represent one set in this picture.
Whats Chaucer got to do with it?
Were finally nearing the end of this project, but the important question still remains, what does Geoffrey Chaucer, author of The Canterbury Tales, Troilus and Criseyde and many other works, have to do with the astrolabe?
Well, as mentioned in the introduction, the astrolabe was an extremely popular instrument during the Middle Ages, which lasted (depending on who you talk to) from about the 4th century to the end of the 14th century A.D. Chaucer just happened to have been born and lived about this time (1343? - 1400) and so, as a learned man of some means (he was a government official, customs officer, justice of peace and was elected to Parliament), he would have been fairly familiar with the astrolabe.
In 1391, Chaucer wrote a work entitled "A treatise on the Astrolabe". The introduction of the work would imply that it was written for his son, however it may have been written instead for the son of a friend, Lewis Clifford. Whoever it was written for, the boy most likely died in 1391, which is why Chaucers work was never finished. "A treatise on the Astrolabe", according to the F.N. Robinson edition, is the oldest known "technical manual" in the English Language and it was compiled from different foreign sources. The introduction, which we have reproduced below, is however, Chaucers very own. Included with the original introduction is an extremely loose translation (the emphasis here is on loose!) of it into modern English:
Now heres a rough (very very very rough) and somewhat edited translation of the above introduction:
Chaucer then begins part 1 of his treatise, which is, as noted above, a description of the parts of the astrolabe. Our own description of the parts of the astrolabe in this project is fairly similar to Chaucers description although Chaucer was no doubt holding an astrolabe as he was writing it. Interspaced among the descriptions of the parts of the astrolabe are reminders and advice for "Litel Lowys", the proper way to use it, hold it, and keep it safe.
Part 2 of the treatise involves instructions for common uses of the astrolabe. The first three such instructions are included below as well as the translation:
Rough translation of the above three instructions from Part 2 of Chaucers treatise:
There are 46 instructions for the astrolabe in Part 2 of Chaucers treatise. They range from finding the altitude of a celestial body to finding the correct date and time, to finding the latitude of a city to finding where North, South, East or West it, to finding which house a zodiac sign is rising into. The instructions Chaucer gives are extremely "hand-on" and quite difficult to reproduce without actually having an astrolabe in your hands. Most calculations and instructions are specified for Oxford, England where "Litel Lowys" most likely lives. Oxford, England sits approximately on the 52nd latitude North and at about 1 longitude west of Greenwich. Thus any plates that Chaucer would have used would have been based on this information.
Unfortunately, Chaucer does not continue to do parts 3,4, and 5 of the treatise. He apparently decides to stop, leaving his work unfinished, and thus his treatise on the astrolabe ends...
And thus our own project ends.
Pictures (in order of appearance):
2) Astrolabe (black background) by James E. Morrison, Janus. http://www.astrolabes.org accessed April 14, 2001
3) Parts of the astrolabe by Martin Brunold. http://www.astrolabe.ch/frameset.htm accessed April 13, 2001
4) A Climate by James E. Morrison, Janus. http://www.astrolabes.org accessed April 14, 2001
5) Stereographic Projection by James E. Morrison, Janus . http://www.astrolabes.org accessed April 14, 2001
"Proof that a Stereographic Projection of a Circle is a Circle" by Yana Zilberberg Mohanty accessed March 30, 2001
"Stereographic Projection" by Bill Casselman Chapter 5 of math 309 text
"Stereographic Projection" by Peter Doyle http://www.geom.umn.edu/docs/doyle/mpls/handouts/node33.html accessed April 6, 2001
"The Astrolabe" by J.D. North Scientific American Journal, Vol: 231,January 1974 pg 96-106