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5. What goes up must come down

In this part of ‘Fun with the Minimap’ we are going to determine the height of various geographical features in Azeroth and the Outland. To do this, it helps to review the relationship between the sides and angles of a right triangle. Collectively, these relationships belong to a branch of mathematics called trigonometry. There are three fundamental trigonometric relations that can be quickly summarized by considering the ‘right triangle’ drawn in the figure below.

The definition of a right triangle is ‘a triangle with a right angle’. A right angle is just a 90 degree angle and is indicated in the figure above by the small square in the lower left corner. The remaining two angles are labeled ‘a1’ in the upper left corner and ‘a2’ in the lower right corner. The side of the triangle opposite to the 90 degree angle is called the hypotenuse and has a length ‘h’. The bottom edge is adjacent to the angle ‘a2’ has a length ‘a’. The left edge is opposite to the angle ‘a2’ and has a length ‘o’. Using the labels in the figure above, the sine of the angle a2 is defined as the length of the opposite side ‘o’ divided by the length of the hypotenuse ‘h’. This can be expressed as sin(a2) = o / h. In a similar fashion the cosine of the angle a2 is cos(a2) = a / h, and the tangent of the angle a2 is tan(a2) = o / a. These expressions are useful because, if the angle a2 is known, we can use a calculator to determine the sine, cosine, and tangent of the angle. Then, if one of the edge lengths ‘h’, ‘a’, or ‘o’ is known, the others are easily determined. For example, if the length ‘a’ and the angle a2 are known, then the length ‘o’ is given by o = a tan(a2). A simple mnemonic device, sohcahtoa, is helpful when trying to remember these expressions.

To relate all this to our goal of using the minimap to measure the height of various peaks in Azeroth and the Outland, it is helpful to think about how the mage ‘Slow Fall’ spell works. Essentially, after casting ‘Slow Fall’ and then jumping off of a tall peak, a mage will float with constant velocity to the ground along a straight line inclined from the horizontal by the angle a2. This angle is determined by how fast the mage is moving before the jump. The steepest angle of descent is achieved by casting ‘Slow Fall’ and then walking off the peak. We’ll call this a ‘walking slow fall.’ The angle of descent becomes shallower when the mage runs off the peak. We’ll call this a ‘running slow fall.’ A ‘mounted slow fall’ exhibits the shallowest angle of descent and occurs by first casting ‘Slow Fall, then mounting, and then riding off the peak. The mounted slow fall yields the greatest ‘Slow Fall’ range. Using a mounted slow fall, it is possible, for example, to jump from the mountains southeast of the Owl Wing Thicket in Winterspring, to the area around Ursolan in the north of Azshara. This is a useful trick when mining rich thorium veins in these two areas. With a bit of trial and error, you can select a heading that will even let you jump from Winterspring and land behind the large stone Timbermaw Hold gate in Azshara. The ‘Slow Fall’ spell requires a [Light Feather] and lasts for 30 seconds. Often, at the end of 30 seconds, a mage is still very high up above the ground. In this case, it is a simple matter to recast ‘Slow Fall’ to remain floating along the original trajectory. As a point of reference, it takes just over three minutes to float down from Teldrassil and requires seven light feathers.

To see how a running slow fall is related to our discussion of the right triangle above, we’ll add a few more labels to our right triangle sketch.

Here we see Prym performing a running slow fall. Starting from a jump site at the top of some peak shown in the upper left of the figure, Prym then floats along a trajectory that corresponds to the hypotenuse of a right triangle, and eventually stops at a landing site to the lower right of the figure. The height of the jump site above the landing site is given by ‘o’ in the sketch above. The horizontal distance Prym drifts from the jump site to the landing site is given by ‘a’. It is this distance ‘a’ that we can measure on our minimap by noting the locations of the jump site and the landing site. To relate the horizontal drift distance ‘a’ to the height ‘o’ of the peak, we use the ‘toa’ part of ‘sohcahtoa’ where tan(a2) = o / a. A bit of algebra yields our peak height o = a tan(a2). The determination of the height ‘o’ therefore requires knowledge of the horizontal distance ‘a’ and the angle a2.

The angle a2 will be the same for all height measurements made with a running slow fall. To determine this angle, just head to the top of tall cliff and cast ‘Slow Fall’. Choose a heading that will take you along the face of the cliff. Then, run off the edge and carefully watch as features of the cliff face pass by. After a bit of experimentation you will observe that the running slow fall trajectory is inclined 45 degrees away from the horizontal direction. In a bit of serendipity, this running slow fall angle can also be measured by observing the plume created by Prym’s Mana-Sphere Shoulderguards. This plume trails off in the opposite direction of motion, and during a ‘Slow Fall’ is aligned with the ‘Slow Fall’ trajectory. This is shown in the figure below where Prym is in the middle of a running slow fall off of Teldrassil.

Here we see the horizon off in the distance providing a well defined horizontal axis. To measure the angle of descent, I superimposed a right isosceles triangle drawn in yellow on the image. Since a right isosceles triangle has two edges of the same length, the two angles adjacent to the edges in the upper left and lower right corners must be equal. Furthermore, since all three angles must sum to 180 degrees and the right angle in the upper right corner is 90 degrees, the two equal smaller angles must themselves sum to 90 degrees and are therefore 45 degrees each. The upper edge of the triangle is aligned along the horizon, and the hypotenuse lies diagonally along the middle of the plume. If the descent angle was different than 45 degrees, the hypotenuse would not fit nicely along the plume. To see how this is related to the angles a1 and a2 we add a second right triangle to our drawing,

Taken together, the yellow and red triangles form a square with 90 degree angles in all four corners. Therefore, a1 + 45 degrees = 90 degrees where a1 = 45 degrees. Also, a2 + 45 degrees = 90 degrees where a2 = 45 degrees. Since tan(45 degrees) = 1, the height of any peak ‘o’ is equal to the horizontal distance ‘a’ traveled during the running slow fall,

o = a tan(a2) = a tan(45 degrees) = a

So, all we have to do to measure the height of a peak in Azeroth or the Outland, is to perform a running slow fall from the top of the peak to the ground and measure the distance ‘a’ from the jump site to the landing site using the techniques outlined in part three of ‘Fun with the Minimap’.

I’ll begin with measuring the height of Teldrassil. The first step is to select a jump site and then perform a running slow fall down to a landing site. In the figure below, the yellow arrow in the upper left points to the jump site, and the yellow arrow in the lower right points to the landing site. The long double headed red arrow is the projection of the running slow fall trajectory onto the minimap,

teldrassil_jump.jpg

The length of long double headed arrow is the horizontal drift distance ‘a’, and by our analysis above, it is equal to ‘o’, the height of Teldrassil above sea level as measured at the jump site. As you can see in the figure above, the jump site is actually displaced a bit out over the edge of Teldrassil because I have walked out onto one of the large branches. The texture map used to depict the branches provides a way to relocate my jump site so that I can repeatedly measure the distance ‘a’. I chose my jump heading by sighting off nearby leaves in the canopy. The particular heading I eventually chose balanced the need to miss several large protruding branches just below the jump site, with the need to not land in deep water. Landing in deep water is bad because of the induced fatigue, and also, because there are not enough features in the deep water to locate my landing site.

To make a measurement of the distance ‘a’, I have to locate the jump site and the landing site on the large minimap I assembled from the game files found in common.mpq. For this purpose, I used screen shots of the ‘in game minimap’. For example, the image below on the left is the ‘in game minimap’ as it appeared in the upper right hand corner of a screen shot taken while Prym was at the jump site, and below on the right is a similar ‘in game minimap’ recorded at the landing site,

To locate the jump site and landing site on the large minimap, I used Photoshop to overlay these images of the ‘in game minimap’ onto the large minimap. The ‘in game minimaps’ were set two clicks in from lowest magnification to match the resolution of the large minimap, however, I did have to scale the ‘in game minimap’ images by 95% to fine tune the fit. Next, I reduced the opacity of the ‘in game minimap’ images to about 60%. This way I could see the large minimap through the semi-transparent ‘in game minimap’ images. Then, all I had to do was adjust the position of the ‘in game minimap’ images to exactly overlay the large minimap landscape features. The silver ‘character arrow’ at the center of the ‘in game minimap’ images then locates the precise position of the jump site and landing site on the large minimap. To complete the process, I used a pair of yellow arrows to mark the jump site and landing site on the large minimap. Then I deleted the images of the ‘in game minimaps’, and measured the distance between the two sites using the techniques outlined in part three of ‘Fun with the Minimap’. I repeated a running slow fall several times starting from the same jump site, and each time I landed to within a pixel of previous landing sites. Since this technique is highly reproducible with very little error I will measure all subsequent heights in Azeroth and the Outland using a single running slow fall.

So, after all of this analysis, just exactly how high above sea level is the Teldrassil jump site? Well, I measure the distance ‘a’ from the jump site to the landing site to be 3870 feet. Using o = a tan(a2), and a jump angle of a2 = 45 degrees, we find that the jump site is 3870 feet (1180 meters) above sea level. That's one tall tree stump. Now, some parts of Teldrassil are a bit lower than the jump site, and some are a bit higher, but 3870 feet serves as a reasonable estimate of the height of Teldrassil above sea level. It is also interesting to note that it took 188.7 seconds to reach the landing site. Using this time I estimate the horizontal drift rate during a running slow fall to be 13.99 Mph. This speed is very nearly the running speed of 14.04 Mph we measured on the Auberdine docks in part four of ‘Fun with the Minimap’ and reflects the fact that the horizontal drift velocity during a ‘Slow Fall’ is determined by the velocity of the mage immediately prior to ‘Slow Falling’. We can also use the time of 188.7 seconds to determine that the rate of descent while slow falling is also essentially the same as the running speed of 14.04 Mph. For comparison, a parachutist with an open canopy descends at roughly 10 Mph.

Now that we have established a method to measure height in Azeroth and the Outland, lets take a look at a couple of other peaks. Since Winterspring is not too far from Darnassus, I flew on over to Everlook and ran south to the mountain peaks just past Owl Wing Thicket to the east,

After a running slow fall, I measured the length ‘a’ to be 2897 feet. The jump site is therefore 2897 feet (883 meters) above sea level. For my next jump, I went south to see how deep Un’Goro Crater is.

Here the distance ‘a’ is 966 feet and the landing site is therefore 966 feet (295 meters) below the jump site. Because the jump site I selected was located on a small peak at the edge of the crater, the actual depth of the crater floor with respect to the desert plains of Tanaris is a bit less than 966 feet. Also, since the desert plain near the crater edge does not appear to be significantly higher than the desert near the Great Sea, we can estimate that the bottom of Un’Goro crater is somewhere around 900 feet below sea level.

So, after measuring a few heights in Kalimdor, I thought I’d head on over to Dun Morogh in the Eastern Kingdoms and measure the height of the peak above the city of Ironforge.

The height of this peak was a bit tricky to measure because the rate of descent after a running slow fall was too rapid to clear the entire mountain. So, I split the descent into two stages. The first stage was a quick 12.5 second ‘Slow Fall’ from Ironforge peak labeled with the blue arrow in the image above, down to the jump site. Using the slow fall descent rate of 14.01 Mph and a time of 12.5 seconds yields a drop of 255 feet for this first stage. I landed approximately 25 feet below the jump site peak as determined by counting the number of body lengths required to climb up to the jump site. I therefore subtracted 25 feet from the first stage drop to obtain an adjusted drop of 230 feet from Ironforge peak to the jump site. Then from the jump site, the distance ‘a’ to the landing site was 2405 feet. When combined with the first stage descent, we find that Ironforge peak is 2634 feet (803 meters) above sea level.

Before heading to the Outland, we’ll measure one more height in the Eastern Kingdoms, the height of the Stonewrought Dam

As shown in the figure above, this is a short jump where ‘a’ = 385 feet and the dam is therefore observed to be 385 feet (117 meters) high. By comparison, the Hoover Dam is 726.4 feet (221 meters) high.

Ok, lets head to the Outland and measure one more height. I thought it would be fun to see how high The Hand of Gul’dun is above the Fel Pits,

After a running slow fall, I measured the Hand of Gul’dun to be 1121 feet (342 meters) above the Fel Pits. I’ve tabulated the all the heights we’ve measured so far in the table below for reference.

LocationHeight (feet)Height (meters)
Teldrassil38701180
Owl Wing Thicket2897883
Un'goro Crater-966-295
Ironforge Peak2634803
Stonewrought Dam385117
Hand of Gul 'Dun1121342

In part one of ‘Fun with the Minimap’ I mentioned that if the whole minimap of Kalimdor were printed at a resolution of 72 pixels per inch, it would be approximately 14 feet by 8 feet. The Eastern Kingdoms (not including Eversong Woods and the Ghostlands) would be approximately 12 feet by 6 feet and the Outland would be about 6 feet by 6 feet. If we placed these maps on a large table, and incorporated our height measurements, we would be able to construct relief maps of Azeroth. For example, Teldrassil would be the highest region of the map and would extend almost nine inches above the table top, while Un’Goro crater would dip just over two inches below the table top. Heh, Blizzard could make a relief map using painted Styrofoam blocks and sell it