First of all, thanks for the nice write-up. It was very readable and your points were quite clear. I apologize in advance for the following wall of text. I tried to explain everything as clearly and precisely as possible, which means it got a little wordy.

The basic misconception I'm seeing here is.... the orthographic projection seems to be a tool for engineers to use, which doesn't typically deal with say, mile-long distances in everyday application. This is **Point 1**.

Perspective occurs at all scales (not just large scales) and orthographic projections are valid and useful at all scales (not just small scales). Orthographic projections very intentionally don't portray perspective in order to allow certain things to be accurately measured.

1. Straight lines in reality are always straight in the diagram.

2. Parallel lines in reality are always parallel in the diagram.

3. Angles parallel to the plane of projection are accurate.

4. Distances parallel to the plane of projection are proportional. Distances in the same direction are proportional.

None of these properties change with scale. If we suddenly decide to add in perspective to the diagram, at any scale, these things will stop being true, and the diagram will no longer be useful for taking measurements.

This is **Point 2**; if all = lines converge, perspective must be taken into consideration - and by the nature of this debate, as **Tom Bishop** stated, it is impossible to get to a point where a orthographic projection would be possible on this scale, so we must rely on perspective, knowing that all parallel lines converge due to perspective.

I

*did* take perspective into consideration. Unlike Tom Bishop and the aptly named P-Brane, I have more than just a vague understanding of perspective. I know

*exactly* how to calculate it. More on this later.

Now, I'm only tentatively holding onto understanding of this at this point, so don't ask me to draw this; but seeing **Point 2** (all = lines converge), that means that due to perspective, something moving parallel to a flat plane far (say a few thousand miles) overhead, would eventually reach that point of convergence, and 'disappear' beyond the horizon, before fading - though it may 'fade' somewhere thereafter - which would explain 'dusk' and 'night time' *quite conveniently* for a Flat Earth; the light source literally extinguished by mechanics of orthographic projection, beyond the horizon.

This kinda-sorta makes sense if you only have a vague understanding of how perspective works. Unfortunately for Mr. Bishop and Mr. Brane, it is relatively easy to calculate

*exactly* how far away an object is based on its height and angle with the horizon. This is where the orthographic diagram becomes useful. See this image:

On top is a side view diagram of a camera and 3 round objects. On bottom is the same 3 objects as seen from the perspective of the camera. The top DOES NOT take into account perspective. The bottom DOES take into account perspective. So, how are they related?

**First: how does perspective arise?** The size of an object in a photo is proportional to the

angular diameter of that object. As an object moves farther away, this angular diameter decreases, which causes the object to appear smaller. The same is true for distances between objects. This is why objects appear smaller as they get farther away. It is also why parallel lines appear to converge as they get farther away.

Look at the orthographic diagram. Notice that the 3 objects are the same size. However, the angular diameter of each object is different (4.2°, 3.0°, 2.3°). The farthest object has the smallest angular diameter. Notice that these angles are easily measurable using the orthographic diagram.

Now look at the bottom picture. Notice that the

*size* of each object in the bottom picture corresponds to the

*angular diameter* of the object from the orthographic diagram. We can convert these angles into sizes using the

field of view of the camera. For example, if the field of view of the camera is 60°, and an object has an angular diameter of 30°, then the object will take up half the picture. (This assumes that the camera doesn't have a huge field of view or lots of optical distortion.)

We can do the exact same process for the

*height* of each object: (skip the next two paragraphs if you already get the point)

Look at the orthographic diagram. Notice that the 3 objects are the same height. However, the angle between the object and the ground decreases as it gets farther away (20°, 14°, 11°). The farthest object has the smallest angular height. Notice that these angular heights are easily measurable using the orthographic diagram.

Now look at the bottom picture. Notice the

*height* of each object in the bottom picture corresponds to the

*angular height* of the object from the orthographic diagram. Once again, we can convert the angles to heights using the field of view of the camera.

Now, please notice 3 things about how we constructed the bottom image:

1. We determined the height and size of the objects from the angles in the orthographic diagram.

2. The size of the objects naturally decreases as they get farther away.

3. The objects naturally approach the horizon as they get farther away.

4. If we draw a line connecting the right edges of each object, and another line connecting the left edges of each object, those two lines will converge below the objects at the horizon. Just like perspective lines! Coincidence? I think not!

**Points 2 and 3 are the effects of what we call perspective. Perspective naturally arises from the angles measured from the orthographic diagram! Neat!**Notice, that there is absolutely no need to overlay any arbitrarily drawn "perspective lines" onto the orthographic diagram in order to calculate the position of each object due to perspective. The perspective lines arise naturally in the bottom picture due to measurements of angles that we took from the orthographic side view diagram.

**Tom Bishop's rebuttal:**So far, Tom Bishop's main argument has been to simply deny the mathematical relationship between the two pictures above. To justify this denial, he has made several... um... entertaining... arguments involving "hidden infinities" and generally denounced the usefulness of orthographic diagrams. (Even though his hero, Rowbotham, uses them quite frequently.)

However, you don't have to take my word for it. Go outside with a camera and a few objects and test the math for yourself. You will find that the mathematical relationship between the angles from the orthographic side view and the size and placement of the objects in the picture is quite accurate for

*any* distance that you manage to test. (Use a relatively narrow field of view for more accuracy. Wide angle lens have too much optical distortion.)

Tom Bishop's

*other* argument is that "maybe the math math suddenly stops working at super long distances that are too long to test". However, he has presented no evidence or logical reasoning that would suggest that the math suddenly stops working at a particular distance. I think we can all agree that this is a horrible argument and merely a desperate attempt to ignore evidence that contradicts his flat earth model.