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Description
*Edit* I have made a video [link] that explains how to use these concepts more clearly than this tutorial does.

Monguro [link] suggested a book to be [link] that had a method of describing 3 point perspective that is remarkably similar to what I have here. The methods are very well presented in that book, and in practice are probably great for actually drawing, but are geometrically not sound.

I have been able to modify my 2 point solution to work for the 3 point solution.

Still don't have a great explanation for where the 3rd point goes, but I have a narrower range than I had before. (The 3rd vanishing point is constrained in the left right dimension by the first two vanishing points, it is almost unconstrained in the up down dimension. The only exception is the area near the horizon, it cannot be within the circle drawn between the two vanishing points.) As much as the idea bothers me, the start of a drawing is a line for a horizon, and then a 3rd vanishing point (if you are close to the object, the point should be close to the horizon. If you are far from the object, the vanishing point is far from the horizon. This has to do with how things distort when they occupy a wide angle of your vision. I do not have a solution worked out for this so just use your judgement.

[Drawing 3 point perspective.:]

1) Draw a circle. Draw a line through the centre of the circle, where the line intersects the circle defines out 2 vanishing points. Locate a 3rd vanishing point above or below this horizon, outside of the circle and horizontally constrained between the first two points.

With the two vanishing points on the horizon, and 3rd vanishing point, we have a triangle.

2) We will now treat each side of this triangle as a horizon, and draw the 2 point square solution up till step 5.

3) Draw the true vertical:
Draw a line perpendicular to the horizon from the 3rd vanishing point to the horizon. This is our true vertical. Due to a quirk of geometry (the parallel objects converge rule), this will always be in our centre of vision, and for the purposes of illustration, in our picture.

you can test this by looking at a tall blocky building. When you look at the ground level, the building is wider that it appears at the top, and when you look at a specific corner, it is exactly vertical. But when you focus on one vertical, pay attention to how the far side of the building gets closer to the corner that you are focusing on, it looks a little tilted doesn't it?

4) Start drawing the object:
By locate a point on the true vertical, and drawing rays through it from the 3 aspect points to the their respective circles.

5) Locate the (6) repetition points.
Draw lines from where the aspect lines intersect with their circles back to their repetition locator points. Where these lines cross the triangle is where the Repetition points are.

6) Draw your cube.
Draw a line from each vanishing point to our object point. Measure off how big you want your cube on one of the these lines, and draw a line from that length back to the appropriate vanishing points. draw lines from vanishing points through the intersections of the aspect lines, to find squares. repeat for all three sides of the front of the cube. Continue for the back if necessary.

And dang-it. noticed something else. What I thought were aspect points are actually aspect locator points. the aspect points are where the line crosses the triangle. At least it is consistent

Description
*Edit* I have made a video [link] that explains how to use these concepts more clearly than this tutorial does.

Monguro [link] suggested a book to be [link] that had a method of describing 3 point perspective that is remarkably similar to what I have here. The methods are very well presented in that book, and in practice are probably great for actually drawing, but are geometrically not sound.

I have been able to modify my 2 point solution to work for the 3 point solution.

Still don't have a great explanation for where the 3rd point goes, but I have a narrower range than I had before. (The 3rd vanishing point is constrained in the left right dimension by the first two vanishing points, it is almost unconstrained in the up down dimension. The only exception is the area near the horizon, it cannot be within the circle drawn between the two vanishing points.) As much as the idea bothers me, the start of a drawing is a line for a horizon, and then a 3rd vanishing point (if you are close to the object, the point should be close to the horizon. If you are far from the object, the vanishing point is far from the horizon. This has to do with how things distort when they occupy a wide angle of your vision. I do not have a solution worked out for this so just use your judgement.

[Drawing 3 point perspective.:]

1) Draw a circle. Draw a line through the centre of the circle, where the line intersects the circle defines out 2 vanishing points. Locate a 3rd vanishing point above or below this horizon, outside of the circle and horizontally constrained between the first two points.

With the two vanishing points on the horizon, and 3rd vanishing point, we have a triangle.

2) We will now treat each side of this triangle as a horizon, and draw the 2 point square solution up till step 5.

3) Draw the true vertical:
Draw a line perpendicular to the horizon from the 3rd vanishing point to the horizon. This is our true vertical. Due to a quirk of geometry (the parallel objects converge rule), this will always be in our centre of vision, and for the purposes of illustration, in our picture.

you can test this by looking at a tall blocky building. When you look at the ground level, the building is wider that it appears at the top, and when you look at a specific corner, it is exactly vertical. But when you focus on one vertical, pay attention to how the far side of the building gets closer to the corner that you are focusing on, it looks a little tilted doesn't it?

4) Start drawing the object:
By locate a point on the true vertical, and drawing rays through it from the 3 aspect points to the their respective circles.

5) Locate the (6) repetition points.
Draw lines from where the aspect lines intersect with their circles back to their repetition locator points. Where these lines cross the triangle is where the Repetition points are.

6) Draw your cube.
Draw a line from each vanishing point to our object point. Measure off how big you want your cube on one of the these lines, and draw a line from that length back to the appropriate vanishing points. draw lines from vanishing points through the intersections of the aspect lines, to find squares. repeat for all three sides of the front of the cube. Continue for the back if necessary.

And dang-it. noticed something else. What I thought were aspect points are actually aspect locator points. the aspect points are where the line crosses the triangle. At least it is consistent