Here is the approach to perspective I used in "Aly’s Lion".
To figure out the scale at any point by measuring to that point from the horizon:
(( 1- (( horizon height from base – y from the horizon) divided by horizon height from base)) times base scale)
To figure the distance:
(1 divided by (1-(( horizon height from base- y from horizon) divided by horizon height from base)) times ground distance)
However, the problem is that it is not convenient to measure from the horizon, or where the eye level falls on the paper, because often you can’t see where it is, and because the measuring of it from there would smudge up the drawing.
So we can measure from the ground instead:
((( the value of the horizon height from base – y from the ground) divided by the value of the horizon height from the base) times the base scale)
This would correspond to a one point perspective. I figured the station point at the corner of her eye on your right side. For the scale of each point of x (what is to your right or your left) , and how it would diminish with distance from the viewer (the z ) I visualize two imaginary lines running through the station point. One is the horizon, which is the y line, and the other is perpendicular to it which is the x. The base scale on this picture was 35/138 of life and the ground is 11 and 1/2 feet away. The way I figured that is by measuring from the corner of the eye to the chin . when the head is level and not tilted up or down this chin to eye corner distance this is usually 20/39 of the chin to crown distance of the head, and the head here is 9/70 of the figure height. The face here is based on Alyson Michalka. I figure her head is 8 and 3/4 inches from chin to crown and that would make her 68 inches tall. I haven’t done a detailed study of her proportions yet though. So it is just my working theory at the moment.
There are some problems with my perspective approach here. Imagine that you are looking at a wall. If it is the same height of your eye line and perpendicular to you then the top of the wall marks your horizon. The ends of the wall to your right and left are farther away from you than is the center, and if you drew a wide enough scene of the wall you would have to diminish the scale to the right and left towards the horizon. The way I drew "Aly’s Lion" wouldn’t do that, that is diminish with the distance of positive X or negative X
So I think that I need to re-think it a bit. I’m thinking about use rho, which is the square root of radius squared plus distance (z) squared. The radius is the square root of x squared plus y squared. On my calculator you could also say that rho is the square root of x squared plus y squared plus z squared. There is a small difference in the results but this is literally not rocket science. It is really hard to mentally fathom a three dimensional triangle but basically rho is the hypotenuse of the three dimensional right triangle where the other sides are x (right or left), y (up or down), and z (the distance straight in front of you.)
So what I would like to do is use rho to determine scale, and have a formula which includes angles of view. Our normal angle of view that is in sharp focus id considered to be the arc tangent of 1/4 from the horizon (the center vertex of our cone of vision) , to the ground ( where the cone intersects it). I think it would be great to draw a scene with a "fish eye" view where the scale diminishes quickly with distance from the station point, or of a telephoto view where the scale diminishes slowly from the station point.
My two limitations are time, and the fact that I am not very good at math. If anyone enjoys a little challenge, and is good at math I would appreciate any suggestions. Thanks
bambootiger 