I apologize for the text changes. I should have edited it in “frontpage” before posting this here, but this is my first submittal so I hope that everyone will make allowances. This is one of the posts I have in my blog on windows live, but it is far older than my blog is and goes back through several computer crashes. I have posted this a number of places on the net before.
The research here has a lot do with my overall research that I have been engaged in on a part time basis for 19 years now. I have studies hundreds, or perhaps thousands of pictures of people to look for what in common mathematical relationships are involved in our, or perhaps just my, perception of beauty, I have found that some relationships described by the same or by very similar rational numbers are a recurring descriptive theme. These include the proportion of head to body and of features within the face. So I have a few theories drawn from this on how to incorporate this relational framework into an image, and also what variations might work well. Up to here it is relatively simple math, but I am fitting it in with a math theory I am working on in the domain of relationships within three dimensional space and the cone of visual perception of the human eye and how to represent this in two dimensions without using the usual method of vanishing points. This was a lot more difficult for me to grasp and I had to study first trigonometry and then spherical trigonometry on my own, then I derived formulas to translate from the left handed view of Cartesian coordinates to spherical coordinates, then worked out a simplified approach which uses a formula expressed with four sets of parentheses. I have an illustration of the math applied to what would correspond to a view of one point perspective with everything drawn to relative scale, if you would like to see it go to
Look at the picture called “Aly’s Lion”
I have the formula expanded now to work out a more complicated view which I will illustrate in the future, but what I would like to do is to express the formula more clearly and concisely. I have an idea now about how I might accomplish it using distributive properties. Of course I am not really good at math and don’t have a lot of spare time so this is proving to be quite a challenge, but for me that is what makes it fun and interesting.
I should have edited it in “frontpage” before posting this here
This site uses textile markup. I don’t think Frontpage would have worked.
I’m hardly familiar with all that you are describing here, but certain aspects seem to be in parallel with some of the workings of 3D CAD software.
Let’s fantasize for a minute about a software solution for this.
First, I hope I’m understanding that what you are basically trying to do is capture the relative scale of objects in a portrait based on their distance from the ‘view’, without using any vanishing points as your reference for their size. It’s really the drawn size of the familiar objects that determine their perceived distance; but the problem is determining what size to draw them to give this illusion in 2 dimensions.
So imagine you had a software tool that allowed you to “place” objects where you wanted them in both an X, Y and Z position. (Z being the ‘depth’ component or distance from the viewer.)
A simple representation of each object would be its extents box. Draw the extents box in the software to the basic real dimensions of what a girl and a lion would measure. Draw the extent boxes for the other objects to their real dimensioned size. The planter in the background, the fountain on the wall, the arches, etc.
Objects represented in 3D software usually have several different transformation matrices attached to them. In the case of manufacturing software there are position and rotation matrices for the relative positions of each object, but there are also view-port matrices, or view transforms that handle rotating and zooming all the objects on the screen. The importance here is the scale component of these transforms. If a certain real sized object has a much greater Z depth in the current view, its scale is adjusted to represent this difference and make it appear farther away.
So imagine “placing” each real-sized drawn extents box for each object in the picture in its 3 dimensional place you want it to be. The scale of each object will adjust via the software. What if the software could tell you the scale factor it calculates for each object after placement? (Let’s assume the girl and the lion have a scale factor of “1”.)
Many 3D softwares have programming API’s that allow you to retrieve this information, but you have to know the programming language and how to set up what I’ve described properly.
If all this were in place, you could then easily apply your “beauty formulas” of symmetry, etc without having to go through painful scale calculations for every variation tried.
I have no idea if this helps you in any way, but the 3D software has to calculate all this in real time, so there must be a way to do this for each object using the transformation matrices and scale calculations that these softwares use.
You might also find this link interesting. (PDF)
Too bad the actual software doesn’t seem to be immediately available.
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