A Geometry/Perspective Question

[gaming_mouse]

Consider the image below. The green screen in the center that the man is watching is a rectangle with an aspect ration of 1:1.73.


We can calculate the coordinates relative to the origin of the entire image (ie, the image's upper left corner = (0,0)) of the three visible coordinates of the green screen: lower-left, upper-left, upper-right. The lower-right is being blocked by the man's head.

The question is: Given ONLY the coordinates of the three visible corners of the green screen, how can we calculate the coordinates of the lower right corner?

EDIT: I realize the coords listed in the picture may not be exactly consistent with the lines they need to fall on, since they were taken from photoshop and I had to eyeball what defined the corner in each particular case. Nevertheless, they should be close enough and I am only interested in the method anyway.

[MtDon]

[ QUOTE ]
Consider the image below. The green screen in the center that the man is watching is a rectangle with an aspect ration of 1:1.73.


We can calculate the coordinates relative to the origin of the entire image (ie, the image's upper left corner = (0,0)) of the three visible coordinates of the green screen: lower-left, upper-left, upper-right. The lower-right is being blocked by the man's head.

The question is: Given ONLY the coordinates of the three visible corners of the green screen, how can we calculate the coordinates of the lower right corner?

EDIT: I realize the coords listed in the picture may not be exactly consistent with the lines they need to fall on, since they were taken from photoshop and I had to eyeball what defined the corner in each particular case. Nevertheless, they should be close enough and I am only interested in the method anyway.


[/ QUOTE ]

I'm not sure, but I don't think the three coordinates are enough to accurately determine the coordinate of the fourth corner of the screen.

In the image, none of the edges of the screen are parallel with any other edge.

A complication is that the image is a photo and not a drawing using classical perspective drawing techniques.

I think you would need to know some or all of the following information:

1. Location of the camera with respect to the screen. (x,y,z)

2. Direction the camera was pointing.

3. Focal length of the camera's lense.

4. Format of the camera, i.e., 35mm, large format, digital, etc.

************************************************** ****

However, with a few more coordinates of points on the bottom edge and the right edge, you should be able to fit curves (possibly straight lines, mabe not) to the edges and find their intersection.

You should also be able to do it graphicly by drawing lines/curves (not necessarily straight lines) for the bottom and right edges and locating their intersection. This would seem to be the easiest way of doing it.

When I wrote this I did not consider the fact that you wrote that you know the aspect ratio of the screen. However, your question seems to imply that this is not to be taken into consideration. If it can be, I haven't thought about it enough to say whether my comments would still hold true. Off hand, I think they still would.

[gaming_mouse]

[ QUOTE ]
I'm not sure, but I don't think the three coordinates are enough to accurately determine the coordinate of the fourth corner of the screen.

[/ QUOTE ]

I'm not sure either, but I'd bet some small amount of money that the 3 points, plus the true aspect ratio of the screen in question (1:1.73) are enough to determine the 4th point.

The fact that it's a photo and not a drawing shouldn't mean that it won't follow the laws of perspective.

But as I said I'm not sure either.....

[RocketManJames]

Maybe I'm wrong here, but I would think that the slopes of the each edge of that frame would remain constant. We would have 4 different lines defined by 4 different slopes. So, I would have two lines defined by the right edge and the bottom edge of the frame. From those two lines, I would be able to guess at the intersection.

But, maybe there's an obvious reason why the slope of each frame edge does not remain constant. Weird warping perhaps?

Nevermind... I misread the problem. You wanted to use ONLY the 3 coordinates given. My bad.

-RMJ

[SuitedSixes]

I haven't thought about Geometry since those three times in college, but don't the slopes of the two opposite edges need to match? So if you find the slope of the far left side, can't you determine the slope of the far right side and solve for the missing point? Or am I not making this complicated enough?

[gaming_mouse]

[ QUOTE ]
I haven't thought about Geometry since those three times in college, but don't the slopes of the two opposite edges need to match? So if you find the slope of the far left side, can't you determine the slope of the far right side and solve for the missing point? Or am I not making this complicated enough?

[/ QUOTE ]

No, the slopes of opposing sides are not parallel. If you extend them they will eventually meet at a vanishing point.

[bholdr]

It's impossible. (i think)

without knowing what the exact position of the camera was relitive to the screen (including the length of the lens, weather or not it is corrected for paralax at all, and so on) all you have there is a triangle. there are an infinite possible number of potential quadralaterals composed of the three points on a triangle and another undefined point; there is no solution.

[gaming_mouse]

[ QUOTE ]
It's impossible. (i think)

without knowing what the exact position of the camera was relitive to the screen (including the length of the lens, weather or not it is corrected for paralax at all, and so on) all you have there is a triangle. there are an infinite possible number of potential quadralaterals composed of the three points on a triangle and another undefined point; there is no solution.

[/ QUOTE ]

I don't think so. Keep in mind that we ALSO know the aspect ratio of the screen. I think this fact and the 3 points determine the quadrilateral. But I'm not 100% sure....

[bholdr]

correct me if i'm wrong: the aspect ratio is just the ratio of the screen height to the width, right?

the problem with your question is that you've provided two dimensional coordinates- with the third coordinate, it's possible, because we could determine the slope of the screen relitive to the camera, but without it, like i said, all we have is a triangle.

[nycplayer]

without establishing where the horizon/eye-level is, this can't be solved, I think.