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How does the size/length of an object vary with distance?

Is it a logarithmic relationship? exponential? linear?

I plotted a curve of the size/length of an object for different distances from the camera, and the curve looked exponential/logarithmic. I was trying to understand the reasoning behind that.

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The relationship is a simple inverse, i.e.

 object size in image = Object size * focal length / object distance from camera

If you keep the same object and the same focal length you get: size = 1/ distance (the =-sign should be proportional-sign).

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Thanks. How is that different when it's object size vs object location image in terms of pixel (x,y) coordinates – fmvpsenior Jul 17 '13 at 16:48
It's not different, pixels are just a different unit than say meters. You can change units by knowing the size of one pixel. – Unapiedra Jul 17 '13 at 17:02

I'm sure this is a duplicate, but I can't find a good answer to the question in the archives so here goes.

The relationship between object size and distance is an inverse linear relationship, i.e. size is 1 / distance. This makes sense when you think about it as if you double the distance the size halves.

This is why you appear to be observing an exponential: the exponent is -1, if you take the reciprocal of the size, your graph should be a straight line.

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Thanks. I am actually plotting the size of the object vs distance in pixels (or location). So I start with the object being on the far left of the image (pixel coord x = 0) and end with the object being on the far right of the image along the same line (pixel coord x=3000). I wonder if the relationship is the same in that case. – fmvpsenior Jul 17 '13 at 16:47

Inversely linear is a good approximation. Imagine a 1,7m tall girl at 1 m distance b. Her head is at point B.

triangle that name corners and sides

How does the size/length of an object vary with distance?

Let the girl walk away from you. Her size a stays the same. She appears smaller, because she is appearing under a smaller angle. Her angular size changes. Try to imagine it with the picture attached. Using arctangent to calculate her angular size is the right way. For small angles you can simplify:

Angular size is inversely proportional to its object distance, without using optical devices.

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