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Is there any way to find the scale of the orthographic camera so that it matches the view field from the perspective camera? So for example if I render a plane in perspective mode and I see a certain part of the plane on that image without knowing how large this part is in blender units, how do I set the orthographic camera to see the exact same image?

Or to put it differently, if two points are exactly one pixel apart on a plane at a certain distance in perspective mode with a certain focal length, I need them to also be one pixel apart at orthographic mode and I'm missing the orthographic scale.

And again, I don't really have any information about the size of the view in blender units, so I hope there's a solution that only needs the distance of the plane to the camera and the focal length of the perspective camera.

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As @ZargulTheWizard has stated, there's a simple solution to this problem. By using the formula object size in image = focal length * object size / object distance and knowing that the object size in image is the sensor size, I can calculate the visible part of the plane in blender units and therefore get the scale factor

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