# Diagonal Angle of View given Horizontal or Vertical angles of view for equidistant lenses

I am struggling to find the correct mathematical relation between the horizontal and diagonal angle of view for a given aspect ratio for equidistant lenses.

For rectilinear lenses, it is possible to obtain vertical and diagonal angles from the horizontal and the aspect ratio. But I am not sure how to do this for equidistant lenses.

I saw here that the relation between horizontal and vertical angle of view is linear with the aspect ratio. By the way, what is the exact relation (that website specifies is approximated)?

• @scottbb I do not see how. I saw that question before posting and I think mine is complete different. I'm not event talking about focal length. – Javi V Jul 27 '18 at 13:06
• What are you actually trying to do? I know you're trying to calculate the angle of view. But to what end? What specific purpose do you have? And perhaps more importantly, to what degree of precision? – scottbb Jul 27 '18 at 13:25
• I saw that question before posting and I think mine is complete different. I'm not event talking about focal length. If you are trying to get the angle of view (as indicated in your comment to @MichaelClark), then you must take focal length into account. Reread that question (especially the accepted answer), and also read the wikipedia article Angle of view to understand the derivation. – scottbb Jul 27 '18 at 13:38
• @scottbb I am doing some analysis independent of the sensor to use. For rectilinear lenses, if you have the horizontal angle of view (sorry for saying FOV), and the aspect ratio of the sensor, it is possible to know vertical and diagonal angles, without considering focal length, sensor size, or anything. I am asking whether it is possible to do the same for equidistant lenses, and what is the formula. Precision is appreciated (I would like to know the exact math behind the formula) but approximated solutions are also OK. – Javi V Jul 27 '18 at 13:55

For lenses with an equidistant mapping function, the mapping function is given by

r = ƒ∙θ

where

• ƒ is the focal length of the lens;
• θ is the angle of an object from the lens's optical axis; and
• r is the linear distance of the image of that object from the center of the camera's sensor.

You state that you know or are given the horizontal angle of view of the camera+lens system (we'll call it θh), and the aspect ratio (horizontal:vertical) of the imaging system, A = h/v. So, the horizontal measurement of the sensor is θh∙ƒ units (probably millimeters).

The vertical dimension of the sensor is just the horizontal dimension divided by the aspect ratio. So the vertical angle of view is just the horizontal angle of view divided by the aspect ratio:

θv = θw / (A)

The length of the diagonal of the sensor is found from the Pythagorean theorem:

d = √(h² + v²)
= √(h² + (h/A)²)
= √(h²(1 + 1/A²))
= h ∙ √(1 + 1/A²)

Because of the equidistant mapping function, the diagonal angle of view is just √(1 + 1/A²) times the given horizontal angle of view.

I saw here that the relation between horizontal and vertical angle of view is linear with the aspect ratio. By the way, what is the exact relation (that website specifies is approximated)?

The relation is exact, to the extent that a particular lens is described by the equidistant mapping function.

• Thanks a lot. Sorry for the trouble with the question at the beginning, but this is very helpful! – Javi V Jul 30 '18 at 13:37
• @JaviV No worries. Happy you found it helpful. – scottbb Jul 30 '18 at 14:55
• to follow up on this, I saw some provides that reach 180 deg FOV horizontal, vertical, and diagonal. This makes me think that your reply applies up to some extent. What do you think? – Javi V Sep 3 '18 at 13:45
• @JaviV No, if the lens uses equidistant mapping, then the mapping function holds. Obviously, any measurements outside of the lens's image circle can't be made. – scottbb Sep 3 '18 at 14:15
• Not sure I follow. Let me show an example: an equidistant lens+ sensor with FOV 80 deg horizontal, 60 deg vertical, would have a diagonal FOV of 100 degrees. But for a 180H x 180V setup, it seems that the diagonal is also 180 deg. From your reply, I understand that this is not possible unless the sensor is actually circular? – Javi V Sep 3 '18 at 16:03