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

Question

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)?

Answer 1

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 θ_h), and the aspect ratio (horizontal:vertical) of the imaging system, A = h/v. So, the horizontal measurement of the sensor is θ_h∙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:

θ_v = θ_w / (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.

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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.