Lens Angle of View

The Angle of View of a lens, or what the lens "sees", is an important element in photography.  With the ubiquitous zoom lenses available today this is more a function of what we see in the viewfinder or on the screen, but for certain applications, such as 360° panoramic photography it is useful to actually know the Angle of View for a particular lens so that the number of shots, and their orientation, can be calculated.

In the era of 35mm Film photography, what the lens sees was expressed as the focal length of the lens.  For example, a 24mm or 28mm lens is a Wide Angle lens, the 24mm being more wide angle than the 28mm, and a 200mm or 300mm lens is a Telephoto or Long lens, the 300mm having a narrower Angle of View than the 200mm.

Although this was the convention, the Angle of View of lenses was often shown graphically, such as in the diagram on the right which shows the Angle of View for a 24mm, 28mm, 35mm, 45mm, 80mm, 120mm, 200mm and 300mm lens when used with a 35mm Film camera.

With the advent of Digital photography came a variety of sensor sizes and formats resulting in the focal length of the lens becoming a meaningless number when considering its Angle of View, unless the sensor dimensions are known. The camera manufacturers, retailers and magazines adopted the convention of quoting the 35mm equivalent so that those familiar with the lenses used for 35mm Film cameras could relate to the Angle of View (e.g. focal length 5.2-26.0mm (29-145mm - 35mm equivalent).  This is fine for those who remember 35mm Film photography!  I feel however that a golden opportunity to adopt the convention of expressing what the lens sees as its Angle of View was missed.  Quoting an Angle of View for a lens for a particular camera would mean that the knowledge of the lens focal length and sensor dimensions would not be necessary in order to understand what the lens sees.

Quoting the Angle of View still has a complication when it comes to calculating the number and orientation of images required for a 360° panorama because the convention is to quote the Angle of View across the diagonal of the image in the same way as TV and VDU screen sizes are expressed.  The information we need for our 360° Panorama calculations are the angle subtended by the height and width of the image rather than the diagonal.

Theoretically, the Angle of View can be calculated using plane trigonometry.  The focal length of the lens and half the dimension in the image forms a right angled triangle so that half the Angle of View = arctangent of half the dimension divided by the focal length.

AoV = 2 x (tan^-1 ( ( d / 2 ) / f ) 

where d is the dimension in the image (e.g. diagonal, width, height)

e.g. For a 45mm lens on a 35mm camera where the diagonal = 43.3mm, width = 36mm and height = 24mm, the respective Angles of View are 51.4°, 43.6° and 29.9°.

This formula works well for the longer focal lengths, but becomes unreliable as the focal length decreases and does not work at all for fisheye lenses, so I decided that the solution was to determine the values empirically.  To do this I set out a semicircle of nails at 10° intervals and photographed them with a camera with a Nikon APS-C sensor (23.6mm x 15.7mm) and with the Nodal Point of the lens at the centre of the semicircle.  The results are shown in the following table:

Comparing the calculated Angles of View with the measured values for the Sigma 10-20mm zoom lens and the Nikon 10.5mm fisheye lens shows that the values calculated trigonometrically for the rectilinear Sigma 10-20mm lens are close enough to the measured values for determining the number and orientation of images for a 360° Panorama, but that we need the measured values when considering the fisheye lenses.  The close correlation of the calculated and measured values for the Sigma 10-20mm lens is due to the lens design and single front Nodal Point of this lens.

The following images show the difference between using the lenses with a 3mm Film SLR camera (Nikon F80) and a DSLR with an APS-C sensor (Nikon D60).  The lenses used to produce these images are designed for use with the DSLR sensor hence the severe vignetting on the images taken with the Nikon F80.  The exception is the Sigma 8mm fisheye, which produces a 180° full circle  image when used with the 35mm camera and a 180° Angle of View across the width of the image with the DSLR.

35mm Film Camera (Nikon F80)	APS-C Digital Camera (Nikon D60)

The following images show the Angle of View for the Sigma 8mm and Nikon 10.5mm Fisheye lenses and the Sigma 10-20mm Zoom Lens set to 10mm then 20mm, for a DSLR with an APS-C sensor, at 10° intervals.

Angle of View of Fisheye Lenses	Angle of View of Rectilinear Lens

		Determining the Nodal Point of a Lens For a lens to be used effectively for Photographic Intersection the location of the Front Nodal Point must be accurately determined.
		Making use of the Nodal Point of a Lens Comments on making use of the knowledge of the behaviour and location of the Nodal point of a lens for various Panorama and Photographic Intersection applications.
		Links to Panorama Related Sites and Software Links include PanoramaStudio, PTGui, Pano2VR, panoramic tripod head suppliers and 360° panoramas
		Focus and Exposure Comments on getting the correct Focus and Exposure for making Panoramas.
		Spherical Panoramas with a Normal Lens Using a normal (rectilinear) lens for Spherical (360°) Panoramas
		Create your own Panoramas Panoramas are straightforward to create from images from just about any camera, with the help of low cost software.  This page is intended to encourage you to create your own Panoramas.