Are 35 & 50 too close together?

Greg7579 wrote:

I will say again (as this is a common question on all DPR Boards) that they are all very different.

This is a basic tenant of photography.

10, 14, 16, 23, 28, 35, 50, 70 and 90 are all very different from each other.

+1

A lens is a simple device that projects a conical region in space in front of the lens into a conical region behind the lens. The angle of view of that cone is determined by the sensor intersecting the cone behind the lens. That is the angle of view for a 35 mm lens on an APSC sensor is the same as the angle of view of a 50 on a FF sensor. It is this resulting cone that is important and it is determined by the focal length of the lens. a simple equation

angle_of_view=2*arctan(d/(2*f))

where d is the dimension of the sensor intersecting the back cone, a.k.a., image circle and f the focal length. Here "d" is normally taken to be the diagonal dimension of the sensor but it is it could be either, diagonal, length or width.

They are all unique. How we see and how our lens renders three dimensions onto two dimensions is determined by projective geometry as every point of light on a line to the center of the sensor is rendered at the same point on the sensor. The field of view is the intersection of the sensor with this cone. No light sources outside the cone will appear on the sensor and every light source on the ray inside will appear.

That is a reason there is a new hot subject in applied mathematics - called applied algebraic geometry because it is the foundation of computer vision or how we make robots see so they can navigate.

My favorite example:

https://photography.tutsplus.com/tutorials/exploring-how-focal-length-affects-images--photo-6508

The woman takes up the same sensor area - to user error. But the perspective of each lens is vastly different.

For any focal length the main subject can be give the same real estate on the sensor. However, everything in front and to the back and will be rendered differently for each focal length.

Of course the whole ideal of distorted perspective is debated ad nauseam on photography forums with no resolution simply because the wrong geometry is being assumed. Projective transforms do not preserve Euclidian distances and all lines intersect (even parallel lines).

https://blog.photoshelter.com/2018/06/a-mathematician-weighs-in-on-lens-compression/

To achieve 3D vision required for computer vision and robotics, like robots moving independently through space or vehicles performing automatic breaking then tools from projective and algebraic geometry are needed. A camera is a device used to intersect a projective cone with a sensor and the properties we see are those that are undergo a projective transform.

https://blog.photoshelter.com/2018/06/a-mathematician-weighs-in-on-lens-compression/

https://hal.inria.fr/inria-00548361/document

https://courses.engr.illinois.edu/cs543/sp2012/lectures/Lecture%2011%20-%20Projective%20Geometry%20and%20Camera%20Models%20-%20Vision_Spring2012.pdf

The reason for the difference in rendering of three dimensions arises out of the different properties of the projective transforms and from a simply fact that in projective space - lengths are not preserved and all lines including parallel lines intersect .