How to enlarge an object by x-times with an optical zoom

Question

My camera employs a 30x optical zoom, with focal length ('f' from here on out): f = 4.3 mm to 129 mm.

If I am at 1x and I see an object as 100 pixels, to enlarge it to 150 pixels, can I set the zoom to 1.5x? This is the main question.

Then, I ask other questions from academic interests:

I understand that a zoom of 1.5x changes the focal length to 1.5*4.3 = 6.45. In "How do zoom, magnification, and focal length relate?", it is said that the the relation between focal length and the zooming-in is:

...quite linear. Double focal length , half width and height of the target in front of you.

So it should be correct to proceed as I initially said. However, I would like more information. Is it exact? What is the math behind it?

Following other answers, I get that varying the focal length varies the angle of view, and hence the size of the objects. On Wikipedia Angle of View there is a derivation for the angle of view. There are two questions: First, the formula is 2*atan(d/(2*f)). So, Wikipedia shows the relationship as not linear...? Second, the derivation assumes a focus at infinity. With a focus not at infinity, the angle of view also depends on the distance of the object! But probably for objects greater than or equal to 1 meter, there is no noticeable difference?

Answer 1

The focal length of a lens is a measurement taken when the lens is imaging an object at infinity. Light rays from such an object arrive parallel. The lens then focuses this image and we take a measurement from a point called the rear nodal to the image plane. This distance is inscribed as the focal length.

Now the lens has limited ability to refract (bend inward) light rays. If the object is closer than infinity, the distance downstream from lens elongates. Thus we must rack the lens forward to obtain focus. We can approximately calculate the back focus if the focal length is known and the distance lens to object.

Assume 30mm mounted and an object 250mm forward of the lens. We approximately calculate the back focus distance by converting both focal length and distance to diopter units.

For the 30mm = 1/30 X 1000 = 33.333d

For the 250mm = 1/250 X 1000 = 4d

We change the sign of the object distance diopter power and add.

33.333 + -4 = 29.33d

We convert back to millimeters

1/29.33 X 1000 = 34.09mm back focus distance

An easy approach: Draw imaginary lines from the top and bottom of the object to the center of the lens. This traces out a triangle. Say the object is 250mm forward of the lens and the object is 10mm in height. The ratio height to distance is 10/250 = 0.04.

Inside the camera the image forming rays trace out a similar triangle, the image triangle has the same angles as the object triangle however the sides and height are different but proportional.

The image triangle back focus distance is the height of the image triangle. The height of image triangle is 34.09mm. An object 10mm in height will image 34.09 X 0.04 = 0.96mm in height.

The magnification is 0.96/10 = 0.96X

As to angle of view:

As to angle of view: Assume compact digital format size 16mm height by 24mm length. The diagonal of this rectangle is 28.8444. Assume 30mm lens

Height angle of view = 29.9°

Length angle of view = 43.6°

Diagonal angle of view = 51.4°

With the object at 250mm distance the back focus of the 30mm lens is 34.09mm

Height angle of view = 26.4°

Length angle of view = 38.8°

Diagonal angle of view = 45.9°