What is optical computing

Calculate reproduction scale (derivation)

The reproduction scale is the answer to the question: How often does the object fit into its picture? If you know the image scale, you can say how long a distance is in the picture, provided that this distance was at the distance with which the image scale was calculated and ran on a plane parallel to the sensor (film).

definition

The image scale β (Greek letter small beta) is the ratio of the image size B. (Image on sensor, film) to the object size G (photographed object):

(1) β = B ⁄ G

If you divide the image size by the image scale, you know how big the depicted object is (formula according to G dissolve):

(2) G = B ⁄ β

Considerations

B. and G are greater than zero, otherwise there would be no picture. One differentiates (B ⁄ G = β may be):

• The reproduction scale is greater than onewhen the image size B. is larger than the item size G. In other words: the object is shown enlarged. Enlarged images are saved as Macro shots designated. (Definition according to Brockhaus encyclopedia [Brockhaus19] and by the authors Kurt Dieter Solf and Otto Croy).
• The reproduction scale is less than 1. The picture size is smaller than the item size. In other words: the object is shown reduced in size. This is the case with close-up and long-distance shots. The boundary between close-up and long-distance photography is not uniformly defined. I use the term Close up for images in the image scale range from 0.1 to 1. I call scales smaller than 0.1 as Remote shooting.
• The reproduction scale is equal to 1. Image and object size are the same, the object is shown in its real size on the sensor (film).

The reproduction scale is given in decimal form or as a fraction which is shortened with the smaller number. In the event of a break, you can see directly how many times smaller or larger than the object the picture is. 1: 5 means the picture (the picture) is 5 times smaller than the object, 5: 1 means the picture is 5 times larger.

Reproduction scale based on the object distance

In this article, the reproduction ratio is calculated using the object distance. This is Not the distance setting on the lens. The Distance adjustment is the distance of the subject from the sensor plane (film plane). The object distance is measured from the main plane on the object side and not from the sensor (film) like the distance setting. The distance to the object can be calculated from the distance setting (symbol legend). With small image scales, the distance setting and the object distance can be set to be the same.

The reproduction scale only applies to the object plane that is in focus.

Derivation:

Figure 1 shows the construction of an image of an object standing on the optical axis.

symbolimportance
α(Greek letter small alpha) equal angle on image and object side
B.Image size
bImage distance
F.Focus
fFocal length
GItem size
GObject distance

Illustration 1:Illustration of an object. The object G stands on the (horizontal) optical axis. The main plane on the image side and the object side (vertical in the middle) are combined for illustration. Rays parallel to the optical axis pass through the focal point on the image side. Rays through the optical center point (point of intersection of the optical axis with the main planes) are not refracted. The image distance b is the distance of the image from the main plane on the image side, the object distance G the distance of the object from the main plane on the object side.

The image scale β is the ratio of image size to item size:

(3) β = B ⁄ G

Because of the similarity of the triangles (Figure 1), the tangent relationship applies:

(4) B ⁄G = b ⁄ g
(also g ⁄ G = b ⁄ B)

The lens equation

(5) 1 ⁄ f = 1 ⁄ b + 1 ⁄ g

dissolved after b:

(6) b = (f * g) ⁄ (g - f)

(6) inserted in (4):

(7) B ⁄ G =
((f * g) ⁄ (g - f)) ⁄ g =
f ⁄ (g - f)

Short:

(8) β = f ⁄ (g - f)

(9) Image scale =
Focal length ⁄
(Object distance - focal length)

Reproduction scale based on the pull-out extension

The extension of the lens is the image distance minus the focal length on the image side.

(10) Image width =
Focal length + extension

(11) Extension extension =
Image distance - focal length

If a subject is closer than infinity, the image distance is greater than the focal length. This is achieved by pulling out the lens (= further away from the sensor, bringing the film) (you can also shorten the focal length). If the subject is infinitely far away, the extract is zero and the image distance is the same as the focal length. The excerpt increases the image width by this.

Application: The lens is set to infinity and attached to a bellows device or extension rings. The length of the bellows device or the intermediate rings is the extension.

If the lens is not set to infinity, the lens equation is used to calculate the image distance based on the object distance and subtract the focal length from this. The result is added to the statement.

In Figure 2 the excerpt is as x drawn. Explanation of the symbols:

symbolimportance
α(Greek letter small alpha) the same size angle on the image side
B.Image size
bImage distance
F.Focus
fFocal length
GItem size
GObject distance
xabstract

Figure 2:Illustration of an object, excerpt (not set to ∞). See Figure 1 for an explanation. The extract x is the distance of the image from the focal point on the image side F.. It is the difference in image width b minus image-side focal length f.

With the help of the angular relationships in similar triangles and the tangent, the image scale = B / G can be derived (see Figure 2):

(12) f ⁄ G = x ⁄ B
(also B ⁄ G = x ⁄ f)

Moved:

(13) B ⁄ G = x ⁄ f = β

In words:

(14) Image scale =
Extension ⁄ focal length

Close-up lenses

Close-up lenses are converging lenses (magnifying glasses) that are screwed onto the lens and shorten its focal length. They reduce the near limit of the lens, you can get closer to the subject.

The formulas for close-up lenses only apply if they are located close to the front lens of the objective. This is the rule in practice. Close-up lenses shorten the focal length of the lens, the image distance remains the same. Because the image scale Extract ⁄ focal length is, as previously shown, a larger image is obtained with the same object distance than without a close-up lens (the numerator = extract remains the same and the denominator = focal length becomes smaller. Result: larger fraction = larger image scale).

Object distance depending on the image distance

If you solve the lens equation for the object distance, you get:

(15) g = f * b ⁄ (b - f)

With the close-up lens in front of the lens, the image distance remains constant, it is only changed via the distance setting. But the focal length is reduced. In mathematical terms, this means: the numerator of the fraction becomes smaller and the denominator larger → The object distance (the fraction) becomes smaller → With a close-up lens in front of the lens, you are closer to the subject.

With a close-up lens in front of it, the lens cannot be set to infinity: at infinity, the lens extension is as long as the focal length. The focal length is shortened by the close-up lens. The minimal The image distance is the old focal length and longer than the new focal length

(16) old focal length minus new focal length

The longer image width acts like an extension for the new focal length. This is how close-up lenses work.

(17) Profit from extension extension =
Old focal length - New focal length

The refractive power of a lens or an objective is in Diopters specified. The diopter D. is the reciprocal of the focal length f in meters:

(18) D = 1 ⁄ f

If you mount several lenses of different or the same focal length directly one behind the other, you can determine the total focal length with the diopter formula: You add up the diopters of the individual lenses and obtain the total refractive power. The total focal length of the assembled lenses is the reciprocal of the total refractive power.

example: Three close-up lenses with diopters 1, 2 and 4 are screwed onto one another. The total dioptric number is 1 + 2 + 4 = 7. The total focal length is 1 ⁄ 7 meters, around 143 mm.

The larger the total diopter number of the lens combination, the shorter the focal length:

(19) Focal length = 1 ⁄ diopter

A fraction (= focal length) becomes smaller, the larger its denominator (= dioptre) is.

Close-up lenses (with high refractive power) can reduce the image quality. The best image quality is achieved with so-called Attachment achromats. These are two lenses mounted directly on top of one another: a diverging lens and a converging lens.

The total focal length of a lens with a close-up lens is the reciprocal of the sum of its refractive power plus the refractive power of the close-up lens.

example: A 50 mm lens has a refractive power of 1 ⁄ 0.05 = 20 diopters. If you put a close-up lens with 5 diopters in front of it, the combination has 25 diopters of refractive power (20 + 5). The focal length is 1 ⁄ 25 = 0.04 meters (40 mm).

symbolimportance
βReproduction scale
bImage distance
f (tot)Focal length lens + close-up lens
f (N)Focal length close-up lens
f (O)Focal length lens
GObject distance
g (O)Distance set on the lens
xDrawer extension

Reproduction scale based on the extension extension:

(20) β = x ⁄ f

Image distance, derived from the lens equation:

(21) b = f * g ⁄ (g - f) =
f (ges) * g ⁄ (g - f (ges))

Total focal length lens + close-up lens:

(22) 1 ⁄ f (ges) =
1 ⁄ f (O) + 1 ⁄ f (N) =
f (O) * f (N) ⁄ (f (O) + f (N))

Extract x:

(23) x = b - f (tot)

Lens equation resolved according to object distance:

(24) g = f (ges) * b ⁄ (b - f (ges))

Image scale with close-up lenses, lens not at infinity

Insert equation (23) into equation (20) for the extract and equation (22) for the focal length. In equation (23) for the image distance b Equation (21) and for the total focal length f (tot) Insert equation (22).

This is the magnification of a lens with close-up lenses that is not set to infinity:

(25) β = (f (O) * (g (O) + f (N)))
⁄ (f (N) * (g (O) - f (O)))

Image scale for close-up lenses, lens at infinity

When set to infinity, the image distance is the same as the focal length of the lens: b = f (O). This image distance is inserted into equation (23): x = f (O) - f (tot). This is inserted into equation (20) and for the focal length equation (22) in equation (20) is the image scale with the close-up lens in front, when the lens is set to infinity:

(26) β = f (O) ⁄ f (N)

Literally:

(27) Scale of the image =
Lens focal length ⁄ close-up lens focal length

Object distance with close-up lenses, lens not at infinity

In equation (24), for the total focal length f (tot) Equation (22) is used and for the image distance b Equation (21).

After forming:

(28) g = f (N) * g (O)
⁄ (g (O) + f (N))

Object distance with close-up lenses, lens at infinity

In equation (24) one sets for the total focal length f (tot) Equation (22) and for the image distance the lens focal length, which is the same as the focal length of the lens at infinity setting.

(29) g = f (N)

If the lens is set to infinity, the object distance is the same as the focal length of the close-up lens. The object width is the same regardless of the focal length of the lens.

The image scale is larger for close-up lenses with a large number of diopters. At the same time you are closer to the subject. With a close-up lens with 5 diopters (focal length 1/5 m = 0.2 m), the object distance is 20 centimeters.

One can Not Assume the frontmost lens of the lens is eight inches from the subject. The object distance is measured from the object-side main plane of the lens. If the main plane facing the object lies outside the lens, the front lens is further away from the subject than calculated. If the main plane on the object side lies within the objective, the objective front lens is closer to the subject.

In practice, longer focal length lenses are often better suited for photographing with close-up lenses: larger image scales are achieved with the same subject distance as with short focal lengths. You can go further from the motif with the same reproduction scale.

Elmar Baumann.

Last update: December 14, 2005