WP's SloMo

Focal length calculator

Image circle, format and crop factor

Easy to use calculator for the distance to object with given focal length and for focal length with given distance to object keeping the condition of format filling imaging. Thus making best utilization of the photoactive area of the imaging medium. (JavaScript required.)
One has to keep at least, however, a distance of 0.3 m (about one foot) (sometimes more than 1 m; about three feet) using standard lenses, otherwise they cannot provide sharp images.

By the way the real sensor size can be estimated when transforming the equations given below. Object size O and object distance d are just measured in a test set, the focal length f is that of the lens inscription:

S = f × O / (d - f); (all values in mm)

Then one discovers the crop-factor is just a sensor property and not a lens characteristic, see [SloMo Info].

Δ

Format filling imaging

Focal distance calculator

Expected is English (imperial) notation: please type in a full stop ».« as decimal punctuation mark and take notice of the different measurement units.
(By the way: 1 inch = 25.4 mm, 1 foot = 304.8 mm, 1 yard = 914.4 mm = 0.9144 m, 1 mile = 1 609.34 m)

Editorial

Please note, the sketch supports constructing the locations and sizes of object and image only and does not describe the optical path.

• d: distance between object scene and camera lens (exact object side located main plane of the imaging system)

• f: focal length inscription on the lens as a rule in millimeters; will be in accordance with reality, if the lens is mounted with the adequate adapter

• O: maximum expansion of the object vertical to the imaging direction; in simple words: what's to be seen. One can select the diagonal as well

• S: minimum expansion of the active sensor area or film vertical to the imaging direction. One can select the diagonal as well. See [SloMo Info] for values.
  By the way: the sensors are much smaller than the format diagonal given in inch seems to claim. Thus a 1 inch sensor never has a diagonal of 25.4 mm, but just one of 15.875 mm (about 5/8 inch). This ratio can be used as »conversion factor« for other formats.

• 2ω: angle of view or field of view 2ω = 2 × arctan (1/2 × O / d) = 2 × arctan (1/2 × S / f'); with the focal length of the lens f' (not really f of the lens inscription, but the flange-back, because usually a camera lens consists of several lenses); [values in mm].

Δ

Depth of field, depth of focus

Evidently one likes to name it bokeh (in words the artificial impression). You will find very detailed info in depth here in the paper of the Carl ZEISS AG.
Each of the following terms, i.e. their sum, give the depth of field (or depth of focus) region where objects are in focus, thus appear sharp. (All in SI units - meters or millimeters.)
The in-focus region in front of the set distance is:

d_front = d × f² / (f² + k × Ø × (d - f))

And the in-focus region behind this distance is:

d_rear = d × f² / (f² - k × Ø × (d + f))

Signs and symbols:

• d = set distance on the focus ring

• f = focal length of the lens

• k = f-stop number

• Ø = diameter of the circle of confusion, as a rule of thumb about the diagonal of the sensor format [in mm] / 1 500

The diameter of the circle of confusion is given by the minimal distance of two spots the human eye is able to identify from a certain distance. A standard value is 2 angle minutes from 250 mm distance, thus 0.0145 mm. Or 6 line pairs per millimeter as well. (Strictly speaking one has to calculate this from the output medium to the sensor or film, resp. backwards.)

Setting d to so-called hyperfocal distance h = f × (f / (k × Ø) + 1) makes the region starting at h/2 to infinity appear in focus. Many simple cameras work in this way at a distance of about 1.2 meters (4 feet).

Δ

Equations of movement

Estimating the size of the interesting area of moving objects and the occurring velocities, here the equations of movement in simple form. (All in SI units meters and seconds.)

Straight movement

Signs and symbols:

• Velocity: v = v(t) [m/s]; t: time [s]; v₀ = v(0)
  
  

• Acceleration: a [m/s²]; covered way: w [m]

v = v₀ + a × t

w = v₀ × t + a/2 × t²

2 × a × w = v² - v₀²

Orbit movement

Signs and symbols:

• Angle velocity: ω = 2 × π × f; frequency: f = K / t = 1/T [1/s]; K: count of rounds; t: time [s]; T: duration of one turn [s]; π = 3.14
  
  

• Orbital radius: r [m]

Orbital velocity: v = r × ω

Free fall and inclined plane

Free fall and inclined plane

Basic principle is the conservation of energy: E_pot = E_kin. Hereby the mass of the object is abridged. Friction is supposed to be negligible. And - important - the object must not significantly gather rotation energy, thus it should preferentially glide down the ramp without friction. (But for a car - model or real - the formula works.)

Signs and symbols:

• Potential energy (energy of level): E_pot = m × g × h; m: mass of the object; Earth gravity acceleration: g = 9.81 m/s²; h: dropping level or altitude of the inclined plane, resp.

• Kinetic energy (energy of movement): E_kin = m/2 × v²; m: mass of the object; v: velocity of the object at impact, i.e. after falling the distance h, or when leaving the inclined plane

v = √(2 × g × h)
h = 1/2 × v² / g

Δ

Special cases

Every theory finds its challenge in reality...

Motion blur

During exposure the object moves the distance

Δw = v × t_shutter

the product of its velocity and time of exposure of each frame.

Example: a car moves with constant velocity v = 64 km/h = 17.78 m/s. With a frame rate F = 1 000 frames/sec the car will change its position in one frame up to

Δw = v × t = v / F = 17.78 m/s / 1 000 frames/sec = 0.01778 m = 17.78 mm

if one does not reduce the time of exposure for each frame by using a shutter. If the image of Δw is smaller than one pixel, the effect will not be visible.

Extending the path by cross movement

Cross movement - diagonal view at the object

If the camera does not perpendicularly look onto the object, but an angle different from 90° lies between camera view and object or its direction of movement, resp., e.g. if a vehicle moves towards the camera, thus the distance to be caught is reduced to about

w_{|_} = w' × sin α

with w_{|_} (= w) as perpendicular component of the drive path w' and the angle α between viewing direction and vehicle drive path w'.

As the figure on the right shows one is possibly able to gain some more path when selecting an appropriate setup.
So one has the chance to capture more details in space and time, thus imaging the object and its movement in higher resolution.

In this setup, however, one must take in consideration depth of field, because the distance to the camera may heavily change.

Small »world formula« from the school days

Also important in the following years, the relationship between the common units:

1 N m = 1 J = 1 kg m² / s² = 1 V A s = 1 W s = 1 V C;
force: Newton [N]; energy = work: Joule [J]; power: Watt [W]; charge: Coulomb [C]