A camera on a robot wrist or a slide stage looks like it should zoom: pull it back and it sees more, bring it in and it resolves finer. On almost every integration the next question is whether the camera holds several fields of view, one per taught position. It does not. A camera with a fixed focal-length lens and a fixed sensor has exactly one angular field of view, and the working distance alone decides how much of it lands in the frame. Below is the scaling law, the arithmetic worked through at three distances, and the cases where the straight line bends.
One lens and one sensor give one field of view per distance
Picture the field of view as a pyramid that starts at the lens and opens toward the part. The pyramid's angle comes from two fixed things: the sensor's active width and the lens's focal length. Because the angle cannot change, the slice you cut at the part grows in exact proportion to the distance. At twice the distance you see twice the width and twice the height — same pyramid, bigger cross-section.
So a camera does not carry multiple fields of view. It has one field of view per working distance: a continuum, not presets. Stepped coverage needs a stepped focal length — a zoom or varifocal lens, or the multi-lens module in a phone camera. A C-mount camera carries one lens, so it climbs one pyramid.
The formula
Use the thin-lens model, with all lengths in millimetres: w and h are the sensor's active width and height, f the focal length, WD the working distance from lens to part, m the magnification and p_px the pixel pitch.
The −f term is the focus correction: a lens focused at WD magnifies by f/(WD − f), not f/WD. The common estimate FOV ≈ w × WD / f drops that term and overstates coverage by 8.7% at WD = 100 mm with an 8 mm lens (92.5 mm predicted against 85.1 mm exact), and by about 1% at WD = 800 mm. As a rule: at ten focal lengths of distance the estimate runs about 11% high; at forty, about 2.6%. Use the exact form at wrist-camera ranges, the estimate for long stations. Where the sensor's width and the focal length come from in the first place is covered in lens field of view: why sensor size and focal length set it.
Worked example: the same camera at 50, 100 and 200 mm
Take a 1/1.7" optical-format sensor — active area about 7.4 × 5.6 mm, 4000 × 3000 pixels — with an 8 mm lens. Pixel pitch: 7400 µm / 4000 px = 1.85 µm. At each distance the scale factor is (WD − f)/f:
| Working distance | Scale (WD − f)/f | Field of view | Object-side pixel |
|---|---|---|---|
| 50 mm | 42/8 = 5.25× | 7.4 × 5.25 = 39 mm × 29 mm | 1.85 × 5.25 = 9.7 µm |
| 100 mm | 92/8 = 11.5× | 7.4 × 11.5 = 85 mm × 64 mm | 1.85 × 11.5 = 21.3 µm |
| 200 mm | 192/8 = 24.0× | 7.4 × 24.0 = 178 mm × 134 mm | 1.85 × 24.0 = 44.4 µm |
Going from 100 mm to 200 mm multiplies the field by 192/92 = 2.09, a little more than double, because the (WD − f) term grows faster than WD itself at short range; from 50 mm to 100 mm the factor is 92/42 = 2.19. Each pixel's footprint at the part grows by the same factor. Coverage and pixel size are two readings of one magnification, and object-space pixel size from magnification works that side in full.
Choosing the working distance from the coverage you need
Solve the same equation backwards when the part fixes the field of view:
To cover a 200 mm wide part with the example camera: WD = 8 × 200 / 7.4 + 8 = 216.2 + 8 = 224.2 mm, and the forward formula gives 7.4 × (224.2 − 8) / 8 = 200 mm. The pixel budget then follows automatically: 200 mm across 4000 pixels is 50 µm per pixel. If that is too coarse for the defect you must see — say object pixels must stay at or under 25 µm — distance is capped at WD = 8 × 25 / 1.85 + 8 = 108.1 + 8 = 116.1 mm, which caps coverage near 7.4 × (116.1 − 8) / 8 = 100 mm and gives exactly 100 mm / 4000 px = 25 µm per pixel. Coverage and pixel size trade against each other through distance alone; when they cannot both be met, the focal length or the sensor is what changes. A complete sizing worked this way for a real part is in working distance and field of view formula for blister inspection.
Off-the-shelf fixed-lens cameras build one point of this curve in at the factory. The MV-SC5060XM-08M-WBN is a 6 MP mono smart camera with a fixed 8 mm M12 lens on a 1/2.4" sensor — the same focal length as the worked example on a narrower sensor, so the same law with a smaller pyramid.
- Interface
- GigE
- Resolution
- 6-10 MP
- Sensor format
- 1/2.4"
- Shutter
- Rolling
What else moves the pyramid: sensor width and focal length
Hold the distance and change the sensor, and coverage scales with active width: at 300 mm with a 12 mm lens, a 1" format sensor (active diagonal about 16 mm) sees roughly 1.7× the width of a 1/1.7" sensor (about 9.5 mm), because 16 / 9.5 ≈ 1.7. Hold the sensor and halve the focal length, and coverage about doubles at every distance. The three cameras below span 1/1.7" through 1.1" to 1"; the same lens on each builds three different pyramids from one mounting height.
| Specification | MV-CU020-90GC | MV-CH120-11UC | MV-CS200-10GC |
|---|---|---|---|
| Sensor type | CMOS, global shutter | CMOS, global shutter | CMOS, rolling shutter |
| Sensor model | GMAX4002 | Sony® IMX304 | Sony® IMX183 |
| Pixel size | 4 μm × 4 μm | 3.45 μm × 3.45 μm | 2.4 μm × 2.4 μm |
| Sensor size | 1/1.7" | 1.1" | 1" |
| Resolution | 2048 × 1200 | 4096 × 3000 | 5472 × 3648 |
| Max. frame rate | 49 fps @ 2048 × 1200 Bayer GB 8 | 30.3 fps @ 4096 × 3000 Bayer RG 8 | 5.9 fps @ 5472 × 3648 |
| Dynamic range | 66.9 dB | 72.2 dB | 65.5 dB |
| SNR | 40.8 dB | 40.2 dB | 41.5 dB |
| Gain | 0 dB to 11.1 dB | 0 dB to 24 dB | 0 dB to 24 dB |
| Exposure time | 4 μs to 10 sec | UltraShort exposure mode: 1 μs to 14 μs; Standard exposure mode: 15 μs to 10 sec | 46 μs to 2.5 sec |
| Exposure mode | Off/Once/Continuous exposure mode | Off/Once/Continuous exposure mode | Off/Once/Continuous exposure auto, supports Timed and Trigger Width exposure mode, and Rolling and Global Reset sensor shutter mode |
| Pixel format | Bayer GB 8/10/10Packed/12/12Packed | Mono 8/10/12; Bayer RG 8/10/10Packed/12/12Packed; YUV422Packed, YUV422_YUYV_Packed; RGB 8, BGR 8 | Mono 8/10/12, Bayer GB 8/10/10p/12/12p, YUV422Packed, YUV422_YUYV_Packed, RGB 8, BGR 8 |
| Price | $309.00 | $2,339.00 · quoted | $739.00 |
All 3 share: Binning Supports 1 × 1, 2 × 2, 4 × 4; Lens mount C-mount; Certification CE, RoHS, KC; Mono/color Color. Values as printed in each product's data sheet; open a part for its drawing and full specifications.
The floor: minimum focus distance
The formula keeps shrinking the field as you approach, but the lens stops first. Inside its minimum object distance, no focus setting gives a sharp image, so that distance defines the smallest field of view the combination can ever deliver — and it is the natural mounting height for a wrist camera that needs the finest pixels. Fixed-focus lenses have no focus ring at all and specify one focus distance instead. The MV-SCA002M-03S-WBN-SR specifies a 3 mm lens and a 240 mm focus distance on a 1/6" sensor: mount it at 240 mm for its sharpest, smallest picture, and expect every millimetre farther to widen coverage by the law above. A lens that will not focus close at all is diagnosed in lens focuses only at close distance: diagnose focus range.
- Interface
- Fast Ethernet
- Resolution
- Up to 3 MP
- Sensor format
- 1/6"
- Shutter
- Global
Where the straight-line rule bends
- Reference plane. WD in the formula runs from the lens's optical centre; data sheets usually measure from the front of the barrel or the flange. The offset is constant, so it shifts every result the same way — measure it once by focusing on a ruler at a known distance.
- Focus breathing. Travelling the focus ring changes effective focal length by a few percent on many lenses, and near the minimum distance the exact formula inherits that error.
- Distortion. The formula assumes a rectilinear projection. Wide-angle barrels bend the pyramid walls outward, so the frame corners cover more than the straight-line number; when you measure in world units, work from the calibrated model as in undistortion scaling parameter: cropping versus field of view.
- Telecentric lenses break the rule on purpose: constant magnification inside the depth of field, so coverage does not grow with distance.
- Variable focal length restores the multiple fields of view: a varifocal lens at two settings is two pyramids, each obeying the same law.
Check the numbers on the bench
- Mount the camera at the computed distance on one of the camera stands and lock the height.
- Lay a ruler in the part plane. Read the visible width at 100 mm and at 200 mm; the ratio should match (200 − f)/(100 − f), i.e. 2.09 for the 8 mm example.
- Divide the visible width by the pixels across it to get the object-side pixel size and compare with p_px × (WD − f)/f.
- Record the working distance with the station documentation, measured from the same reference the lens data sheet uses.
- Repeat the ruler check after any lens, spacer or adapter change — flange distance moves the whole pyramid.
If two heights must show one identical world region, that matching procedure is its own article: magnification at different working distances: match one world region.
When one station really needs two fields of view
Three working answers. Move the camera — a wrist or a Z stage between two taught heights gives two fields of view, each on the same pyramid; several wrist camera snapshot positions to cover a large table plans exactly that. Change the optics — a varifocal lens, or a second camera with a different focal length. Or size one distance to the decision that matters most and let a second camera handle the other: coverage for finding, pixels for judging. What no set-up does is make one fixed camera hold two fields of view at once; the pyramid has one angle.
Size your own pair the same way. Pick the sensor format and mount in the area scan cameras category or the fixed-lens smart cameras category, match a C-mount lens to the pyramid you computed, and the camera finder filters by sensor format, mount and interface. The systems topic page collects the neighbouring set-up geometry, and the configurator below cross-checks camera and lens against each other.
The configurator matches a camera, lens, light, light controller and cables to each other, step by step, and lists only parts that fit what you already picked.
Questions
Does a camera have multiple fields of view?
No. One lens and one sensor define a single angular field of view, and the camera shows a slice of it at every working distance. Coverage grows smoothly with distance rather than stepping between presets. A picture that changes in steps comes from a change in focal length — a zoom or varifocal lens, or a multi-lens camera module — not from distance itself.
How much wider is the field of view when I double the working distance?
Almost exactly double. The exact factor is (WD2 − f) / (WD1 − f). With an 8 mm lens, moving from 100 mm to 200 mm gives 192 / 92 = 2.09, about 9% more than double, and 50 mm to 100 mm gives 92 / 42 = 2.19. From 500 mm to 1000 mm the factor is 992 / 492 = 2.02, within 1% of two.
Why is my measured field of view smaller than the calculated one?
First apply the exact (WD − f) form; the quick estimate alone overstates coverage by about 9% at 100 mm with an 8 mm lens. Then check three things: the reference plane you measured the distance from (front of barrel versus flange), any region-of-interest crop that reads fewer pixels than the sensor offers, and focus breathing, which shifts effective focal length a few percent through the focus travel. Distortion correction is only needed for world-unit measurements.
How close can I mount a wrist camera to the part?
Down to the lens's minimum object distance, and no closer: inside it, no focus setting produces a sharp image. That distance sets the smallest field of view the camera can ever deliver, so start there and add margin for part height variation and depth of field. Fixed-focus smart cameras list a single focus distance instead of a range; treat that value as the mounting target.
Do telecentric lenses change this behaviour?
Yes, deliberately. A telecentric lens holds magnification constant within its depth of field, so its field of view stays the same while the working distance changes — coverage does not grow as the part moves. That is why they suit measurement tasks where part position varies. The trade-offs are larger front elements, longer barrels and a format-matched design rather than a generic C-mount choice.