Hyperfocal distance
The hyperfocal distance is the minimum focusing distance for a lens that places the far limit of the depth of field at infinity, ensuring that all objects from half that distance to infinity appear acceptably sharp when the lens is focused there.[1][2] In photography and optics, this concept maximizes the usable depth of field for a given lens aperture and focal length, making it particularly valuable for landscape and architectural shots where sharpness across a wide range is desired.[3] The hyperfocal distance is calculated using the formula H = \frac{f^2}{N \cdot c}, where f is the lens focal length, N is the f-number (aperture setting), and c is the acceptable circle of confusion diameter, which depends on factors like sensor size and viewing conditions.[1][3] Longer focal lengths increase the hyperfocal distance, requiring the camera to be positioned farther from the subject to achieve the same depth of field, while smaller apertures (higher N) decrease it, extending sharpness closer to the lens.[3][2] Historically rooted in geometric optics and thin lens approximations, the hyperfocal distance underpins fixed-focus camera designs, where the lens is preset to this point for optimal performance across typical subject distances without adjustment.[2] Modern applications extend to digital photography, where software tools and depth-of-field calculators incorporate it to guide focus stacking and zone focusing techniques.[4]Basic Concepts
Definition
The hyperfocal distance, denoted as H, is defined as the closest distance at which a lens can be focused to keep objects at infinity acceptably sharp, such that all objects from H/2 to infinity fall within the depth of field.[1] When the lens is focused precisely at this distance, the near limit of the depth of field aligns at approximately half the hyperfocal distance, while the far limit extends to infinity, optimizing the overall range of sharpness in the image.[2] This concept serves to maximize the depth of field in photographic scenarios where extensive sharpness from foreground elements to distant horizons is desired, such as in landscape or architectural imaging.[5] By setting the focus at the hyperfocal distance, photographers can achieve the broadest possible zone of acceptable focus without needing to adjust for specific subject distances, making it especially valuable for fixed-focus systems or scenes with varied depths. To understand hyperfocal distance, it is essential to grasp basic optics principles: lens focusing involves adjusting the lens-to-sensor distance to place the image of a subject onto the focal plane, the surface where light rays from the subject converge sharply.[2] Depth of field refers to the range of object distances in a scene that appear acceptably sharp in the final image when the lens is focused at a specific point, influenced by factors like aperture and focal length.[6] The acceptability of sharpness is typically evaluated using the circle of confusion criterion, which sets the threshold for perceived blur.[2]Acceptable Sharpness
Acceptable sharpness in photography refers to the level of detail resolution that aligns with the human eye's visual acuity, typically defined by the 20/20 vision standard, which corresponds to an angular resolution of approximately 1 arcminute or 1/60th of a degree when viewing prints or screens at standard distances such as 25 cm.[7] This perceptual limit determines the threshold beyond which blur becomes noticeable, ensuring that the image appears sufficiently focused to the observer under normal viewing conditions.[8] The circle of confusion (CoC) serves as the quantitative criterion for acceptable sharpness, representing the maximum diameter of a blurred spot on the image sensor or film that still projects as a sharp point when the final image is viewed.[9] For traditional 35mm film or full-frame digital sensors, a standard CoC value is approximately 0.03 mm, though it ranges from 0.025 mm to 0.035 mm depending on specific conventions.[10] This value ensures that defocused light rays from a point source do not exceed the eye's resolution limit after enlargement and viewing.[8] Several factors influence the CoC size, primarily the sensor or film diagonal, as it scales with the image format to maintain consistent perceived sharpness across different systems.[11] A common standard sets the CoC to 1/1500 of the sensor diagonal for full-frame formats (yielding about 0.029 mm for a 43.3 mm diagonal), accommodating typical print enlargements to 8x10 inches viewed at 25 cm.[11] Viewing conditions, such as distance and magnification, further adjust the CoC; closer inspection or larger displays demand smaller CoC values to preserve sharpness.[12] The CoC directly governs depth of field by establishing the tolerance for defocus blur on the image plane, where a larger CoC permits greater deviation from the focal plane before sharpness degrades, thereby extending the range of in-focus distances and influencing hyperfocal distance computations.[9] For digital sensors versus film, CoC standards remain equivalent when based on output viewing conditions, as both mediums are evaluated by the final image's perceptual quality rather than capture medium differences.[13]Calculation Methods
Standard Formula
The standard formula for hyperfocal distance H in photography and optics is given by H = \frac{f^2}{N c} + f where f is the focal length of the lens, N is the f-number (also known as the relative aperture), and c is the diameter of the circle of confusion.[14][15] This equation determines the closest focusing distance at which the depth of field extends to infinity, assuming the lens is focused on that point. The variable f represents the focal length, typically measured in millimeters (mm), which is the distance from the lens optical center to the image plane when focused at infinity. The f-number N is a unitless ratio defined as N = f / D, where D is the effective diameter of the lens aperture in mm; smaller values of N correspond to larger apertures and shallower depth of field. The circle of confusion c, also in mm, quantifies the maximum acceptable blur diameter on the image plane that is perceived as sharp by the human eye, as defined in prior discussions of acceptable sharpness.[14][16] This formula operates under the thin lens approximation, where the lens is treated as a single refractive surface with negligible thickness, and relies on Gaussian optics principles for paraxial rays (rays close to the optical axis). It assumes the distant object is at infinity and that the image sensor or film plane is fixed. Units for H are typically in meters (m), though calculations often use consistent mm scaling for f and c before conversion. For instance, with f = 50 mm, N = 8, and c = 0.03 mm, H \approx 10.5 m.[14][15] A common simplification arises when H is much larger than f, such as in wide-angle lenses or stopped-down apertures, allowing the approximation H \approx \frac{f^2}{N c}; this neglects the +f term without significant loss of accuracy for most practical scenarios.[14][16]Derivation
The derivation of the hyperfocal distance relies on geometric optics principles, specifically the thin lens approximation and similar triangles to model defocus blur. Consider a thin lens of focal length f. When focused on an object at distance u = H (the hyperfocal distance), the image forms at v' > f according to the lens equation \frac{1}{v'} + \frac{1}{H} = \frac{1}{f}, or v' = \frac{f H}{H - f}. However, with the image plane fixed at v = f (the position for infinity focus), rays from infinity (parallel to the optical axis) converge toward the focal point at f but are intercepted at the plane, while for the focused setup, the defocus for infinity arises from the mismatch. To achieve focus at H, the lens or image plane would be adjusted to v', but for fixed plane at f, we consider the equivalent defocus for infinity rays. The aperture diameter is d = f / N, where N is the f-number.[15][17] The step-by-step derivation begins with the thin lens equation, \frac{1}{v'} + \frac{1}{H} = \frac{1}{f}, where distances are positive in the photography convention (object to lens H, image to lens v'). Solving for v' gives v' = \frac{f H}{H - f}. The defocus shift is then \delta = v' - f = \frac{f^2}{H - f}. For rays from infinity, which would focus at f but with the focus set for H (effective plane shift), the blur circle diameter b at the image plane is b = \frac{\delta}{N}, using the divergence of the marginal rays beyond the intended focus point for infinity. Substituting yields b = \frac{f^2}{N (H - f)}. Setting the acceptable blur b = c (circle of confusion) results in H - f = \frac{f^2}{N c}, or H = f + \frac{f^2}{N c}. This defines the hyperfocal distance as the focusing distance where defocus blur for infinity is exactly c, ensuring sharpness from approximately H/2 to infinity under the symmetric DOF approximation.[15][18] Two alternative definitions of the blur circle lead to equivalent results under the small-angle approximation. In Definition 1, the blur is calculated from the sagittal ray, which lies in the plane perpendicular to the meridional plane (containing the optical axis and chief ray), intersecting the image plane to form the spot's extent in the sagittal direction; this accounts for the full cone of rays from the aperture edge. In Definition 2, the paraxial ray (near the optical axis, with small angles) is used, approximating the blur diameter via first-order ray tracing through the lens center and a parallel marginal ray. Both yield H = f + \frac{f^2}{N c} when angles are small (\theta \ll 1 radian), as the sagittal and paraxial contributions converge, neglecting higher-order terms.[19][18] This derivation assumes the paraxial approximation, valid for small ray angles relative to the optical axis, which linearizes the lens equation and ignores aberrations like spherical distortion. Diffraction effects are neglected, treating light as geometric rays rather than waves, which holds for large apertures but fails near the diffraction limit. The +f term is often omitted because H \gg f for typical photographic scenarios, simplifying computation without significant error; however, it becomes relevant in close-up or macro photography where H approaches f.[15][17][18]Practical Applications
Usage Examples
A practical example of hyperfocal distance application involves a 35 mm lens on a full-frame camera (using a circle of confusion of 0.03 mm) set to f/11, resulting in a hyperfocal distance of approximately 3.7 meters.[20] Focusing at this distance places the near limit of acceptable sharpness at about 1.85 meters (roughly half the hyperfocal distance), extending sharpness to infinity, which is ideal for landscape photography where foreground details and distant horizons must both appear in focus. Hyperfocal distances vary significantly with lens focal length and aperture. For a wide-angle 24 mm lens at f/8 on full-frame, the hyperfocal distance is about 2.4 meters, facilitating expansive scene coverage from nearby elements like rocks or flowers to the horizon.[20] In contrast, a telephoto 200 mm lens at f/16 yields a hyperfocal distance of approximately 83 meters, shifting the focus point far out and limiting utility for close subjects but ensuring distant compression remains sharp.[20] Aperture adjustments further influence this: for the 35 mm lens, f/8 produces a hyperfocal distance of 5.1 meters, while stopping down to f/16 reduces it to 2.6 meters, drawing the near sharp point closer for tighter compositions without losing infinity focus.[20] The following table illustrates hyperfocal distances for common focal lengths across full-frame and APS-C formats (circle of confusion 0.03 mm and 0.02 mm, respectively), highlighting how sensor size affects results; values are rounded for practicality.| Focal Length (mm) | Format | f/8 (m) | f/11 (m) | f/16 (m) |
|---|---|---|---|---|
| 24 | Full-frame | 2.4 | 1.7 | 1.2 |
| 24 | APS-C | 3.6 | 2.6 | 1.8 |
| 35 | Full-frame | 5.1 | 3.7 | 2.6 |
| 35 | APS-C | 7.7 | 5.6 | 3.8 |
| 50 | Full-frame | 10.4 | 7.6 | 5.2 |
| 50 | APS-C | 15.6 | 11.4 | 7.8 |
| 200 | Full-frame | 167 | 121 | 83 |
| 200 | APS-C | 250 | 182 | 125 |