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Fig 1. Reference scale on a windshield

To the right is a photo of the reference scale resting on a car windshield. Enough information can be gleaned from that image of the reference scale to determine the tilt of the windshield, and the angle of view of the camera: the windshield tilts back 56°, and the camera's gaze was at an angle of 52° from the right and looking down at an angle of 38°.

This is possible because when photographed obliquely, the circular target on the reference scale will appear in photos as an ellipse, from which we can determine the angle between the camera's view and the axis of the disc on the reference scale.

Furthermore, if the horizontal axis of the perspective disc on the reference scale was aligned to horizontal (ie., was level), and the camera was also horizontal (level, though possibly angled looking down or up), it's possible to find the angles of rotation (horizontal and vertical) of the disc in the real world with respect to the ground, and the angle by which the camera was pointing down or up.

Use the calculator on this web page (below) to analyse an ellipse in one of your own photos.

Photoshop can be used to obtain the major/minor axes lengths and tilt from an image by matching an ellipse path generated by Photoshop to one in a photograph (more accurate than 'eyeballing' it). Here's the procedure I use with Photoshop:

Fig 2. The circular target of figure 1, with measurements

- Enlarge the section of photo containing the ellipse to maximum
- Select the Ellipse tool. Set the Ellipse tool options to Paths (second icon in the Window->Options toolbar)
- Drag the tool to create a circle roughly the size of the ellipse (no need to be accurate yet)
- Select Edit->Free Transform Path (or Ctrl-T, Windows). In one operation (important), use translate, stretch (vertical and horizontal), and rotate (counter-clockwise) to match the ellipse path to the ellipse in the photograph. Then read the ellipse dimensions and angle from the Information palette (at Windows->Info). If you rotate counter-clockwise, the angle in the Info palette will be -tilt (as defined above); if you went clockwise, tilt = 90 - angle.
- (The ellipse path is no longer needed and can be deleted)

Next, the tilt of the reference scale's horizontal is easily measured using Photoshop's 'ruler' tool (the angle of a ruler line with respect to horizontal is displayed in the Info panel and on the options bar). If you measure left to right, the angle displayed by Photoshop will be relative to horizontal and negative -- convert that to positive for 'tilt of scale' in the form above. For better accuracy, measure your ellipse from its extreme left to right (instead of its center to right).

As an example, measurements of the reference scale in figure 1 (upper right) on a car windshield, shown in figure 2, have been pre-entered into the calculator below, with the results displayed.

The way the math works out, there is more than one solution (ie., set of rotations) for a given ellipse, usually corresponding to a 'left' or 'right' view. Which one is best is usually obvious from the rest of your photo, but in orthographic projection, the ellipses in isolation are identical and indistinguishable (to see this, click the 'Orthographic' button above).

For the windshield analysis above, the two aliases correspond to the scale being viewed from the left or right. Looking at the photo containing the reference scale, it looks like the alias with the right side of the reference scale toward the camera is best, which corresponds to the second alias in the results. Also, we could pick that same alias because we know windshields lean back; we expect a positive vertical rotation.

Sometimes there are circles 'in the wild' with horizontals marked, such as the clock faces in the photo below, of London's Big Ben clock tower. The two clocks in the right-most image have been marked up using Photoshop with the ellipses (red), their major axis (white), and the horizontal (green).

The measurements of the ellipse thus obtained are as follows:

Left clock | Right clock | |
---|---|---|

Minor axis (length, pixels) | 63 px | 58 px |

Major axis (length, pixels) | 86 px | 86 px |

Tilt of major axis (degrees clockwise) | 8° | -7° |

Tilt of horizontal (degrees clockwise) | -7° | 7° |

Do those measurements make sense? Assuming the clock faces are vertical, the vertical rotation ('tilt back') should be zero. Assuming the tower plan is a regular rectangle, and is viewed from a less-than-infinite distance, the sum of the horizontal rotations ('push right'), ignoring their signs, should approach but not reach 90 degrees (if it were viewed from infinity, it'd be exactly 90 degrees). The camera dips should be equal (because the camera is looking at both clocks at the same angle) and negative (because it is looking upwards at them).

Entering Big Ben's clock measurements in the form above, the results and expectations are summarized below:

Left clock | Right clock | Expectation | |
---|---|---|---|

Vertical rotation (top tilted back) | 0° | 1° | 0° |

Horizontal rotation (right tilted back) | -42° | 47° | 90° (|-42°| + |47°| = 89°) |

Camera dip (looking down) | -8° | -7° | Approx equal and negative |

The results are close to expectations. Using this information, we could probably figure out from where the photo was taken.

- A circle is just a special-case ellipse, where the minor and major axes are the same length.
- The major axis of an image ellipse is in a plane perpendicular to the line of sight; the minor axis is parallel.
- The major axis is 'true length' but the minor axis is shorter than 'true' (unless a circle, in which case both are true).
Fig 3. A close range (~1 cm), wide-angle photo of disc; red lines drawn to show ellipse axes & tilt

Regardless what lens is used or the distance from which it is photographed, unless face-on, a disc perimeter will always appear in a photo as an ellipse.

- If the angle of line of sight of a camera is not the same across the frame (ie., if the photo is taken from less than an infinite distance), the center of an obliquely photographed disc will not exactly match the center of the image ellipse, by an amount that increases with decreasing camera distance from the subject. The disc center will be on the minor axis farther from the camera than the ellipse center. This is noticeable at short camera-subject distances, eg., within low multiples of the disc diameter, but drops off rapidly and becomes small beyond about 50 multiples of the disc diameter.
Figure 3 is a close-up photo (taken ~1 cm from the end of the lens) of a reference scale disc (2 cm dia), with red lines drawn afterwards on the axes of the image ellipse; note the substantial distance between the ellipse center and the disc center. The two centers are quite close when a camera is more than about a meter from the reference scale, as in the case of figure 2 above. The effect is independent of the lens focal length and is a consequence of perspective projection.

For the purposes of the calculations in the form above, a photo was taken 'far enough' from a subject if the ellipse center essentially coincides with the center of the crosshairs of the reference scale disc, as in figure 2 (and not as in figure 3), as is likely the case for distances over about a meter.

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