Angular diameter
The apparent size (also apparent diameter, angle of view ) of an object is the angle at which it is perceived by an observer.
The figure illustrates the relationship between the apparent size α, distance r and true extension g of an object. It can be deduced from the following relationship between the three variables:
Is the viewing distance r to an object whose size is equal to g, the object always appears at an angle α of
In geodesy, by means of an object with a standardized size g, for example, a vertically arranged slat can be calculated from the apparent size α the distance r:
In astronomy, g can be calculated with a known distance r of an object whose approximate true extent:
For small angles <1 ° can also be approximated ( radians):
And thus for the apparent size in minutes of angle:
The error is 1 ° 1/10000, at 0.1 °, only 1/1000 of 000
One problem here is that astronomical objects such as diffuse nebulae often have irregular shapes or such as galaxies usually have an angle of inclination. Therefore, various methods for measuring distance may result in different distances r. This results in the calculation with the same apparent size α slightly different expansions g Due to the vast distances however this plays a relatively minor role in practice. In observations of objects at cosmological distances shown above the connection between the apparent size, extent and distance of an object is also greatly complicated by the curvature of space.
Vertical and horizontal viewing angle
In photography using the vertical and horizontal angle of view of an object. The vertical viewing angle εv an object we define by the fixed from the eye object is often named a horizontal prone rectangle, then pulls the two emanating from the eye beams to the end points of the vertical line through the rectangle center point and determines the angle between these rays. Analogously, the horizontal visual angle εh the angle between the two beams from the eye to the end points of the horizontal line through the rectangle center point.
If we choose the Cartesian coordinate system whose origin lies in the center of the rectangle whose y- and z-axes are the vertical and horizontal axis of symmetry of the rectangle and in which there is the viewer in the half-space x > 0, then, these two visual angle for the rectangle with the vertical side length Gv = 2γv and the horizontal side length Gh = 2γh for any observer point (x, y, z) determine trigonometry:
Due to the rotational symmetry of the graph of the vertical visual angle εv (x, y, z) during the rotation about the y- axis (cylinder symmetry) may be restricted to the investigation, the x, y plane. For the Sehwinkelfunktionen as functions only of the plane coordinates x and y we obtain the following terms and the function graph shown in the pictures:
Maximum visual angle of an object for a camera
For the full and sharp image of a fixed predetermined object using a camera the camera location is restricted to an acceptability range Z. This area Z is described by four inequalities, in which enter the camera parameters:
1) εv (x, y, z ) ≤ αv,
2) εh (x, y, z ) ≤ αh,
3) ρ (x, y, z) = ≥ d = gmin - f
4) x > 0,
Where αv AoV, αh, the horizontal angle, gmin is the minimum object distance and f are the fixed fixed focal length of the camera.
If you look in this area Z is a site where εv, the vertical viewing angle or horizontal viewing angle εh of the object for the camera is at a maximum, so this gives each a nonlinear optimization problem, where the objective function by the to maximizing visual angle and its allowable range by Z is. If you want to, however, find a mounted on a camera crane camera in a place where both the vertical and the horizontal viewing angle is the maximum, so this leads to the solution of the maximization problem, in which both visual angle to maximize the objective functions simultaneously ( " multi-objective optimization " ).
If we restrict ourselves in the simultaneous maximization of both visual angle εv and εh to the x, y plane, the edge of the admissibility region Z is formed by two of the following three arcs:
Kd:
Kh:
Kv:
With ηh = γh / tan ( αh / 2) = tan αv wv, xv = γv / wv, rv = γv • (1 wv2 ) 1/2, ξv = xv rv = γv / tan ( αv / 2), 0 < αh, αv < π.
For the determination of the optimality range Os to the simultaneous maximization of both visual angle εv and εh the cases I) 0 < αv < π / 2, II) αv = π / 2, III) π / 2 < αv < π and to each of the sub-cases 1
) R = max { d, ηh } ≤ ξv, 2) ξv