無奈這個『一維鏡頭』只有『單點』之象觀，無論『視野』 之『角度』大小︰

Angle of view

In photography, angle of view (AOV)^[1] describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the more general term field of view.

A camera’s angle of view can be measured horizontally, vertically, or diagonally.

It is important to distinguish the angle of view from the angle of coverage, which describes the angle range that a lens can image. Typically the image circle produced by a lens is large enough to cover the film or sensor completely, possibly including some vignetting toward the edge. If the angle of coverage of the lens does not fill the sensor, the image circle will be visible, typically with strong vignetting toward the edge, and the effective angle of view will be limited to the angle of coverage.

A camera’s angle of view depends not only on the lens, but also on the sensor. Digital sensors are usually smaller than 35mm film, and this causes the lens to have a narrower angle of view than with 35mm film, by a constant factor for each sensor (called the crop factor). In everyday digital cameras, the crop factor can range from around 1 (professional digital SLRs), to 1.6 (consumer SLR), to 2 (Micro Four Thirds ILC) to 4 (enthusiast compact cameras) to 6 (most compact cameras). So a standard 50mm lens for 35mm photography acts like a 50mm standard “film” lens even on a professional digital SLR, but would act closer to an 80mm lens (1.6 x 50mm) on many mid-market DSLRs, and the 40 degree angle of view of a standard 50mm lens on a film camera is equivalent to a 28 – 35mm lens on many digital SLRs.

In 1916, Northey showed how to calculate the angle of view using ordinary carpenter’s tools.^[2] The angle that he labels as the angle of view is the half-angle or “the angle that a straight line would take from the extreme outside of the field of view to the center of the lens;” he notes that manufacturers of lenses use twice this angle.

In this simulation, adjusting the angle of view and distance of the camera while keeping the object in frame results in vastly differing images. At distances approaching infinity, the light rays are nearly parallel to each other, resulting in a “flattened” image. At low distances and high angles of view objects appear “foreshortened”.

Derivation of the angle-of-view formula

Consider a rectilinear lens in a camera used to photograph an object at a distance  , and forming an image that just barely fits in the dimension,  , of the frame (the film or image sensor). Treat the lens as if it were a pinhole at distance  from the image plane (technically, the center of perspective of a rectilinear lens is at the center of its entrance pupil):^[7]

Now  is the angle between the optical axis of the lens and the ray joining its optical center to the edge of the film. Here  is defined to be the angle-of-view, since it is the angle enclosing the largest object whose image can fit on the film. We want to find the relationship between:

the angle 

the “opposite” side of the right triangle, (half the film-format dimension)

the “adjacent” side,  (distance from the lens to the image plane)

Using basic trigonometry, we find:

 

which we can solve for α, giving:

To project a sharp image of distant objects, needs to be equal to the focal length,  , which is attained by setting the lens for infinity focus. Then the angle of view is given by:

where 

Note that the angle of view varies slightly when the focus is not at infinity (See breathing (lens)), given by  rearranging the lens equation.

Macro photography

For macro photography, we cannot neglect the difference between and  . From the thin lens formula,

.

From the definition of magnification,  , we can substitute  and with some algebra find:

Defining  as the “effective focal length”, we get the formula presented above:

where .

A second effect which comes into play in macro photography is lens asymmetry (an asymmetric lens is a lens where the aperture appears to have different dimensions when viewed from the front and from the back). The lens asymmetry causes an offset between the nodal plane and pupil positions. The effect can be quantified using the ratio (P) between apparent exit pupil diameter and entrance pupil diameter. The full formula for angle of view now becomes:^[5]

─── 摘自《光的世界︰矩陣光學六辛》

 

難免受限於

笛沙格定理

笛沙格（Desargues）定理說明：在射影空間中，有六點A,B,C,a,b,c。Aa,Bb,Cc共點若且唯若AB∩ab,BC∩bc,CA∩ca共線。

在射影幾何的對偶性來看，笛沙格定理是自對偶的。

笛沙格（Desargues）定理說明：在射影空間中，有六點A,B,C,a,b,c。Aa,Bb,Cc共點若且唯若AB∩ab,BC∩bc,CA∩ca共線。

在射影幾何的對偶性來看，笛沙格定理是自對偶的。

 

也。

讀者或可藉著『嘗試』體驗矣！

Desargues’ Theorem

Let A₁B₁C₁ and A₂B₂C₂ be two triangles. Consider two conditions:

1. Lines A₁A₂, B₂B₂, C₁C₂ joining the corresponding vertices are concurrent.
2. Points ab, bc, ca of intersection of the (extended) sides A₁B₁ and A₂B₂, B₁C₁ and B₂C₂, C₁A₁ and C₂A₂, respectively, are collinear.

Desargues’ Theorem claims that 1. implies 2. It’s dual asserts that 1. follows from 2. In particular, the dual to Desragues’ theorem coincides with its converse.

(In the applet below each of the triangles as well as each of the vertices is draggable.)

Created with GeoGebra

Two triangles that satisfy the first condition are said to be perspecitve from a point. Two triangles that satisfy the second condition are said to be perspecitve from a line. Desargues‘ theorem thus claims that two triangles perspective from a point are perspective from a line. Its dual asserts that two triangles perspective from a line are also perspective from a point.

Monge’s theorem can be derived from that of Desargues and in fact is the latter in disguise. The existence of an orthic axis of a triangle is also an immediate consequence of Desargues’ Theorem. In a somewhat disguised form Desargues’ Theorem establishes a relationship between a triangle and a cevian triangle of a point not on a triangle itself.

Curiously, Desragues’ theorem admits an intuitive proof if considered as a statement in the 3-dimensional space, but is not as easy in the 2-dimensional case, where it is often taken as an axiom.

Following is the proof (kindly supplied by Hubert Shutrick) that adopts the 3-dimensional perspective. (The proof fails in some exceptional configurations, e.g., when A₁, A₂, B₁, and B₂ are collinear. These are simple enough to be treated individually and will not be considered here. We simply restrict the theorem to two triangles in general position, i.e., assuming that A₁≠A₂, B₁≠B₂, C₁≠C₂ and that, similarly, no two corresponding side lines coincide.)

 

聽聞此恰是『平面國』之『點投派』將『投影線』『抽象公設化』的時候，所遭遇『四元 □ ○ 完備』困難哩◎

 Complete Quadrangle

If the four points making up a quadrilateral are joined pairwise by six distinct lines, a figure known as a complete quadrangle results. A complete quadrangle is therefore a set of four points, no three collinear, and the six lines which join them. Note that a complete quadrilateral is different from a complete quadrangle.

The midpoints of the sides of any complete quadrangle and the three diagonal points all lie on a conic known as the nine-point conic. If it is an orthocentric quadrilateral, the conic reduces to a circle.

Complete Quadrilateral

The figure determined by four lines, no three of which are concurrent, and their six points of intersection (Johnson 1929, pp. 61-62). Note that this figure is different from a complete quadrangle. A complete quadrilateral has three diagonals (compared to two for an ordinary quadrilateral). The midpoints of the diagonals of a complete quadrilateral are collinear on a line (Johnson 1929, pp. 152-153).

A theorem due to Steiner (Mention 1862ab, Johnson 1929, Steiner 1971) states that in a complete quadrilateral, the bisectors of angles are concurrent at 16 points which are the incenters and excenters of the four triangles. Furthermore, these points are the intersections of two sets of four circles each of which is a member of a conjugate coaxal system. The axes of these systems intersect at the point common to the circumcircles of the quadrilateral.

Newton proved that, if a conic section is inscribed in a complete quadrilateral, then its center lies on (Wells 1991). In addition, the orthocenters of the four triangles formed by a complete quadrilateral lie on a line which is perpendicular to . Plücker proved that the circles having the three diagonals as diameters have two common points which lie on the line joining the four triangles’ orthocenters (Wells 1991).