We have three kinds of mathematical perspective:
- Algebraic Perspective [non-visual class]: involves the application of algebraic formulae.
- Geometrical Perspective [analytical visual class]: employs graphical calculation.
- Projection Perspective [projective visual class]: employs spatial projection.
Algebraic perspective refers to mathematical images composed (solely) of letters and symbols. Accordingly, we have categorised this as symbolic perspective; hence, it is a non- visual class of perspective [discrete analysis].
Geometrical perspective employs analytic geometry, also called coordinate geometry, or the use of algebraic symbolism and methods to represent and solve spatial problems [continuous analysis]. Analytic geometry refers to spatial modelling using graphical points, lines, and multi-dimensional objects and shapes, surfaces, and solids. Ergo, due to the inherent links with analytical geometry, we have categorised geometrical perspective as a form of visual perspective (1st type).
Another form of visual perspective (1st type) is projection perspective, which refers to applying projective principles to create an image or view of a spatial scene or object. This type employs either descriptive geometry (i.e. parallel perspective: plan, front-elevation, side- elevation views within a machine/technical drawing), or projective geometry (e.g. linear perspective), to produce images of spatial reality.
Perspective and Continuous Analysis
The ancient Greek experience of space was a tactile one, making a close physical connection between space and Forms in space. In the Western tradition, we have incorporated this tactile concept into geometry, whereby the subject of optical/technical perspective, despite being ostensibly a visual subject, is intimately tied to geometrical ideas.
We can identify several kinds of space, including natural/physical space, geometrical space, perspective or image space, instrument space, represented space (analogue and digital), and, finally, imaginary space. Whereby each kind of space is linked to a corresponding type of spatial reality. We have also binned these into two different kinds of space, being natural and artificial space, which correspond to the two basic kinds of perspective, natural and artificial.
Once again, we can also have a synthetic perspective, meaning a combination of natural and artificial, but this is a combinational idea that can often be separated out into contributing spatial types. Also, we named synthetic, composite, blended, and mixed perspective types, which can, and often do, result in the generation of unusual new kinds of space that exhibit special properties.
Despite our wish to clarify and simplify the subject from an optical/technical perspective <IMAGING CLASS>, introducing all these different types of space/ perspective may seem unnecessarily complex. However, it is important to realise that perspective is inherently complex, partly due to the variety and composite/ interacting nature of the visual scenarios (and imaging processes) involved.
One way of simplifying matters is to remind ourselves that normally we are concerned with the nature and geometry only of the space present in the target or object space (primarily). That is, we can consider all of the intermediate perspective categories/methods/principles operating, and hence any intermediate spaces, including the final image space, as a representation or model of the target optical space, and thus being of secondary concern and waypoints on the journey towards understanding/mapping the target space and the Forms contained therein.
But why does this help?
Well, we can avoid the often arbitrary geometry of intermediate space(s) and focus on decoding the target space, which normally follows the well-defined principles of Euclidean geometry.
Geometry of Physical Space
Patently, perspective views/images can help us to understand and map the geometry of the physical environment. But the details of how this occurs and the scope, scale, and degree of accuracy/precision to which a perspective processes aids in such a comprehension are dependent on several factors, including the type/form of perspective employed, plus the number and variety of images captured, to say nothing of geometrical factors such as optical assembly, projection and observation modes, etc.
Unfortunately, multiple factors and diverse elements lie behind how perspective images are created, viewed, perceived, and interpreted.
Patently parallel perspective images are often easier to decode, especially when the object or scene contains straight lines, sets of parallels, right-angled corners, etc., which can (often) be aligned with a front elevation, side elevation, or plan. However, an image is often perspectival, or ‘optically warped’, and we humans must then employ knowledge of factors such as depth cues, perspective phenomena, and the (typically) simplified geometry or likely structural Forms being observed, and thus to overcome the equivalency/correspondence problem.
Coordinate Reference Frames
Normally, for a perspective image, and in local and visual terms, we do assume a type of Euclidean reference frame or coordinate system in which we have identified the 6 cardinal directions, north, south, west, east, and up/down. Whereby in a geometrical reference frame, we can identify an object’s visual position, angle, and scale/size, etc.
However, we humans use both local and global reference systems, including various types of positional geometry (e.g., latitude and longitude on Earth’s surface) and astronomical systems, as explained below. In astronomy, various coordinate systems are used for identifying the positions of celestial objects (satellites, planets, stars, galaxies, etc.) relative to a fixed reference frame.
Coordinate systems in astronomy can specify an object’s relative position in three-dimensional space and also plot its apparent direction on a celestial sphere, especially when the object’s distance is vast or unknown.The primary celestial sphere is a relatively fixed imaginary sphere that surrounds Earth and is used to calculate the positions of objects in the night sky (at a specific time).
Spherical coordinates, projected onto a celestial sphere, are similar to the geographic coordinate system used on the Earth’s surface. Also, in changing sidereal coordinates (ref. Earth’s rotation), a celestial sphere is centred on the observer. Azimuth is measured eastward from the north point (sometimes from the south point) of the horizon; altitude is the angle above the horizon.Whereby each system is named after the origin/projection plane. Astronomical systems include alt-az, equatorial, ecliptic, galactic and super-galactic.
Structural Order as the Goal of Perspective
The foundation of visual/optical/technical perspective is mathematics, and in particular geometry; dealing with visual images that are composed of standard mathematical structures such as points, lines, and regular solids.
Humans interpret images in terms of measurement and structure, or ordered Forms, because this is the only way we can deal with the complexity/uncertainty of reality. This is how we cope with (in actuality) unknown shapes/sizes/ locations/orientations of spatial objects in spatial reality.The problem relates to the correspondence/equivalence issue mentioned earlier, in which humans have no direct visual means of unambiguously interpreting an image.
As an alternative, we can use our memory of an irregular object’s shape and size to interpret its distance, aspect, or true shape. However, in general, the interpretation of a perspective image/view relies on making certain assumptions about the level of order or structural order present in the object space, and that is a primary cause of the image itself (in combination with the projection method/imager employed).
In short, we represent disorder by using order. How so? Well, when looking at a spatial image, our mind tries to match recognised shapes: being straight, regular curved lines, plus flat planes, and regular solids onto what it sees by recognising the same from known examples. For example, recognition of converging parallels, or warped squares, rectangles, and circles, etc. In other words, the interpretation of a perspective image involves seeking out the perspective degradation of regular forms, including perspective frameworks and/or metric grids.
And even highly irregular shapes are perceived in this way by (mentally) testing or looking to see if the irregular shape is comprised of smaller regular shapes, or is in some way resting upon, or is aligned next to, a regular or flat plane, for example, and from which we can gain an impression of location and/or size, angle, and/or physical extent, etc.
Geometric Forms
Geometry is the overlaying of abstract shapes (Forms) onto nature, and often in an approximate way to represent (and simplify) the overwhelming complexity of reality (ref. physical forms).
Geometry corresponds to the measurement/representation of the degree of order, and (by absence/omission) its direct corollary, disorder/complexity, present in any spatial reality. Order is the degree of regular, repeating, redundant, predictable, symmetrical, periodic, monotonous, crystal-like patterning, plus reducible, compressible data present in any region of space. Perspective is a highly ordered visual method that seeks out order/disorder (regular/irregular spatial Forms).
Geometric Form is the perceived shape or external structure of a thing. The ordering of an object’s parts is its structure—a type of form within form. Whereby structure (in general) is a definite visible arrangement or visible pattern of parts (points/lines/planes/solids); and is a measure of the degree of order imbued by a Form. Whereby, disorder is always present (to a degree) and becomes apparent according to the magnification and/or measuring scale employed. Structure functions across different hierarchies (atomic to macro) and exhibits various ordering types, such as matter/energy patterns and symmetry.
Coordinate or Anlaytic Geometry
We can now summarise what we have learned in a simple statement.
Analytic geometry, or coordinate geometry, studies geometry using a coordinate system and is applied in physics, engineering, and space science. Coordinate geometry (or analytic geometry) bridges algebra and geometry, using a coordinate system to define and analyse the properties of geometric shapes on a plane. It allows us to solve complex geometric problems using numerical and algebraic equations.
The Cartesian Plane
- Axes: The plane is divided by two perpendicular axes: the horizontal x-axis and the vertical y-axis.
- The Origin: The point of intersection (0,0) where both axes meet.
- Coordinates: Points are represented as ordered pairs (x,y), where x defines horizontal distance and y defines vertical distance
In fact, analytical geometry extends to forms of geometry that lie outside the Cartesian plane and are encompassed by the more general 3-D concept of Euclidean geometry. Patently, perspective concerns a spatial reality that extends into the third dimension or depth. If we apply coordinate geometry, it would sometimes involve planes that, in one way or another, extend into depth (z-axis) or a plane (e.g. the vertical picture plane) that relates to such a plane (e.g., the ground plane).