A Lesson on Perspective & The Cube..

This lesson we masted perspective and the Cube. Simple, Yet effective.

Perspective (from Latin perspicere, to see through) in the graphic arts, such as drawing, is an approximate representation, on a flat surface (such as paper), of an image as it is perceived by the eye. The two most characteristic features of perspective are that objects are drawn:

- Smaller as their distance from the observer increases

- Foreshortened: the size of an object's dimensions along the line of sight are relatively shorter than dimensions across the line of sight.

The Cube

A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and of 3-sided trapezohedron. The cube is dual to the octahedron. It has cubical symmetry (also called octahedral symmetry). A cube is the three-dimensional case of the more general concept of a hypercube, which exists in any dimension.

Regular Hexahedron

The cube is unique among the Platonic solids for being able to tile space regularly. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry).

Compound of Three Cubes

An Aternately truncated cube

Related polyhedra

The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron. These two together form a regular compound, the stella octangula. The intersection of the two forms a regular octahedron. The symmetries of a regular tetrahedron correspond to those of a cube which map each tetrahedron to itself; the other symmetries of the cube map the two to each other.

One such regular tetrahedron has a volume of ⅓ of that of the cube. The remaining space consists of four equal irregular polyhedra with a volume of 1/6 of that of the cube, each.
The rectified cube is the cuboctahedron. If smaller corners are cut off we get a polyhedron with 6 octagonal faces and 8 triangular ones. In particular we can get regular octagons (truncated cube). The rhombicuboctahedron is obtained by cutting off both corners and edges to the correct amount.

A cube can be inscribed in a

dodecahedron so that each vertex of the cube is a vertex of the dodecahedron and each edge is a diagonal of one of the dodecahedron's faces; taking all such cubes gives rise to the regular compound of five cubes.

Perspective

A cube in two-point perspective.

Perspective works by representing the light that passes from a scene through an imaginary rectangle (the painting or photograph), to the viewer's eye. It is similar to a viewer looking through a window and painting what is seen directly onto the windowpane. If viewed from the same spot as the windowpane was painted, the painted image would be identical to what was seen through the unpainted window. Each painted object in the scene is a flat, scaled down version of the object on the other side of the window.[1] Because each portion of the painted object lies on the straight line from the viewer's eye to the equivalent portion of the real object it represents, the viewer cannot perceive (sans depth perception) any difference between the painted scene on the windowpane and the view of the real scene.

Related concepts:
Some concepts that are commonly associated with perspective include:
foreshortening (see later)
horizon line
vanishing points

All perspective drawings assume a viewer is a certain distance away from the drawing. Objects are scaled relative to that viewer. Additionally, an object is often not scaled evenly: a circle often appears as an ellipse and a square can appear as a trapezoid. This distortion is referred to as foreshortening.
Perspective drawings typically have an—often implied—horizon line. This line, directly opposite the viewer's eye, represents objects infinitely far away. They have shrunk, in the distance, to the infinitesimal thickness of a line. It is analogous to (and named after) the Earth's horizon.

Any perspective representation of a scene that includes parallel lines has one or more vanishing points in a perspective drawing. A one-point perspective drawing means that the drawing has a single vanishing point, usually (though not necessarily) directly opposite the viewer's eye and usually (though not necessarily) on the horizon line. All lines parallel with the viewer's line of sight recede to the horizon towards this vanishing point. This is the standard "receding railroad tracks" phenomenon. A two-point drawing would have lines parallel to two different angles. Any number of vanishing points are possible in a drawing, one for each set of parallel lines that are at an angle relative to the plane of the drawing. A master in this thing was Johannes Vermeer

Perspectives consisting of many parallel lines are observed most often when drawing architecture (architecture frequently uses lines parallel to the x, y, and z axes). Because it is rare to have a scene consisting solely of lines parallel to the three Cartesian axes (x, y, and z), it is rare to see perspectives in practice with only one, two, or three vanishing points; even a simple house frequently has a peaked roof which results in a minimum of six sets of parallel lines, in turn corresponding to up to six vanishing points.
In contrast, natural scenes often do not have any sets of parallel lines. Such a perspective would thus have no vanishing points.

Rays of light travel from the object, through the picture plane, and to the viewer's eye. This is the basis for graphical perspective.

One-point perspective

One vanishing point is typically used for roads, railroad tracks, or buildings viewed so that the front is directly facing the viewer. Any objects that are made up of lines either directly parallel with the viewer's line of sight or directly perpendicular (the railroad slats) can be represented with one-point perspective.

One-point perspective exists when the painting plate (also known as the picture plane) is parallel to two axes of a rectilinear (or Cartesian) scene — a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the painting plate (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the painting plate are drawn as parallel lines. All elements that are perpendicular to the painting plate converge at a single point (a vanishing point) on the horizon. Two-point perspective

Two-point perspective can be used to draw the same objects as one-point perspective, rotated: looking at the corner of a house, or looking at two forked roads shrink into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Looking at a house from the corner, one wall would recede towards one vanishing point, the other wall would recede towards the opposite vanishing point.

Two-point perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the z-axis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a one-point and two-point perspective.

Three-point perspective

Three-point perspective is usually used for buildings seen from above (or below). In addition to the two vanishing points from before, one for each wall, there is now one for how those walls recede into the ground. This third vanishing point will be below the ground. Looking up at a tall building is another common example of the third vanishing point. This time the third vanishing point is high in space.

Three-point perspective exists when the perspective is a view of a Cartesian scene where the picture plane is not parallel to any of the scene's three axes. Each of the three vanishing points corresponds with one of the three axes of the scene. Image constructed using multiple vanishing points.

One-point, two-point, and three-point perspectives appear to embody different forms of calculated perspective. The methods required to generate these perspectives by hand are different. Mathematically, however, all three are identical: The difference is simply in the relative orientation of the rectilinear scene to the viewer.

Zero-point perspective

Due to the fact that vanishing points exist only when parallel lines are present in the scene, a perspective without any vanishing points ("zero-point" perspective) occurs if the viewer is observing a nonlinear scene. The most common example of a nonlinear scene is a natural scene (e.g., a mountain range) which frequently does not contain any parallel lines. A perspective without vanishing points can still create a sense of "depth," as is clearly apparent in a photograph of a mountain range (more distant mountains have smaller scale features).

My drawing showing perspective.

Sheet handed out in class.