Descriptive Geometry/Central Projections
Station Points
editCentral projections are projections from one plane to another where the first plane’s point and the image on the second plane lie on a straight line from a fixed point not on either plane.^{1}
The projecting rays pass through one point called the center of projection, also known as the station point (or vantage point). This is essentially the reference point from which all the lines and projections in the problem are related to. The station point can be high (bird’s eye view) or low (worm’s eye view). It is the base point (the most important point) to understand and generate the composition of the subject.^{2}
References: 1. http://dictionary.reference.com/browse/central+projection 2. http://www.creativeglossary.com/artperspective/stationpoint.html
Line of View
editCentral projection is the projection of a plane onto another plane from a certain point not on either of those planes. The line of view extends from the center of projection to the first plane, then to the second plane in a direct straight line. Depending on the relationship of the two planes, the projected image will appear differently. If the planes are parallel, the projection will retain the same shape at a different scale, as the ratio between points will remain constant. When the planes are nonparallel, the betweenness is no longer kept and vanishing lines are constructed. Lines that are not parallel to the vanishing line will have a vanishing point on both planes. Any lines on the first plane that intersect at a point on the vanishing line will be projected as parallel lines onto the second plane. An example of a central projection problem can be seen below, where the shadow of a hexahedron is projected onto a plane, as seen in the top and front view.

Central projection between parallel planes.

Central projection between nonparallel planes. Lines that intersect on the vanishing line are projected as parallel lines.
Vanishing points
editThe vanishing point is a point on the horizon line where parallel lines of a certain object move away towards from the observer and appear to join. An object constructed on these lines becomes smaller and smaller until it converges into one point, so it appears to have ‘vanished.’ To find the vanishing point of an object, simply extend parallel lines of the object until all lines meet at one point. In perspective drawings using the vanishing point, an object can have numerous vanishing points, although traditionally most have up to three. In one point perspective, all lines parallel to the picture plane are drawn as parallel horizontal or vertical lines while lines perpendicular to the picture plane converge into the vanishing point on the horizon line. In two point perspective, two vanishing points will appear to the left and right of the object where two individual sets of lines will converge towards, and one set of lines parallel to the picture plane. An example of such construction can be observed to the right.
Finally in three point perspective, none of the three axes will be parallel to the picture plane, allowing for a third vanishing point to appear below or above the object.

An example of a one point perspective drawing.

An example of a two point perspective drawing.
Introduction to Perspective
editA perspective drawing is a two dimensional representation of the way in which scene or object is seen by the eye. This representation, while not perfectly accurate, does give the effect of foreshortening as the object or lines travel away from the viewer and towards the vanishing point which coincides with the perception of the ocular nerve endings. In order to create a perspective, one must possess a top view and a side view of the object or scene. Through these two views, the perspective can be created by citing each point within the views. A onepoint perspective has four major factors within it. Firstly, the horizon line establishes the height of the vanishing point. The names of different perspective constructions (onepoint, twopoint, and threepoint) address how many vanishing points exist in the particular construction. The second major factor is the ground line, which is in front view; all objects are created relative to this line. The third major factor is the station point, which is the location of the viewer relative to the objects being viewed. The fourth and final major element of perspective is the picture plane, which is also seen in top view. This is an imaginary plane perpendicular to the line of sight. The picture plane acts as the two dimensional surface that all points must be brought to in order to make the two dimensional perspective. Otherwise, the image would become an axonometric view.
OnePoint Perspective
editA onepoint perspective is a simple representation of a scene or object whose lines are exactly perpendicular or parallel with the line of vision. That means that it must be perpendicular or parallel with the picture plane in top view and the ground line as seen in front view. The perpendicular lines (traveling away from the ground line) will travel toward a vanishing point (which lies on the horizon line) while the ones parallel to the ground line will remain at the same degree. The parallel lines however will change positions in the image depending on how far away from the ground line they are. A line that is five inches away from the ground line will appear drastically further away in perspective than a line that is one inch away from the ground line (see Figure 1.1).
Steps to creating a 1Point Perspective:
1: Cite points along the picture plane and draw vertical construction lines to perspective view. Then draw construction lines from the side view, parallel to the ground line, and draw passed the vertical lines in the perspective view.
2: Where the vertical and horizontal lines intersect in the corresponding points to the object are the points of this plane in perspective. Mark each point.
3: Connect these points after they have been established.
4: Take points that do not lie on the picture plane and draw a line from each of them to the station point.
5: Where this line crosses the picture plane is the cited point.
6: Draw a vertical line from the cited point down to the perspective view.
7: Now go to the side view. The construction lines that were drawn from it to the perspective plane (the ones that established the plane that sits on the picture plane) should be a guide. Take each point as it lies on the plane in construction and project it back to the vanishing point.
8: Where this line intersect the vertical line is the new point in perspective
9: Do this for all point not lying on the picture plane
10: Connect the appropriate points, understand the depth of the image, and you have a perspective!

step 1 for constructing a 1point perspective

final 1point perspective constructed view
TwoPoint Perspective
editA twoPoint Perspective has the same rules and structure as a onepoint perspective, however, a twopoint perspective is seeing an object at an angle not parallel to the picture plane. This view is much more focused on a corner of the objects as opposed to an entire side of the objects. Instead of one line receding and the other standing at the same degree, both sets of lines would vanish to opposite vanishing points.
Instructions to constructing a 2Point Perspective:
1: Cite points along the picture plane and draw vertical construction lines to perspective view. Then draw construction lines from the side view, parallel to the ground line, and draw passed the vertical lines in the perspective view.
2: Where the vertical and horizontal lines intersect in the corresponding points to the object are the points of this plane in perspective. Mark each point.
3: Connect these points after they have been established.
4: Take points that do not lie on the picture plane and draw a line from each of them to the station point.
5: Where this line crosses the picture plane is the cited point.
6: Draw a vertical line from the cited point down to the perspective view.
7: Now go to the side view. The construction lines that were drawn from it to the perspective plane (the ones that established the plane that sits on the picture plane) should be a guide. Take each point as it lies on the plane in construction and project it back to the respective vanishing points (the lines at an angle off to the right side of the page will go to the Right side vanishing point, and likewise the left angled ones will go to the left).
8: Where this line intersect the vertical line is the new point in perspective
9: Do this for all point not lying on the picture plane
10: Connect the appropriate points, understand the depth of the image, and you have a perspective!

step 1 for constructing a 2point perspective

final 2point perspective constructed view
ThreePoint Perspective
editA threepoint perspective construction is used when the picture plane is not parallel to the x, y, or zaxes. This usually occurs in a situation where one is looking up at a scene or looking down at something from above. There are three vanishing points, and each line vanishes at the point associated with its axis. For example, all lines in the zaxis (moving vertically) will vanish at the zaxis vanishing point.
Instructions to constructing a 3point perspective of a cube:
1: Shown is the cube in top and front view. The two vanishing points (VPr and VPl) were made by drawing a line perpendicular to the line between the station point and the center of the cube and drawing lines from the station point parallel to the sides of the cubes. The locations of the intersections is where the vanishing points are.

step one to a 3point perspective drawing of a cube
2: Draw a folding line parallel to the line between the station point and center of the cube and project all points using transfer distances. To find the location of the vanishing points, draw a line parallel to the folding line from the station point in view 1. This will give you the line between the two vanishing points in point view (therefore, they appear as the same vanishing point). Draw another line perpindicular to the folding line. Then draw a horizontal line from VP. Where the perpendicular line from SP and the horizontal meet is the location of the vertical vanishing point (VPv). The horizontal line drawn is the picture plane (PP).

step two to a 3point perspective drawing of a cube
3: Draw another folding line perpendicular to the picture plane, and project all points across. Draw lines from the station point to each point of the cube in views 1 and 2. Mark where these lines cross the picture plane. In the case of point 3, extend the line until it reaches the picture plane. Extend the folding line down and transfer the points along the picture plane in view one to the lower part of the folding line between views 1 & 2. This will prepare you to make the final 3point perspective view.

step three to a 3point perspective drawing of a cube
4: Project the points along the picture plane down from view 2. Then project the points along the folding line horizontally. Where one line intersects its corresponding line, mark the point. Connect the points and calculate visibility to create the final view.

step four to a 3point perspective drawing of a cube
Practice Problems
edit1. You are in a helicopter looking at the clock tower of a building. Given the top and front views of the building, construct a 3point perspective showing what you see.

Problem 1

Answer to problem 1
2. Given the top and front views, construct a 2point perspective.

Problem 2

Problem 2 answer
3. Given the top and front views, construct a 3point perspective.

Problem 3

Problem 3 answer