Projective+Geometry

What exactly is projective geometry? Projective geometry is the study of geometric properties that are invariant under projection. In older literature, projective geometry is sometimes called "higher geometry," "geometry of position," or "descriptive geometry". What is meant by projection? A very simple example is as follows. Look at a checker board head on. All of the lines are parallel. Turn that same board at an angle keeping your perspective the same and what you see is quite different. The lines are no longer parallel. From a geometrical standpoint, what you are seeing is a projection of the lines of the checkerboard onto another plane. Projective geometry is the study of the properties of these lines after they have been projected. Early projective geometers found that while lengths, areas and angles were not maintained, there were properties of points and lines which were unaffected or invariant in projection. Using these discoveries they were able to construct new ways to solve old problems, and that is how this now highly regarded genre of geometry arose.

The properties that are meaningful in projective geometry are those that are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation and translation; the first issue for geometers is what kind of geometric language would be adequate to the novel situation. It is not possible to talk about angle in projective geometry as it is in Euclidean, because angle is an example of a concept not invariant under projective transformations, as is seen clearly in perspective drawing. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which parallel lines can be said to meet in a point of infinity, once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing.

Projective geometry is an elementary non-metrical form of geometry, meaning that it is not based on a concept of distance. Projective geometry is about distortions, transformations or things as they appear after projective process. The geometry that was done at school was Euclidean geometry which is about things as they are, and has to do with measurements and relations of distances and angles. Below image is an example of the projection of a right rectangular prism on to a place.



The initial understanding of the effects of perspective geometry was developed in the context of artistic drawing. From the 15th century onwards, the problem of understanding and precisely constructing the effects of perspective viewing has been considered a key aspect of artistic drawing. The goal is to create a realistic impression of the depth on a two-dimensional surface where the central phenomenon which must be accounted for is convergence of parallel lines. A brief treatment of the perspective construction will prove in establishing the effect of perspective on geometric properties.

The figure above is a 2-D construction of perspective viewing of parallel lines on the ground which illustrates the formation of a vanishing point. The image of the points far from O approach v, and for V, which is at infinity, the image is at vanishing point v. So,we can observe that as the points recede to points at infinity, their corresponding image point converges to the same vanishing point. It is obvious from the figure that the points equally spaced on the ground are not equally spaced on the image plane. This demonstrates that neither distance nor ratios of distance are preserved under perspective viewing.

Have a look at this amazing animation that explains the remarkable fact that any two distinct lines meet in a unique point and also a hyperbola is projectively the same thing as an ellipse. [|The Real Projective Plane]

It is interesting to consider what happens when there is a chain of perspective views. For example: If the scene to be projected itself contains a perspective image. Another important example of a chain of perspective views occurs in outdoor scenes where shadows are projected by a point light source.

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