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The word geometry means measurement of the earth. All are familiar
with the basics of Euclidean geometry, which describes the
threedimensional world we live in. In Euclidean geometry, objects have
length and intersecting lines determine angles. In particular two
lines are said to be parallel if they
lie in the same plane and never meet. Moreover, these properties do
not change when an Euclidean transformations (translation and
rotation) are applied. Euclidean geometry describes our world so well
that it is tempting to conclude that it is the only type of geometry.
However, when we consider the imaging process of a camera, it becomes
clear that Euclidean geometry is insufficient: lengths and angles are
no longer preserved, and parallel lines may intersect.
By exploring these "new" viewpoint one realizes that Euclidean
geometry is actually a subset of what is known as projective geometry.
Projective geometry is an ancient branch of geometry which officialy started in the 19th centry, but that has its roots way back in history. An historical account can be found here One interesting area of application is computer vision, more eprecisely camera calibration, stereo, object recognition, scene reconstruction, mosaicing, image synthesis, and the analysis of shadows. These are problems in which invariant descriptions between images are important. The composition of two perspective projections is not necessarily a perspective projection but it is projective transformation. In other words the set of projective transformations form a group, whereas perspective projections do not. Some arising questions, to be explored in a "kandidat examensarbete" are: References: Coxeter, H. S. M., 2003. Projective Geometry, 2nd ed. Springer Verlag. ISBN 978-0-387-40623-7. | |||