Projective geometry is a branch of geometry that studies the geometric properties that remain unchanging under the projective transformation, which is the kind of transformation that maintains inmediation (for example, points that lie on one line remaining located on one line after transformation). Unlike the Euklides geometry that focuses on length, angle, and distance, projective geometry is more interested in more fundamental structures, such as dots, lines, and fields, without paying attention to size or angle.
Projective geometry first appeared in the Renaissance, when artists began exploring ways to describe three-dimensional objects in two-dimensional fields. One of the great innovations of art in this time was development of perspective techniques, which allowed artists to describe objects in such a way that it seemed realistic, with parallel lines that seemed to meet at a lost point in the distance.
Pioneer in development of mathematical theory behind the perspective is Girard Desargues (1591-1661), a mathematician and French architect. Desargues is known for developing a basic idea in projective geometry, especially about the relationship between the lines - lines that seem to meet at infinite distances. Desargues' theory becomes the basis of many modern concepts in projective geometry.
Another significant advance in projective geometry is carried on by Jean@@ (178- 1867), a French mathematician who expanded the idea of Desargues and developed a projective transformation theory. His work in projective geometry became the foundation for further studies in this field.
One of the most fundamental concepts in projective geometry is Projective space. In Euklides space, two parallel lines never met. However, in projective space, parallel lines are viewed at "point in infinity." Projective space thus adds points in infinity to expand the Euklides space, so all lines finally meet at a certain point.
For example, projective line (Determine with P1mathbb {P} ^ 1P1) is an extension of a real line of Rmathbb {R} R, with an additional one point in infinity. Likewise, projective field (used to be symbolized with P2mathbb {P} ^ 2P2) expand the two-dimensional field of R2mathbb {R} ^ 2R2 by adding "line at infinity," which consists of all points where the parallel lines meet.
In projective geometry, there is no concept of "paralelism" in the meaning of Euklides. All lines are considered to meet, both in real and in infinite points. This led to some significant differences in the way we understand the relationship between points, lines and fields.
For example:
One of the most important aspects of projective geometry is Projective transformation. Projective transformations are transformations that map points on lines, lines on fields, or more common, objects in projective space, by maintaining interactions (for example, if two intersect lines before transformation, they will remain intersect after transformation).
This transformation could be viewed as an extension of the transformation of the afine, where we not only consider the translation, rotation, and dilation, but also the depiction in perspective, where the dots are much more closely visible.
Theorem Desargues is one of the fundamental theorem in projective geometry. This theorem states that if two ABCABCABC triangles and A B.A. A'B'B'B'A'A
This theorem doesn't always apply in Euklides geometry, but it always applies in projective geometry, which shows the strength and generality of this geometry.
Pappus Theorem is another important theorem in projective geometry. This theorem states that if we had two lines with three different points on each line (let's say A, B, CA, B, CA, B, C, on the first line and A-C-C-C-A-A-A-A-A-A, B-A-A, C-CB is on the second line), then the number of intersections of the AB 'AB' AB 'AB and A-BA-BA-B-BA-C-AC-AC-AC-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C, and-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-1-1-1-C-1-1-C-1-1-1-1-1-C-C-1-C-1-1-C-1-1-1-1-C-1-1-1-1-1-1-1-1-1-1-1-
Pappus' theorem is another example of the nature of the incision that's guarded in projective geometry.
Projective geometry has extensive applications in modern science and technology. Some important applications include:
In graphical computer, projective geometry is used to project three-dimensional objects into a two-dimensional field of computer screen. This is the foundation of engineering rendering perspective, where objects that are closer to the camera seem larger than those that are further away. Projective transformation is used to calculate how objects in 3D space are projected to the 2D screen.
In photography and image processing, projective geometry is used to fix the distortions of perspective in the picture. For example, when an image of a building is taken from a particular angle, it may look skewed or distorted. By using projective techniques, these distortions can be corrected so that buildings appear upright and straight.
In robotics and computer vision, projective geometry is used to understand and model how robots or computer vision systems see the world around them. Projected techniques are used to extract information about the shape and position of objects in the real world from images or videotape.
Projective geometry also has important applications in algebra geometry and Number theory, where projective space is often used to study solutions to polynomial equations and other geometric objects. Many of the modern results in numbers theory and algebra geometry involve the use of projective space to simplify and understand complex structure problems.
Projective geometry is a deep and rich field of mathematics with applications that extends from art and design to modern technology like computer graphics, computer vision and robotics. By studying the fundamental structure that remains unchanging under the projective transformation, projective geometry unfolds new insights in understanding the relationship between point, line and space.
Theoremate- keytheorem like Theorem Desargues and Pappus Theorem Show that projective geometry offers a new perspective on these geometric properties that can't be explained in Euklides geometry frame. His modern applications in various disciplines show relevance and the use of this theory in technology and science.
Source: Coxeter, H. S. M.. John Wiley & Sons.
