What do you see at the left? Two pictures of the same cat? Yes, you are right; the "above picture" is the same as the "down picture" except that upper picture is a one-to-one transformation of the lower one.
The concave picture has exactly the same amount of points and colors as the flat picture lying in the "ground", which can be mathematically described as the X-Z plane.
We can think of the flat picture of the cat as a set of an organized square array of points, each one with a precisely defined position and a precisely defined color. That is what makes us see a cat and not a dog, or a bird, etc.
The concave picture above the flat one contains exactly the same amount of points, and exactly the same amount of colors, and exactly the same colors. That's what makes the concave picture as if the flat picture of the cat was "inflated" from below.
Calculated deformations is part of the beauty of the point-to-point transformations.
Most of the time, transformations are treated abstractly. Like in the next figure, where we can see the same dome-shaped surface, devoid of any particular photo transformation, this time looking from the negative side of the Y-axis toward the positive side of the Y-axis. We are looking the dome "from below".
There are other special and interesting mathematical functions with the particular property that the transformations are angle-preserving.
An example of angle-preserving one-to-one (1-1) transformations is the transcomplex trigonometric sine function shown below.
|In a transformation we have two main "objects": the domain, and the transformed domain. In some special mappings we can study some of the properties of one of the "objects" by studying the properties of the other.|
The mathematical transformation used here is the Transcomplex sine trigonometric.
In the figure above we can see how the transcomplex sine function maps a rectangular strip along the X-axis into a curved surface called a circular helicoid. Each small square of the XZ-plane is mapped to another deformed "square" in the surface. When the squares of the XZ-plane are made infinitesimally small, the squares of the helicoid also become perfect squares, thus preserving the right-angles of the flat strip.
In another application like the cat above, we see below how Da Vinci's Mona Lisa is transformed into a "helicoidal Mona Lisa". The presence of the XZ-plane is omitted in the figure of the contortioned Mona Lisa; the two "objects" are shown as separated images.
A section of Da Vinci's Mona Lisa.
The same section of Da Vinci's Mona Lisa, but this time transformed under the same mathematical function as the figure above.
Transformations ---specially the one-to-one transformations (those that are angle preserving)--- are useful in the study of heat conduction along materials, and in the study of air flow along the wings of airplanes and all types of aircrafts.
To see more examples of transformations download the EBook: The Golden EBook of Graphs of Mathematical Functions, or go to the EBook Catalog for more EBooks.
Learn more about conformal transformation in the article: What are conformal mappings?
Maybe Descartes never imagined how far was going to evolve his system and how much beauty is in his coordinate system. What if we add coloring to the math plots? What if we add real-life picture transformations to the mathematical equations? What are one-to-one (1-1) transformations? How can a 2-dimensional figure be transformed into a 3-dimensional surface? This book has some elementary answers to those profound questions.