# Which Of These Properties Of A Rigid Transformation Is Exclusive To Translations

Computer Graphics: What is a rigid body transformation?

In computer graphics there are many different kinds of transformation which includes translation, rotation, scale, similarity (reflection) affine, homography and rigid. Rigid transformation includes combination of rotation translation and reflection. These transformations are mainly used for image registration.Please check the link for details.http://www.math.tau.ac.il/~dcor/...

Why would we want to transform the coordinates of a vector to another basis? Are there any real life examples where this may be necessary?

The fact that linear algebra is invariant under a change of basis is probably one of the most important facts in all of mathematics.One reason is that, for most problems, the "standard" axes are meaningless.  For instance, if I'm given a 5,000-dimensional data set describing how much people like a long list of movies along with some demographic information about these people, then the meaningful axes of the data set are the people's personality traits.  Obviously you want to transform to that basis to understand the situation, which you'd perhaps do using principal component analysis.In a completely unrelated domain, I might have a geometric problem that could be simplified substantially by rotating everything so that everything has nice coordinates.This is a very basic computational technique, one I use dozens of times every day either explicitly or implicitly.  It's hard to describe individual applications because I'd have to describe the problems themselves and all the unrelated steps leading up to the change of basis and the unrelated steps following.  Suffice to say, though, that a huge percentage of problems in all areas of mathematics can be reduced to understanding the structure of a particular matrix group, and this notion of change of basis is one of the most fundamental things that lets you understand a matrix group.To put things another way, most of the time I'm working with vectors I don't even think of them as having coordinates.

Is it possible for a 3x3 matrix to not be able to be inverted if it is a transformation matrix over 2D points, composed of translation, scale, and rotation operations?

Ah, a refreshing question!In terms of the conditionings in adjoint operations relating to the solving of the equation systems of the matrises - there is a condition based on the determinant of the system.If the determinant is 0 - then a specific solution in terms of solving is eliminated.The fundamental conclusion of which - means it cannot be inversed.The inherent representative nature of the actual points and their dimensional stature may yet prove to yield a fundamental interplay - alas - to my knowledge, if the square matris pattern of Cubic scaling (i.e 2x2 yields the same ruleset as 3x3 in terms of Singularities and determinants) -Then nothing of stature should fundamentally change on the front of capacity of inversion.As for the inherent prospect of if it is composed of translations, scalings, rotations - fundamentally - the topological implications of the system and the co-ordinate representation of the integration of the system is not fundamentally very directly applicative to the rule set of solving.Of which is more related unto rule integrations, eliminations, structural compositions and defined by properties of uniqueness, etc.So - fundamentally - to answer your question:Under the presumption of that the determinant is not 0 -Then possibly, yes - ASSUMING - that the inherent factors of composition in terms of operations akin to Translation/Scale/Rotation and operative statures - does not inherently interfere with the trait composition of rule disposition.I quite enjoyed reasoning about this -Thank you, for this A2A.

Time travel... can it be done..? if so, how? Explain fully..?

Only time travel into the future is possible, the past is not.

By constantly accelerating, like in a space ship with unlimited fuel, ( same effect as gravitational acceleration), time will slow down for you compared to a stationary observer. Gravitational fields have the same effect.
So in your space ship, you can accelerate contineously ( if unlimited fuel), and you will observe your speed growing faster and faster, and appear to exceed the speed of light by many times. To you, time is passing by normally. How is this possible? from your perspective, as your velocity increases, the thickness of planets appear to get thinner, and thinner. As you approach the speed of light, a planet will appear like a paper cutout hanging in space. You will appear to be traveling many times the velocity of light.
However, an observer who is at rest, will see you accelerate, and approach the speed of light, but never quite get there. If he could see your watch, and compare to his watch, he would note that your watch has nearly stopped, and is slowing down further, the more you accelerate. You looking back at the clock on earth would see the clocks running unbelievably fast, but yours would appear to be keeping time normally. If you projected a light beam forward of your speeding ship, it would move forward at the speed of light, even though your ship is supposedly moving nearly at the speed of light already.
Time dilation, and length contraction. Derived from special relativity.

The other way is to fly through the center of a spinning black hole shaped like a donut. The gravitational field would be so strong, that time would slow down for you, as seen by an observer. The field would probably be so strong, that you would be squeezed into a strand the diameter of spagettie. So that may not be the best approach.

These would work, but a time machine in a room on earth would not work, except for gravitational effects. If you could create a time machine, you would not disappear, but would appear frozen in time.

The gravitational effects on time have been observed by placing an atomic clock in orbit, and the identical one on Earth, in the gravitational field. The clock on arth ticks slower than the one in orbit. Same is true for airline passengers. Time moves a bit faster for them, the world clock is slower while they are traveling.
( just a few Femto seconds or so )

I have a question about geometry!!?

Which property of a rigid transformation is exclusive to rotations?

A.The measures of the angles and sides in the preimage are preserved.

B.A line segment connecting the image to the preimage forms a 90° angle with the center of rotation.

C.The distance from each point on the image to the center of rotation is preserved.

D.All the points on the image move along a parallel line when the preimage is transformed.

In hindu religion, at which point did the flexible varna system as mentioned in Bhavagat Gita did become a rigid caste system?