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?
From the very beginning.Bhagavad Gita does not preach a flexible varna system. It urges Arjuna to fight because he is a Kshatriya. And it was Kshatriya’s Dharma or duty by birth to fight to attain Moksha. This is the central theme.Bhagavad Gita - Chapter 2, Verse 47कर्मण्येवाधिकारस्ते मा फलेषु कदाचन |मा कर्मफलहेतुर्भूर्मा ते सङ्गोऽस्त्वकर्मणिBG 2.47: You have a right to perform your prescribed duties, but you are not entitled to the fruits of your actions. Never consider yourself to be the cause of the results of your activities, nor be attached to inaction.While this verse is addressed to Arjuna, the verse actually is addressed to rest of the Shudra castes.You have a right to perform your prescribed duty - It is not only a duty, it is a birth-right. No else has that right. This is root of the ‘deserving’ argument.Not entitled to returns - Shudra’s were not entitled to the fruits of their labor. Shudra’s were not paid in cash, but in kind like food. Cash is a flexible they could for example pay the fees for education.Never consider yourself to be the cause of the results of your activities - Shudra’s efforts is not the cause for good harvest. Now, imagine some one telling consider your exam marks is not result of your efforts.Nor be attached to inaction - But hastily, don’t stop doing it. You have to do your work without salary.The whole concept was while the rest of Varnas lived in luxury in rich palaces, temples, singing and dancing in wealth you read so much about in Indian literature, the Shudra castes alone slugged it out for others as laborers, iron workers, miners, fishermen, dhobis, carcass removers etc.The Mahabharata preaches that Karna is not eligible to become Dronacharya’s student because he was not a Kshatriya. Kripacharya refused to allow Karna participate in a contest of skills because he was not a Kshatriya by birth.If it was a ‘flexible’ system at the time of the Baghavad Gita (and Mahabharata), why was Karna refused admission without examining his ‘nature’? He was not even allowed to participate in the exam.If it was a ‘flexible’ system, why was Ekalavya not taught archery?
What is the syllabus for JEE Mains?
The JEE Main syllabus for mathematics, Physics, and chemistry are must to know for all the candidates who are preparing for the exam. The syllabus is prepared by the conducting body. The first step towards JEE Main preparation is to know the syllabus soo all the aspirants can plan accordingly.JEE Main syllabus 2019/2020The important topics that all the candidates find in the JEE Main syllabus for 2019 and 2020 are the same topics that are candidates find in the CBSE, ICSE and state board books. It is adviced to the candidates that they must keep practicing the topics of 11th and 12th level education.Exam Tip: To score well in the examination cover all the important topics of physics, chemistry, and mathematics of classes XI and XII.I am providing syllabus and books for studying for the JEE Main exam. The details are as follows-:JEE Main syllabus for paper 1 (Mathematics)This syllabus is recommended by CBSE for JEE Main entrance.JEE Main syllabus for paper 1 (physics)Note-: This syllabus contains 2 sections A & B. Part A contains theory and its weightage is 80% and part B contains practical and its has 20% weightage.This is recommended by CBSE that you should go through.JEE Main syllabus for paper 1 ( chemistry )NOTE -: This syllabus is divided into 3 sections A, B, and C. Section A contains Physical chemistry. Section B contains inorganic chemistry. Section C contains organic chemistry.This syllabus is recommended by CBSE. Go through itI am also providing you the syllabus for JEE Main paper 2 ( Aptitude test )It is also into 2 parts which are as follows :-Let me also tell you which topics you should study from 11th and 12th ( physics, chemistry, and mathematics).I will also tell you which books you should go for preparing JEE Mains.Hopefully, This will help everyone in preparing their best for JEE Main exam.SOURCE:-