Affine transformations preserve parallel lines, and include rotations, scaling, shears and translations. Linear transformations can’t perform translations, but this can be achieved if we go to a higher dimension.
In this animation, a planar (2D) shape lying on the plane z = 1 is translated by means of a linear transformation in three dimensions: a shear along the z axis.
Rotations can be performed normally, also around the z axis. Both operations are now linear in 3D, and can be represented by a single 3x3 affine transformation matrix acting on two dimensions.
The “shadow” of the shape illustrates the relative position between the two images, on the planes z = 0 and z = 1, and was included to better visualize the shear.
Whoever, at any time, has undertaken to build a new heaven has found the strength for it in his own hell.Friedrich Nietzsche (via likeafieldmouse)
Every form of addiction is bad, no matter whether the narcotic be alcohol, morphine or idealism.Carl Jung (via likeafieldmouse)
The Lunar Module “Spider” ascent stage is photographed from the Command/Service Module on the fifth day of the Apollo 9 earth-orbital mission. The Lunar Module’s descent stage had already been jettisoned.
The moon is framed between the Orbiter’s OMS pod and the Earth limb over the Atlantic Ocean as seen from the aft windows onboard Discovery on mission STS-95.
“When I arrived at this fantastic club 13 years ago I would never have believed that I would be fortunate enough to play so many games and enjoy sharing in so much success. This club has become part of my life and I have so many people to thank for the opportunity. Firstly, Ken Bates, who put his neck on the line to sign me as a young player and without him I would not have even begun this experience. Roman Abramovich, the man who saved our Club and took us all to new levels. His desire to push the Club to the top of the football world has rubbed off on everyone. All the managers and coaches who have helped me develop my game during the time I have been here. I have learnt from every one of them. All the brilliant team mates who I have been lucky enough to train and play alongside for so long. Not just their football qualities but also the friendships I have gained along the way. I’d also like to give a special mention to all the staff and people behind the scenes at the Club who do not receive the glory but without them the Club would simply not function. The Club will move forward, and as a Chelsea man I have no doubt that with the quality of the players that are there, they will continue with the success that we have all enjoyed over the past seasons. Finally and most importantly, I would like to thank the Chelsea fans. I believe they are the greatest fans in world football. You have supported me from the moment I arrived. Not only on the pitch, but in difficult moments in my personal life off the pitch. I will never forget. I feel honoured to have shared so many special times with you. Bolton, Amsterdam, Munich, the list goes on and on. I, and we as a team, could not have achieved those moments without your support. Whatever the next challenge is for me I will always be among you and have Chelsea in my home and in my heart. Hopefully I will get to see you all soon to say goodbye properly. In the meantime thank you for the memories and keep making history!” THANK YOU FOR EVERYTHING, LEGEND!!!