Why does the application of quaternions in frame based hierarchies prevent Gimbal lock?
452210
Egads.  It's been quite awhile.  I remember some papers at SIGGRAPH a decade or so ago on this topic.\n\nAlright, here's a stab at the explanation.\n\nWith Euler mechanics or Euler angles, there are three reference angles: yaw, pitch, and roll. It is possible to rotate an object using these three reference angles such that two line up causing the inability to do an arbitrary rotation.  Say you roll to the right 90 degrees and this lines up the yaw axis with your pitch axis -- gimble lock results.\n\nWith quaternions, you don't have three reference angles.  You have only one plane and an arbitrary axis of rotation.  Each move creates a new plane and axis of rotation.  As such, there are no additional axes of rotation with which to collide because there is only one axis of rotation and it's position is arbitrary where as with Euler angles, the reference angles are fixed.\n\nI don't know if I've done it justice, but maybe this explanation and the references below will help you stitch together an understanding.
984859
internal rotation on shouldwer and elbow bent
358834
KNOW YOUR RELATED RATES FOR CALC!! thats one thing that killed me and know your 2-D rotation around a line intergrals.
535174
There is often a wide divergence between theory and application.  I remember one application where lengthy mechanical linkages and cables were used to transmit rudder position for a huge seagoing ship in the even that the steering gear room became flooded (and electronic indication was lost).  The system had too much lost motion to indicate rudder position even though it looked good in theory.  I suggested connecting a vertical rod to the rudders that pierced the overhead into the ("unflooded") compartment above which could be connected to a reliable electronic transmitter (similar to existing "flooded" transmitters) and that worked much better.  Even electronic control systems can exhibit unexpected hysteresis when built.
705060
If I understand the question, you are giving a velocity to the cube and don't want it to touch the hemisphere at any time in its flight. I think you want the minimum, not the maximum hor. velocity to do this, right? Because any arbitrarily high velocity will do it. I'll use a planar analysis where the hemisphere is a semicircle.\nAssuming the semicircle is centered at x=y=0, the trajectory will be a parabola y=r-kx^2 and at y=0, |x| must be >r. Another constraint arises when we consider the region of the semicircle near the top. Here the semicircle approximates a parabola, since 1-cos(small angle) approaches small angle^2/2. Thus deltay = r(1-cos(small angle)) = r-y approaches r*small angle^2/2 = r(x/r)^2/2, or x approaches sqrt(2/r*deltay*r^2) = sqrt(2r*deltay).\nDownward acceleration is g and horizontal velocity is v. When y=0, 0.5gt^2=r, so t=sqrt(2r/g), and x=vt must be >r. Thus the 1st condition (for clearance at the bottom) is v > r/sqrt(2r/g), or\nv > sqrt(rg/2).\nThe 2nd condition requires that when r-y=deltay is small, then and x=vt must be >sqrt(2r*deltay) where t=sqrt(2deltay/g), or  \nv > sqrt(2r*deltay) / sqrt(2deltay/g) or \nv > sqrt(rg).\nThis latter is the more stringent of the two requirements and thus is the minimum clearance velocity.
483398
r u 4 r3al ?!! \n\nwhat mapping ? what equasion?  deckard maybe ? \n\nmore info please.
147563
Look at how a bug does it.\nThe side opposite the direction you wish to turn takes steps slightly longer than the side toward the turn.\nFor very quick turning, the legs take strides in opposite directions.
792270
Hi I assume you mean an unsymmetrical item. I also assume you want a best fit. I have done similar things on a computer but not by hand. The computer finds three point most distant from the centroid and uses these points as pivots for three orthogonal planes. Not sure of the rest.
709071
On some shot locations it has to do with the rotational angle to where it is you're searching. The other thing you may try is doing a very slow zoom, as this changes the focusing aperature. I have similar problems with my view cam when I zoom in to fast. The slower rate allows for the focus to establish more defined.
826674
use conservation of angular momentum in these kind of problems\nNOT energy conservation\nif you dont know how to use the conservation equation, better read up on examples
733431
Because there is no absolute reference frame for motion, you can only define uniform motion with respect to a specified coordinate frame.\n\nFor example, a ball falling in a vacuum is not in uniform motion with respect to an earth-fixed reference frame; the ball accelerates.\n\nWhen the forces on an obejct sum to zero, that object is in uniform motion with respect to a reference frame that is non-accelerated. That means that if you compare gravity acceleration between the a frame fixed to the object, and the frame you are wondering about, if the gravity is the same, the frames are not accelerated with respect to each other.\n\nOthers have suggested that uniform motion requires absence of forces. This is not quite true - it requires absence of *net* forces. For example, a car traveling a constant velocity on a highway is not in a force-free environment: you can identify forces of air friction, propulsion from the wheels, weight, and support from the pavement. However it is considered to be in uniform motion with respect to a reference frame fixed to the earth, because all of those forces sum to zero, and because the apparent gravity in a frame fixed to the car is the same as in a frame fixed to the earth.
692261
You need to switch to the Projection Matrix when you are specifying the projections. \ne.g.\n glMatrixMode( GL_PROJECTION );\n glLoadIdentity();\n gluPerspective( 60.0, aspectRatio, 1.0, 30.0 );\n\n\nThe model view matrix is used for the rotation, scaling and translation stuff. So, you need to do glMatrixMode(GL_MODELVIEW) before you do your transformations.
393990
It's not a characteristic of the human eye, which does not have a "frame rate". The effect occurs with devices like TV and movie cameras which have frame rates of 16 to 60 Hz. Any data sampling system is prone to such aliasing. If spokes pass a given point at exactly the frame rate, or at exactly 2, 3, etc. times tha frame rate, the wheel appears to be stopped. A slight change in wheel speed plus or minus makes the apparent wheel speed that plus or minus amount.
863293
In the opposite direction of the applied force or motion
785138
I desperately need to see a computer motion graphic of birds called terns in a flock as they twist and turn in the air.  I need the triangular space BETWEEN the birds marked in the motion graphic like little sails.  The triangular sails shift and redefine themselves as the flock of birds fly and twist and undulate.  I need to see just all those triangles and not the birds themselves.  If someone could do that for me, I could probably answer the Great and Mysterious Secret of the Universe.   : )   I'm looking at Chaos in nature. I can pretty much see it with my own eyes.  But, gosh, I wish I had the computer skills to make that happen like I describe above.
1105920
They add up to 360 degrees. (If you walked around the quadrilateral you would end up facing the same way as when you started, hence you went through 360 degrees. For a re-entrant or crossed-over quadrilateral, make that 360k degrees)
649942
Sorry, it's not 3D.\n\nMove to one side. Do any of the elements in the photograph move in relation to each other? Do the items in the foreground now hide different elements in the background? If the image was 3D then they would.