Game Engine Architecture

639 t 1 (which is just v(t 1 )) and extrapolating it linearly to the next time step t 2. As we can see in Figure 12.23, linear e ...

640 12. Collision and Rigid Body Dynamics terms represent the error inherent in the method. For example, the explicit Euler equa ...

641 able on the topic. (For a start, check out htt p://en.wikipedia.org/wiki/Verlet_ integration.) The regular Verlet method is ...

642 12. Collision and Rigid Body Dynamics locity-dependent. If it is velocity-dependent, then we must approximate next frame’s v ...

643 Angular: α ω ωθ () () () (); t dt dt t = == t^ Linear: a v vr () () () (). t dt dt t = == t 12.4.5.3. Moment of Inerti ...

644 12. Collision and Rigid Body Dynamics mass .) This is illustrated in Figure 12.25. The torque N caused by a force F ap- plie ...

645 Angular: ω ω θ θω () () (); () () () ; t Nt I t tt 21 1 211 =+ =+ net Δ t Δ t t Linear: vv F r rv () () () ; () () (). t t m ...

646 12. Collision and Rigid Body Dynamics . xx xy xz yx yy yz zx zy zz III III III ⎡⎤ ⎢⎥ =⎢⎥ ⎢⎥⎣⎦ I The elements lying along the ...

647 defi nes the axis of rotation, scaled by the two-dimensional angular velocity ωθuu= of the body about the u-axis. Hence, ωω ...

648 12. Collision and Rigid Body Dynamics () () () (). () () xxxxyxzx yyxyyyzy zzxzyzzz Lt I I I t Lt I I I t Lt I I I t ⎡ ⎤ ⎡ ⎤ ...

649 When solving for the orientation given the angular velocity, we have a problem: The angular velocity is a three-element vec ...

650 12. Collision and Rigid Body Dynamics The orientation quaternion q(t) should be renormalized periodically to reverse the eff ...

651 12.4.7.2. Impulsive Collision Response When two bodies collide in the real world, a complex set of events takes place. The b ...

652 12. Collision and Rigid Body Dynamics primed kinetic energy sum becomes zero, and the bodies stick together aft er the colli ...

653 () 21 12 ˆˆ( 1). p 11 mm ==ε+ ⋅ − ⋅ + pn vnvnn Notice that if the coeffi cient of restitution is one (perfectly elastic coll ...

654 12. Collision and Rigid Body Dynamics Unfortunately, because penalty forces respond to penetration (i.e., rela- tive positio ...

655 can have friction as well. For example, a rusted hinge or axle might resist be- ing turned by introducing a friction torque. ...

656 12. Collision and Rigid Body Dynamics spurious contacts because it will think that the object is about to hit the edge of th ...

657 When an object really does stop moving (fi nds itself in a state of equilib- rium ), there is no reason to continue integrat ...

658 12. Collision and Rigid Body Dynamics the simulation island design is one we’re likely to continue to see in future versions ...