9-math

To now,the engine still has limits:

• It can't describe the way objects rotate.
• Robust general-purpose collision detection system

Center of Mass

\[p_{cofm} = \frac{1}{m}\sum m_{i}p_{i} \]

where \(p_{cofm}\) is the position of the center of mass, \(m\) is the total mass of the object,\(m_{i}\) is the mass, and \(p_{i}\) is the position of particle i.

Angular Velocity and Acceleration

The angular velocity is a three-element vector that can be decomposed into an axis and rate of angular change:

\[\dot{\theta} = r\widehat{a} \]

where \(a\) is the axis around which the object is turning and r is the rate at which it is
spinning, which (by convention) is measured in radians per second.

The velocity of a point on an object depends on both its linear and angular velocity:

\[\dot{q} = \dot{\theta} \times (q - p) + \dot{p} \qquad \qquad [9.5] \]

where \(\dot{q}\) is the velocity of the point, q is the position of the point in world coordinates, p is the position of the origin of the object, and \(\theta\) is the angular velocity of the object.

注意此处，点的速度 和 角速度