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Unofficial Hints for Problems

Waterloo '08 Spring E - Balance:

For the centroid ~(C_x, C_y)~ of a polygon, given points, the formula is:

$$\displaystyle C_x = \frac{1}{6A} \sum_{i=0}^{n-1} (x_i + x_{i+1})(x_i y_{i+1} - x_{i+1} y_i)$$

$$\displaystyle C_y = \frac{1}{6A} \sum_{i=0}^{n-1} (y_i + y_{i+1})(x_i y_{i+1} - x_{i+1} y_i)$$

A is the signed area of the polygon, and its formula is given by the Shoelace formula:

$$\displaystyle A = \frac{1}{2} \sum_{i=0}^{n-1} (x_i y_{i+1} - x_{i+1} y_i)$$

Rotations in 3 Dimensions

Rodrigues' Rotation Formula is helpful here:

$$\displaystyle \mathbf{v}_{\text{rot}} = \mathbf{v} \cos \theta + (\mathbf{k} \times \mathbf{v}) \sin \theta + \mathbf{k} (\mathbf{k} \cdot \mathbf{v})(1 - \cos \theta)$$